Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
can essentially be considered as doing computations similar to those of
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the
theory of equations
In algebra, the theory of equations is the study of algebraic equations (also called "polynomial equations"), which are equations defined by a polynomial. The main problem of the theory of equations was to know when an algebraic equation has an ...
. For example, the
fundamental theorem of algebra
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
belongs to the theory of equations and is not, nowadays, considered as belonging to algebra (in fact, every
proof must use the
completeness of the real numbers
Completeness is a property of the real numbers that, intuitively, implies that there are no "gaps" (in Dedekind's terminology) or "missing points" in the real number line. This contrasts with the rational numbers, whose corresponding number li ...
, which is not an algebraic property).
This article describes the history of the theory of equations, called here "algebra", from the origins to the emergence of algebra as a separate area of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
.
Etymology
The word "algebra" is derived from the
Arabic
Arabic (, ' ; , ' or ) is a Semitic languages, Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C ...
word الجبر ''al-jabr'', and this comes from the treatise written in the year 830 by the medieval Persian mathematician,
Muhammad ibn Mūsā al-Khwārizmī
Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persians, Persian polymath from Khwarazm, who produced vastly influential works in Mathematics ...
, whose Arabic title, ''
Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala'', can be translated as ''The Compendious Book on Calculation by Completion and Balancing''. The treatise provided for the systematic solution of
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
and
quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
s. According to one history, "
is not certain just what the terms ''al-jabr'' and ''muqabalah'' mean, but the usual interpretation is similar to that implied in the previous translation. The word 'al-jabr' presumably meant something like 'restoration' or 'completion' and seems to refer to the transposition of
subtracted terms to the other side of an equation; the word 'muqabalah' is said to refer to 'reduction' or 'balancing'—that is, the cancellation of like terms on opposite sides of the equation. Arabic influence in Spain long after the time of al-Khwarizmi is found in ''
Don Quixote
is a Spanish epic novel by Miguel de Cervantes. Originally published in two parts, in 1605 and 1615, its full title is ''The Ingenious Gentleman Don Quixote of La Mancha'' or, in Spanish, (changing in Part 2 to ). A founding work of Wester ...
'', where the word 'algebrista' is used for a bone-setter, that is, a 'restorer'." The term is used by al-Khwarizmi to describe the operations that he introduced, "
reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation.
Stages of algebra
Algebraic expression
Algebra did not always make use of the symbolism that is now ubiquitous in mathematics; instead, it went through three distinct stages. The stages in the development of symbolic algebra are approximately as follows:
* Rhetorical algebra, in which equations are written in full sentences. For example, the rhetorical form of
is "The thing plus one equals two" or possibly "The thing plus 1 equals 2". Rhetorical algebra was first developed by the ancient
Babylonians
Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c. 1 ...
and remained dominant up to the 16th century.
* Syncopated algebra, in which some symbolism is used, but which does not contain all of the characteristics of symbolic algebra. For instance, there may be a restriction that subtraction may be used only once within one side of an equation, which is not the case with symbolic algebra. Syncopated algebraic expression
first appeared in
Diophantus
Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...
' ''
Arithmetica'' (3rd century AD), followed by
Brahmagupta
Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical trea ...
's ''
Brahma Sphuta Siddhanta
Brahma ( sa, ब्रह्मा, Brahmā) is a Hindu god, referred to as "the Creator" within the Trimurti, the trinity of supreme divinity that includes Vishnu, and Shiva.Jan Gonda (1969)The Hindu Trinity Anthropos, Bd 63/64, H 1/2, pp. 21 ...
'' (7th century).
* Symbolic algebra, in which full symbolism is used. Early steps toward this can be seen in the work of several
Islamic mathematicians
Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important progress was made, such as full ...
such as
Ibn al-Banna
Ibn al‐Bannāʾ al‐Marrākushī ( ar, ابن البناء المراكشي), full name: Abu'l-Abbas Ahmad ibn Muhammad ibn Uthman al-Azdi al-Marrakushi () (29 December 1256 – 31 July 1321), was a Moroccan polymath who was active as a math ...
(13th-14th centuries) and
al-Qalasadi (15th century), although fully symbolic algebra was developed by
François Viète
François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...
(16th century). Later,
René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...
(17th century) introduced the modern notation (for example, the use of ''x''—
see below) and showed that the problems occurring in
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
can be expressed and solved in terms of algebra (
Cartesian geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineerin ...
).
Equally important as the use or lack of symbolism in algebra was the degree of the equations that were addressed.
Quadratic equations
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not quadr ...
played an important role in early algebra; and throughout most of history, until the early modern period, all quadratic equations were classified as belonging to one of three categories.
*
*
*
where
and
are positive.
This trichotomy comes about because quadratic equations of the form
with
and
positive, have no positive
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
.
In between the rhetorical and syncopated stages of symbolic algebra, a geometric constructive algebra was developed by classical
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
and
Vedic Indian mathematicians in which algebraic equations were solved through geometry. For instance, an equation of the form
was solved by finding the side of a square of area
Conceptual stages
In addition to the three stages of expressing algebraic ideas, some authors recognized four conceptual stages in the development of algebra that occurred alongside the changes in expression. These four stages were as follows:
*Geometric stage, where the concepts of algebra are largely geometric. This dates back to the
Babylonians
Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c. 1 ...
and continued with the
Greeks
The Greeks or Hellenes (; el, Έλληνες, ''Éllines'' ) are an ethnic group and nation indigenous to the Eastern Mediterranean and the Black Sea regions, namely Greece, Cyprus, Albania, Italy, Turkey, Egypt, and, to a lesser extent, oth ...
, and was later revived by
Omar Khayyám.
*Static equation-solving stage, where the objective is to find numbers satisfying certain relationships. The move away from the geometric stage dates back to
Diophantus
Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...
and
Brahmagupta
Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical trea ...
, but algebra didn't decisively move to the static equation-solving stage until
Al-Khwarizmi
Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astronom ...
introduced generalized algorithmic processes for solving algebraic problems.
*Dynamic function stage, where motion is an underlying idea. The idea of a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
began emerging with
Sharaf al-Dīn al-Tūsī
Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī ( fa, شرفالدین مظفر بن محمد بن مظفر توسی; 1135 – 1213) was an Iranian mathematician and astronomer of the Islamic Golden Age (during the ...
, but algebra did not decisively move to the dynamic function stage until
Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
.
*Abstract stage, where mathematical structure plays a central role.
Abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
is largely a product of the 19th and 20th centuries.
Babylon
![Plimpton 322](https://upload.wikimedia.org/wikipedia/commons/c/c2/Plimpton_322.jpg)
The origins of algebra can be traced to the ancient
Babylonians
Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c. 1 ...
, who developed a positional
number system
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
that greatly aided them in solving their rhetorical algebraic equations. The Babylonians were not interested in exact solutions, but rather approximations, and so they would commonly use
linear interpolation
In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.
Linear interpolation between two known points
If the two known poi ...
to approximate intermediate values.
[ "Babylonian mathematicians did not hesitate to interpolate by proportional parts to approximate intermediate values. Linear interpolation seems to have been a commonplace procedure in ancient Mesopotamia, and the positional notation lent itself conveniently to the rile of three. ..a table essential in Babylonian algebra; this subject reached a considerably higher level in Mesopotamia than in Egypt. Many problem texts from the Old Babylonian period show that the solution of the complete three-term quadratic equation afforded the Babylonians no serious difficulty, for flexible algebraic operations had been developed. They could transpose terms in an equations by adding equals to equals, and they could ]multiply
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
both sides by like quantities to remove fraction
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
s or to eliminate factors. By adding to they could obtain for they were familiar with many simple forms of factoring. ..gyptian algebra had been much concerned with linear equations, but the Babylonians evidently found these too elementary for much attention. ..In another problem in an Old Babylonian text we find two simultaneous linear equations in two unknown quantities, called respectively the "first silver ring" and the "second silver ring."" One of the most famous tablets is the
Plimpton 322 tablet, created around 1900–1600 BC, which gives a table of
Pythagorean triples
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
and represents some of the most advanced mathematics prior to Greek mathematics.
Babylonian algebra was much more advanced than the Egyptian algebra of the time; whereas the Egyptians were mainly concerned with linear equations the Babylonians were more concerned with quadratic and
cubic equation
In algebra, a cubic equation in one variable is an equation of the form
:ax^3+bx^2+cx+d=0
in which is nonzero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
s.
The Babylonians had developed flexible algebraic operations with which they were able to add equals to equals and
multiply
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
both sides of an equation by like quantities so as to eliminate
fraction
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
s and factors.
They were familiar with many simple forms of
factoring,
three-term quadratic equations with positive roots, and many cubic equations,
although it is not known if they were able to reduce the general cubic equation.
[ "There is no record in Egypt of the solution of a cubic equations, but among the Babylonians there are many instances of this. ..Whether or not the Babylonians were able to reduce the general four-term cubic, ''ax''3 + ''bx''2 + ''cx'' = ''d'', to their normal form is not known."]
Ancient Egypt
![Egyptian A'h-mosè or Rhind Papyrus (1065x1330)](https://upload.wikimedia.org/wikipedia/commons/4/4b/Egyptian_A%27h-mos%C3%A8_or_Rhind_Papyrus_%281065x1330%29.png)
Ancient Egyptian algebra dealt mainly with linear equations while the Babylonians found these equations too elementary, and developed mathematics to a higher level than the Egyptians.
The Rhind Papyrus, also known as the Ahmes Papyrus, is an ancient Egyptian papyrus written c. 1650 BC by Ahmes, who transcribed it from an earlier work that he dated to between 2000 and 1800 BC. It is the most extensive ancient Egyptian mathematical document known to historians. The Rhind Papyrus contains problems where linear equations of the form
and
are solved, where
and
are known and
which is referred to as "aha" or heap, is the unknown.
[ "The Egyptian problems so far described are best classified as arithmetic, but there are others that fall into a class to which the term algebraic is appropriately applied. These do not concern specific concrete objects such as bread and beer, nor do they call for operations on known numbers. Instead they require the equivalent of solutions of linear equations of the form or , where a and b and c are known and x is unknown. The unknown is referred to as "aha," or heap. ..The solution given by Ahmes is not that of modern textbooks, but one proposed characteristic of a procedure now known as the "method of false position," or the "rule of false." A specific false value has been proposed by 1920s scholars and the operations indicated on the left hand side of the equality sign are performed on this assumed number. Recent scholarship shows that scribes had not guessed in these situations. Exact rational number answers written in Egyptian fraction series had confused the 1920s scholars. The attested result shows that Ahmes "checked" result by showing that 16 + 1/2 + 1/8 exactly added to a seventh of this (which is 2 + 1/4 + 1/8), does obtain 19. Here we see another significant step in the development of mathematics, for the check is a simple instance of a proof."] The solutions were possibly, but not likely, arrived at by using the "method of false position", or ''
regula falsi'', where first a specific value is substituted into the left hand side of the equation, then the required arithmetic calculations are done, thirdly the result is compared to the right hand side of the equation, and finally the correct answer is found through the use of proportions. In some of the problems the author "checks" his solution, thereby writing one of the earliest known simple proofs.
Greek mathematics
![P](https://upload.wikimedia.org/wikipedia/commons/8/8d/P._Oxy._I_29.jpg)
It is sometimes alleged that the
Greeks
The Greeks or Hellenes (; el, Έλληνες, ''Éllines'' ) are an ethnic group and nation indigenous to the Eastern Mediterranean and the Black Sea regions, namely Greece, Cyprus, Albania, Italy, Turkey, Egypt, and, to a lesser extent, oth ...
had no algebra, but this is inaccurate.
By the time of
Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
, Greek mathematics had undergone a drastic change. The Greeks created a geometric algebra where terms were represented by sides of geometric objects,
usually lines, that had letters associated with them,
[ "In the arithmetical theorems in Euclid's ''Elements'' VII–IX, numbers had been represented by line segments to which letters had been attached, and the geometric proofs in al-Khwarizmi's ''Algebra'' made use of lettered diagrams; but all coefficients in the equations used in the ''Algebra'' are specific numbers, whether represented by numerals or written out in words. The idea of generality is implied in al-Khwarizmi's exposition, but he had no scheme for expressing algebraically the general propositions that are so readily available in geometry."] and with this new form of algebra they were able to find solutions to equations by using a process that they invented, known as "the application of areas".
[ "Whether deduction came into mathematics in the sixth century BCE or the fourth and whether incommensurability was discovered before or after 400 BCE, there can be no doubt that Greek mathematics had undergone drastic changes by the time of Plato. ..A "geometric algebra" had to take the place of the older "arithmetic algebra," and in this new algebra there could be no adding of lines to areas or of areas to volumes. From now on there had to be strict homogeneity of terms in equations, and the Mesopotamian normal form, = b, were to be interpreted geometrically. ..In this way the Greeks built up the solution of quadratic equations by their process known as "the application of areas," a portion of geometric algebra that is fully covered by Euclid's ''Elements''. ..The linear equation , for example, was looked upon as an equality of the areas and rather than as a proportion—an equality between the two ratios and . Consequently, in constructing the fourth proportion in this case, it was usual to construct a rectangle OCDB with the sides ''b'' = OB and ''c'' = OC (Fig 5.9) and then along OC to lay off OA = ''a''. One completes the rectangle OCDB and draws the diagonal OE cutting CD in P. It is now clear that CP is the desired line for rectangle OARS is equal in area to rectangle OCDB"] "The application of areas" is only a part of geometric algebra and it is thoroughly covered in
Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
's ''
Elements''.
An example of geometric algebra would be solving the linear equation
The ancient Greeks would solve this equation by looking at it as an equality of areas rather than as an equality between the ratios
and
The Greeks would construct a rectangle with sides of length
and
then extend a side of the rectangle to length
and finally they would complete the extended rectangle so as to find the side of the rectangle that is the solution.
Bloom of Thymaridas
Iamblichus
Iamblichus (; grc-gre, Ἰάμβλιχος ; Aramaic: 𐡉𐡌𐡋𐡊𐡅 ''Yamlīḵū''; ) was a Syrian neoplatonic philosopher of Arabic origin. He determined a direction later taken by neoplatonism. Iamblichus was also the biographer of ...
in ''Introductio arithmatica'' says that
Thymaridas Thymaridas of Paros ( el, Θυμαρίδας; c. 400 – c. 350 BCE) was an ancient Greek mathematician and Pythagorean noted for his work on prime numbers and simultaneous linear equations.
Life and work
Although little is known about the life ...
(c. 400 BCE – c. 350 BCE) worked with simultaneous linear equations.
[ Thymaridas of Paros, an ancient Pythagorean already mentioned (p. 69), was the author of a rule for solving a certain set of simultaneous simple equations connecting unknown quantities. The rule was evidently well known, for it was called by the special name ..the 'flower' or 'bloom' of Thymaridas. ..The rule is very obscurely worded, but it states in effect that, if we have the following equations connecting unknown quantities namely ..Iamblichus, our informant on this subject, goes on to show that other types of equations can be reduced to this, so that they rule does not 'leave us in the lurch' in those cases either."] In particular, he created the then famous rule that was known as the "bloom of Thymaridas" or as the "flower of Thymaridas", which states that:
If the sum of quantities be given, and also the sum of every pair containing a particular quantity, then this particular quantity is equal to of the difference between the sums of these pairs and the first given sum.[ "Thymaridas (fourth century) is said to have had this rule for solving a particular set of linear equations in unknowns:]
If the sum of quantities be given, and also the sum of every pair containing a particular quantity, then this particular quantity is equal to of the difference between the sums of these pairs and the first given sum."
![euclid-proof](https://upload.wikimedia.org/wikipedia/commons/5/5e/Euclid-proof.jpg)
or using modern notation, the solution of the following system of
linear equations in
unknowns,
is,
Iamblichus goes on to describe how some systems of linear equations that are not in this form can be placed into this form.
Euclid of Alexandria
Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
(
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
: ) was a
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
mathematician who flourished in
Alexandria
Alexandria ( or ; ar, ٱلْإِسْكَنْدَرِيَّةُ ; grc-gre, Αλεξάνδρεια, Alexándria) is the second largest city in Egypt, and the largest city on the Mediterranean coast. Founded in by Alexander the Great, Alexandria ...
,
Egypt
Egypt ( ar, مصر , ), officially the Arab Republic of Egypt, is a transcontinental country spanning the northeast corner of Africa and southwest corner of Asia via a land bridge formed by the Sinai Peninsula. It is bordered by the Mediter ...
, almost certainly during the reign of
Ptolemy I (323–283 BCE).
[ "but by 306 BCE control of the Egyptian portion of the empire was firmly in the hands of Ptolemy I, and this enlightened ruler was able to turn his attention to constructive efforts. Among his early acts was the establishment at Alexandria of a school or institute, known as the Museum, second to none in its day. As teachers at the school he called a band of leading scholars, among whom was the author of the most fabulously successful mathematics textbook ever written—the ''Elements'' (''Stoichia'') of Euclid. Considering the fame of the author and of his best seller, remarkably little is known of Euclid's life. So obscure was his life that no birthplace is associated with his name."] Neither the year nor place of his birth
have been established, nor the circumstances of his death.
Euclid is regarded as the "father of
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
". His ''
Elements'' is the most successful
textbook
A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbooks are produced to meet the needs of educators, usually at educational institutions. Schoolbooks are textboo ...
in the
history of mathematics
The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments ...
.
Although he is one of the most famous mathematicians in history there are no new discoveries attributed to him; rather he is remembered for his great explanatory skills. The ''Elements'' is not, as is sometimes thought, a collection of all Greek mathematical knowledge to its date; rather, it is an elementary introduction to it.
''Elements''
The geometric work of the Greeks, typified in Euclid's ''Elements'', provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations.
Book II of the ''Elements'' contains fourteen propositions, which in Euclid's time were extremely significant for doing geometric algebra. These propositions and their results are the geometric equivalents of our modern symbolic algebra and trigonometry.
Today, using modern symbolic algebra, we let symbols represent known and unknown magnitudes (i.e. numbers) and then apply algebraic operations on them, while in Euclid's time magnitudes were viewed as line segments and then results were deduced using the axioms or theorems of geometry.
[ "Book II of the ''Elements'' is a short one, containing only fourteen propositions, not one of which plays any role in modern textbooks; yet in Euclid's day this book was of great significance. This sharp discrepancy between ancient and modern views is easily explained—today we have symbolic algebra and trigonometry that have replaced the geometric equivalents from Greece. For instance, Proposition 1 of Book II states that "If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments." This theorem, which asserts (Fig. 7.5) that AD (AP + PR + RB) = AD·AP + AD·PR + AD·RB, is nothing more than a geometric statement of one of the fundamental laws of arithmetic known today as the distributive law: In later books of the ''Elements'' (V and VII) we find demonstrations of the commutative and associative laws for multiplication. Whereas in our time magnitudes are represented by letters that are understood to be numbers (either known or unknown) on which we operate with algorithmic rules of algebra, in Euclid's day magnitudes were pictured as line segments satisfying the axions and theorems of geometry. It is sometimes asserted that the Greeks had no algebra, but this is patently false. They had Book II of the ''Elements'', which is geometric algebra and served much the same purpose as does our symbolic algebra. There can be little doubt that modern algebra greatly facilitates the manipulation of relationships among magnitudes. But it is undoubtedly also true that a Greek geometer versed in the fourteen theorems of Euclid's "algebra" was far more adept in applying these theorems to practical mensuration than is an experienced geometer of today. Ancient geometric "algebra" was not an ideal tool, but it was far from ineffective. Euclid's statement (Proposition 4), "If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments, is a verbose way of saying that ,"]
Many basic laws of addition and multiplication are included or proved geometrically in the ''Elements''. For instance, proposition 1 of Book II states:
:If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments.
But this is nothing more than the geometric version of the (left)
distributive law
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmetic, ...
,
; and in Books V and VII of the ''Elements'' the
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
and
associative laws for multiplication are demonstrated.
Many basic equations were also proved geometrically. For instance, proposition 5 in Book II proves that
and proposition 4 in Book II proves that
Furthermore, there are also geometric solutions given to many equations. For instance, proposition 6 of Book II gives the solution to the quadratic equation
and proposition 11 of Book II gives a solution to
''Data''
''Data'' is a work written by Euclid for use at the schools of Alexandria and it was meant to be used as a companion volume to the first six books of the ''Elements''. The book contains some fifteen definitions and ninety-five statements, of which there are about two dozen statements that serve as algebraic rules or formulas.
Some of these statements are geometric equivalents to solutions of quadratic equations.
For instance, ''Data'' contains the solutions to the equations
and the familiar Babylonian equation
[ "Euclid's ''Data'', a work that has come down to us through both Greek and the Arabic. It seems to have been composed for use at the schools of Alexandria, serving as a companion volume to the first six books of the ''Elements'' in much the same way that a manual of tables supplements a textbook. ..It opens with fifteen definitions concerning magnitudes and loci. The body of the text comprises ninety-five statements concerning the implications of conditions and magnitudes that may be given in a problem. ..There are about two dozen similar statements serving as algebraic rules or formulas. ..Some of the statements are geometric equivalents of the solution of quadratic equations. For example ..Eliminating we have or from which The geometric solution given by Euclid is equivalent to this, except that the negative sign before the radical is used. Statements 84 and 85 in the Data are geometric replacements of the familiar Babylonian algebraic solutions of the systems which again are the equivalents of solutions of simultaneous equations."]
Conic sections
A
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
is a curve that results from the
intersection of a
cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines con ...
with a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
. There are three primary types of conic sections:
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s (including
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
s),
parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...
s, and
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
s. The conic sections are reputed to have been discovered by
Menaechmus :''There is also a Menaechmus in Plautus' play, ''The Menaechmi''.''
Menaechmus ( el, Μέναιχμος, 380–320 BC) was an ancient Greek mathematician, geometer and philosopher born in Alopeconnesus or Prokonnesos in the Thracian Chersonese, w ...
(c. 380 BC – c. 320 BC) and since dealing with conic sections is equivalent to dealing with their respective equations, they played geometric roles equivalent to cubic equations and other higher order equations.
Menaechmus knew that in a parabola, the equation
holds, where
is a constant called the
latus rectum
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a speci ...
, although he was not aware of the fact that any equation in two unknowns determines a curve.
He apparently derived these properties of conic sections and others as well. Using this information it was now possible to find a solution to the problem of the
duplication of the cube
Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related probl ...
by solving for the points at which two parabolas intersect, a solution equivalent to solving a cubic equation.
[ "If OP = ''y'' and OD = ''x'' are coordinates of point P, we have ).OV, or, on substituting equals, ]
''y''2 = R'D.OV = AR'.BC/AB.DO.BC/AB = AR'.BC2/AB2.''x''
Inasmuch as segments AR', BC, and AB are the same for all points P on the curve EQDPG, we can write the equation of the curve, a "section of a right-angled cone," as where is a constant, later to be known as the latus rectum of the curve. ..Menaechmus apparently derived these properties of the conic sections and others as well. Since this material has a string resemblance to the use of coordinates, as illustrated above, it has sometimes been maintains that Menaechmus had analytic geometry. Such a judgment is warranted only in part, for certainly Menaechmus was unaware that any equation in two unknown quantities determines a curve. In fact, the general concept of an equation in unknown quantities was alien to Greek thought. ..He had hit upon the conics in a successful search for curves with the properties appropriate to the duplication of the cube. In terms of modern notation the solution is easily achieved. By shifting the curring plane (Gig. 6.2), we can find a parabola with any latus rectum. If, then, we wish to duplicate a cube of edge we locate on a right-angled cone two parabolas, one with latus rectum and another with latus rectum ..It is probable that Menaechmus knew that the duplication could be achieved also by the use of a rectangular hyperbola and a parabola."
We are informed by
Eutocius
Eutocius of Ascalon (; el, Εὐτόκιος ὁ Ἀσκαλωνίτης; 480s – 520s) was a Palestinian-Greek mathematician who wrote commentaries on several Archimedean treatises and on the Apollonian ''Conics''.
Life and work
Little is ...
that the method he used to solve the cubic equation was due to
Dionysodorus
Dionysodorus of Caunus ( grc-gre, Διονυσόδωρος ὁ Καύνειος, c. 250 BC – c. 190 BC) was an ancient List of Greek mathematicians, Greek mathematician.
Life and work
Little is known about the life of Dionysodorus. Pliny the E ...
(250 BC – 190 BC). Dionysodorus solved the cubic by means of the intersection of a rectangular hyperbola and a parabola. This was related to a problem in
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
' ''On the Sphere and Cylinder''. Conic sections would be studied and used for thousands of years by Greek, and later Islamic and European, mathematicians. In particular
Apollonius of Perga's famous ''
Conics
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a speci ...
'' deals with conic sections, among other topics.
China
Chinese mathematics dates to at least 300 BC with the ''
Zhoubi Suanjing
The ''Zhoubi Suanjing'' () is one of the oldest Chinese mathematical texts. "Zhou" refers to the ancient Zhou dynasty (1046–256 BCE); "Bì" literally means "thigh", but in the book refers to the gnomon of a sundial. The book is dedicated to ...
'', generally considered to be one of the oldest Chinese mathematical documents.
[ "estimates concerning the ''Chou Pei Suan Ching'', generally considered to be the oldest of the mathematical classics, differ by almost a thousand years. ..A date of about 300 B.C. would appear reasonable, thus placing it in close competition with another treatise, the ''Chiu-chang suan-shu'', composed about 250 B.C., that is, shortly before the Han dynasty (202 B.C.). ..Almost as old at the ''Chou Pei'', and perhaps the most influential of all Chinese mathematical books, was the ''Chui-chang suan-shu'', or ''Nine Chapters on the Mathematical Art''. This book includes 246 problems on surveying, agriculture, partnerships, engineering, taxation, calculation, the solution of equations, and the properties of right triangles. ..Chapter eight of the ''Nine chapters'' is significant for its solution of problems of simultaneous linear equations, using both positive and negative numbers. The last problem in the chapter involves four equations in five unknowns, and the topic of indeterminate equations was to remain a favorite among Oriental peoples."]
''Nine Chapters on the Mathematical Art''
![九章算術](https://upload.wikimedia.org/wikipedia/commons/8/88/%E4%B9%9D%E7%AB%A0%E7%AE%97%E8%A1%93.gif)
''Chiu-chang suan-shu'' or ''
The Nine Chapters on the Mathematical Art
''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest sur ...
'', written around 250 BC, is one of the most influential of all Chinese math books and it is composed of some 246 problems. Chapter eight deals with solving determinate and indeterminate simultaneous linear equations using positive and negative numbers, with one problem dealing with solving four equations in five unknowns.
''Sea-Mirror of the Circle Measurements''
''Ts'e-yuan hai-ching'', or ''Sea-Mirror of the Circle Measurements'', is a collection of some 170 problems written by
Li Zhi Li Zhi may refer to:
*Emperor Gaozong of Tang (628–683), named Li Zhi, Emperor of China
*Li Ye (mathematician) (1192–1279), Chinese mathematician and scholar, birth name Li Zhi
*Li Zhi (philosopher) (1527–1602), Chinese philosopher from the M ...
(or Li Ye) (1192 – 1279 CE). He used ''fan fa'', or
Horner's method
In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Horn ...
, to solve equations of degree as high as six, although he did not describe his method of solving equations.
[ "Li Chih (or Li Yeh, 1192–1279), a mathematician of Peking who was offered a government post by Khublai Khan in 1206, but politely found an excuse to decline it. His ''Ts'e-yuan hai-ching'' (''Sea-Mirror of the Circle Measurements'') includes 170 problems dealing with ..ome of the problems leading to equations of fourth degree. Although he did not describe his method of solution of equations, including some of sixth degree, it appears that it was not very different form that used by Chu Shih-chieh and Horner. Others who used the Horner method were Ch'in Chiu-shao (c. 1202 – c. 1261) and Yang Hui (fl. c. 1261 – 1275). The former was an unprincipled governor and minister who acquired immense wealth within a hundred days of assuming office. His ''Shu-shu chiu-chang'' (''Mathematical Treatise in Nine Sections'') marks the high point of Chinese indeterminate analysis, with the invention of routines for solving simultaneous congruences."]
''Mathematical Treatise in Nine Sections''
''Shu-shu chiu-chang'', or ''
Mathematical Treatise in Nine Sections
The ''Mathematical Treatise in Nine Sections'' () is a mathematical text written by Chinese Southern Song dynasty mathematician Qin Jiushao in the year 1247. The mathematical text has a wide range of topics and is taken from all aspects of th ...
'', was written by the wealthy governor and minister
Ch'in Chiu-shao (c. 1202 – c. 1261). With the introduction of a method for solving simultaneous
congruences
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done wi ...
, now called the
Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
, it marks the high point in Chinese .
Magic squares
![Yanghui triangle](https://upload.wikimedia.org/wikipedia/commons/e/ea/Yanghui_triangle.gif)
The earliest known
magic square
In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number ...
s appeared in China.
[ "The Chinese were especially fond of patters; hence, it is not surprising that the first record (of ancient but unknown origin) of a magic square appeared there. ..The concern for such patterns left the author of the ''Nine Chapters'' to solve the system of simultaneous linear equations ..by performing column operations on the matrix ..to reduce it to ..The second form represented the equations = 24, and from which the values of and are successively found with ease."] In ''Nine Chapters'' the author solves a system of simultaneous linear equations by placing the coefficients and constant terms of the linear equations into a magic square (i.e. a matrix) and performing column reducing operations on the magic square.
The earliest known magic squares of order greater than three are attributed to
Yang Hui
Yang Hui (, ca. 1238–1298), courtesy name Qianguang (), was a Chinese mathematician and writer during the Song dynasty. Originally, from Qiantang (modern Hangzhou, Zhejiang), Yang worked on magic squares, magic circles and the binomial theo ...
(fl. c. 1261 – 1275), who worked with magic squares of order as high as ten.
''Precious Mirror of the Four Elements''
''Ssy-yüan yü-chien''《四元玉鑒》, or ''Precious Mirror of the Four Elements'', was written by
Chu Shih-chieh
Zhu Shijie (, 1249–1314), courtesy name Hanqing (), pseudonym Songting (), was a Chinese mathematician and writer. He was a Chinese mathematician during the Yuan Dynasty. Zhu was born close to today's Beijing. Two of his mathematical works ha ...
in 1303 and it marks the peak in the development of Chinese algebra. The
four elements
Classical elements typically refer to earth, water, air, fire, and (later) aether which were proposed to explain the nature and complexity of all matter in terms of simpler substances. Ancient cultures in Greece, Tibet, and India had simi ...
, called heaven, earth, man and matter, represented the four unknown quantities in his algebraic equations. The ''Ssy-yüan yü-chien'' deals with simultaneous equations and with equations of degrees as high as fourteen. The author uses the method of ''fan fa'', today called
Horner's method
In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Horn ...
, to solve these equations.
[ "The last and greatest of the Sung mathematicians was Chu Chih-chieh ( fl. 1280–1303), yet we know little about him-, ..Of greater historical and mathematical interest is the ''Ssy-yüan yü-chien'' (''Precious Mirror of the Four Elements'') of 1303. In the eighteenth century this, too, disappeared in China, only to be rediscovered in the next century. The four elements, called heaven, earth, man, and matter, are the representations of four unknown quantities in the same equation. The book marks the peak in the development of Chinese algebra, for it deals with simultaneous equations and with equations of degrees as high as fourteen. In it the author describes a transformation method that he calls ''fan fa'', the elements of which to have arisen long before in China, but which generally bears the name of Horner, who lived half a millennium later."]
The ''Precious Mirror'' opens with a diagram of the arithmetic triangle (
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although o ...
) using a round zero symbol, but Chu Shih-chieh denies credit for it. A similar triangle appears in Yang Hui's work, but without the zero symbol.
There are many summation equations given without proof in the ''Precious mirror''. A few of the summations are:
[ "A few of the many summations of series found in the ''Precious Mirror'' are the following: ..However, no proofs are given, nor does the topic seem to have been continued again in China until about the nineteenth century. ..The ''Precious Mirror'' opens with a diagram of the arithmetic triangle, inappropriately known in the West as "pascal's triangle." (See illustration.) ..Chu disclaims credit for the triangle, referring to it as a "diagram of the old method for finding eighth and lower powers." A similar arrangement of coefficients through the sixth power had appeared in the work of Yang Hui, but without the round zero symbol."]
:
:
Diophantus
Diophantus
Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...
was a
Hellenistic
In Classical antiquity, the Hellenistic period covers the time in Mediterranean history after Classical Greece, between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire, as signified by the Battle of Actium in ...
mathematician who lived c. 250 CE, but the uncertainty of this date is so great that it may be off by more than a century. He is known for having written ''Arithmetica'', a treatise that was originally thirteen books but of which only the first six have survived.
[ Uncertainty about the life of Diophantus is so great that we do not know definitely in which century he lived. Generally he is assumed to have flourished about 250 CE, but dates a century or more earlier or later are sometimes suggested ..If this conundrum is historically accurate, Diophantus lived to be eighty-four-years old. ..The chief Diophantine work known to us is the ''Arithmetica'', a treatise originally in thirteen books, only the first six of which have survived."] ''Arithmetica'' has very little in common with traditional Greek mathematics since it is divorced from geometric methods, and it is different from Babylonian mathematics in that Diophantus is concerned primarily with exact solutions, both determinate and indeterminate, instead of simple approximations.
[ "In this respect it can be compared with the great classics of the earlier ]Alexandrian Age
In Classical antiquity, the Hellenistic period covers the time in Mediterranean history after Classical Greece, between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire, as signified by the Battle of Actium in 3 ...
; yet it has practically nothing in common with these or, in fact, with any traditional Greek mathematics. It represents essentially a new branch and makes use of a different approach. Being divorced from geometric methods, it resembles Babylonian algebra to a large extent. But whereas Babylonian mathematicians had been concerned primarily with '' approximate'' solutions of ''determinate'' equations as far as the third degree, the ''Arithmetica'' of Diophantus (such as we have it) is almost entirely devoted to the ''exact'' solution of equations, both ''determinate'' and ''indeterminate''. ..Throughout the six surviving books of ''Arithmetica'' there is a systematic use of abbreviations for powers of numbers and for relationships and operations. An unknown number is represented by a symbol resembling the Greek letter ζ (perhaps for the last letter of arithmos). ..It is instead a collection of some 150 problems, all worked out in terms of specific numerical examples, although perhaps generality of method was intended. There is no postulation development, nor is an effort made to find all possible solutions. In the case of quadratic equations with two positive roots, only the larger is give, and negative roots are not recognized. No clear-cut distinction is made between determinate and indeterminate problems, and even for the latter for which the number of solutions generally is unlimited, only a single answer is given. Diophantus solved problems involving several unknown numbers by skillfully expressing all unknown quantities, where possible, in terms of only one of them."
It is usually rather difficult to tell whether a given Diophantine equation is solvable. There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation. He also considered simultaneous quadratic equations. Also, no general method may be abstracted from all Diophantus' solutions.
In ''Arithmetica'', Diophantus is the first to use symbols for unknown numbers as well as abbreviations for powers of numbers, relationships, and operations;
thus he used what is now known as ''syncopated'' algebra. The main difference between Diophantine syncopated algebra and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponentials.
[ "The chief difference between Diophantine syncopation and the modern algebraic notation is the lack of special symbols for operations and relations, as well as of the exponential notation."] So, for example, what we would write as
:
which can be rewritten as
:
would be written in Diophantus's syncopated notation as
:
where the symbols represent the following:
Unlike in modern notation, the coefficients come after the variables and that addition is represented by the juxtaposition of terms. A literal symbol-for-symbol translation of Diophantus's syncopated equation into a modern symbolic equation would be the following:
:
where to clarify, if the modern parentheses and plus are used then the above equation can be rewritten as:
:
''Arithmetica'' is a collection of some 150 solved problems with specific numbers and there is no postulational development nor is a general method explicitly explained, although generality of method may have been intended and there is no attempt to find all of the solutions to the equations.
''Arithmetica'' does contain solved problems involving several unknown quantities, which are solved, if possible, by expressing the unknown quantities in terms of only one of them.
''Arithmetica'' also makes use of the identities:
:
India
The Indian mathematicians were active in studying about number systems. The earliest known
Indian mathematical documents are dated to around the middle of the first millennium BC (around the 6th century BC).
The recurring themes in Indian mathematics are, among others, determinate and indeterminate linear and quadratic equations, simple mensuration, and Pythagorean triples.
[ "The ''Livavanti'', like the ''Vija-Ganita'', contains numerous problems dealing with favorite Hindu topics; linear and quadratic equations, both determinate and indeterminate, simple mensuration, arithmetic and geometric progressions, surds, Pythagorean triads, and others."]
''Aryabhata''
Aryabhata
Aryabhata ( ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the ''Aryabhatiya'' (which ...
(476–550) was an Indian mathematician who authored ''
Aryabhatiya''. In it he gave the rules,
:
and
:
''Brahma Sphuta Siddhanta''
Brahmagupta
Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical trea ...
(fl. 628) was an Indian mathematician who authored ''
Brahma Sphuta Siddhanta
Brahma ( sa, ब्रह्मा, Brahmā) is a Hindu god, referred to as "the Creator" within the Trimurti, the trinity of supreme divinity that includes Vishnu, and Shiva.Jan Gonda (1969)The Hindu Trinity Anthropos, Bd 63/64, H 1/2, pp. 21 ...
''. In his work Brahmagupta solves the general quadratic equation for both positive and negative roots. In indeterminate analysis Brahmagupta gives the Pythagorean triads
but this is a modified form of an old Babylonian rule that Brahmagupta may have been familiar with. He was the first to give a general solution to the linear Diophantine equation
where
and
are
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s. Unlike Diophantus who only gave one solution to an indeterminate equation, Brahmagupta gave ''all'' integer solutions; but that Brahmagupta used some of the same examples as Diophantus has led some historians to consider the possibility of a Greek influence on Brahmagupta's work, or at least a common Babylonian source.
[ "he was the first one to give a ''general'' solution of the linear Diophantine equation where and are integers. ..It is greatly to the credit of Brahmagupta that he gave ''all'' integral solutions of the linear Diophantine equation, whereas Diophantus himself had been satisfied to give one particular solution of an indeterminate equation. Inasmuch as Brahmagupta used some of the same examples as Diophantus, we see again the likelihood of Greek influence in India—or the possibility that they both made use of a common source, possibly from Babylonia. It is interesting to note also that the algebra of Brahmagupta, like that of Diophantus, was syncopated. Addition was indicated by juxtaposition, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, as in our fractional notation but without the bar. The operations of multiplication and evolution (the taking of roots), as well as unknown quantities, were represented by abbreviations of appropriate words. ..Bhaskara (1114 – c. 1185), the leading mathematician of the twelfth century. It was he who filled some of the gaps in Brahmagupta's work, as by giving a general solution of the Pell equation and by considering the problem of division by zero."]
Like the algebra of Diophantus, the algebra of Brahmagupta was syncopated. Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, similar to our modern notation but without the bar. Multiplication, evolution, and unknown quantities were represented by abbreviations of appropriate terms.
The extent of Greek influence on this syncopation, if any, is not known and it is possible that both Greek and Indian syncopation may be derived from a common Babylonian source.
Bhāskara II
Bhāskara II (1114 – c. 1185) was the leading mathematician of the 12th century. In Algebra, he gave the general solution of
Pell's equation
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinates, ...
.
He is the author of ''
Lilavati'' and ''Vija-Ganita'', which contain problems dealing with determinate and indeterminate linear and quadratic equations, and Pythagorean triples
and he fails to distinguish between exact and approximate statements.
[ "In treating of the circle and the sphere the ''Lilavati'' fails also to distinguish between exact and approximate statements. ..Many of Bhaskara's problems in the ''Livavati'' and the ''Vija-Ganita'' evidently were derived from earlier Hindu sources; hence, it is no surprise to note that the author is at his best in dealing with indeterminate analysis."] Many of the problems in ''Lilavati'' and ''Vija-Ganita'' are derived from other Hindu sources, and so Bhaskara is at his best in dealing with indeterminate analysis.
Bhaskara uses the initial symbols of the names for colors as the symbols of unknown variables. So, for example, what we would write today as
:
Bhaskara would have written as
:: .
_ .
: ''ya'' 1 ''ru'' 1
::: .
: ''ya'' 2 ''ru'' 8
:::: .
: Sum ''ya'' 1 ru ''9''
where ''ya'' indicates the first syllable of the word for ''black'', and ''ru'' is taken from the word ''species''. The dots over the numbers indicate subtraction.
Islamic world
![Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala](https://upload.wikimedia.org/wikipedia/commons/2/23/Image-Al-Kit%C4%81b_al-mu%E1%B8%ABta%E1%B9%A3ar_f%C4%AB_%E1%B8%A5is%C4%81b_al-%C4%9Fabr_wa-l-muq%C4%81bala.jpg)
The first century of the
Islam
Islam (; ar, ۘالِإسلَام, , ) is an Abrahamic religions, Abrahamic Monotheism#Islam, monotheistic religion centred primarily around the Quran, a religious text considered by Muslims to be the direct word of God in Islam, God (or ...
ic
Arab Empire saw almost no scientific or mathematical achievements since the Arabs, with their newly conquered empire, had not yet gained any intellectual drive and research in other parts of the world had faded. In the second half of the 8th century, Islam had a cultural awakening, and research in mathematics and the sciences increased.
[ "The first century of the Muslim empire had been devoid of scientific achievement. This period (from about 650 to 750) had been, in fact, perhaps the nadir in the development of mathematics, for the Arabs had not yet achieved intellectual drive, and concern for learning in other parts of the world had faded. Had it not been for the sudden cultural awakening in Islam during the second half of the eighth century, considerably more of ancient science and mathematics would have been lost. ..It was during the caliphate of al-Mamun (809–833), however, that the Arabs fully indulged their passion for translation. The caliph is said to have had a dream in which Aristotle appeared, and as a consequence al-Mamun determined to have Arabic versions made of all the Greek works that he could lay his hands on, including Ptolemy's ''Almagest'' and a complete version of Euclid's ''Elements''. From the Byzantine Empire, with which the Arabs maintained an uneasy peace, Greek manuscripts were obtained through peace treaties. Al-Mamun established at Baghdad a "House of Wisdom" (Bait al-hikma) comparable to the ancient Museum at Alexandria. Among the faculty members was a mathematician and astronomer, Mohammed ibn-Musa al-Khwarizmi, whose name, like that of Euclid, later was to become a household word in Western Europe. The scholar, who died sometime before 850, wrote more than half a dozen astronomical and mathematical works, of which the earliest were probably based on the ''Sindhad'' derived from India."] The Muslim
Abbasid
The Abbasid Caliphate ( or ; ar, الْخِلَافَةُ الْعَبَّاسِيَّة, ') was the third caliphate to succeed the Islamic prophet Muhammad. It was founded by a dynasty descended from Muhammad's uncle, Abbas ibn Abdul-Muttalib ...
caliph
A caliphate or khilāfah ( ar, خِلَافَة, ) is an institution or public office under the leadership of an Islamic steward with the title of caliph (; ar, خَلِيفَة , ), a person considered a political-religious successor to th ...
al-Mamun
Abu al-Abbas Abdallah ibn Harun al-Rashid ( ar, أبو العباس عبد الله بن هارون الرشيد, Abū al-ʿAbbās ʿAbd Allāh ibn Hārūn ar-Rashīd; 14 September 786 – 9 August 833), better known by his regnal name Al-Ma'mu ...
(809–833) is said to have had a dream where Aristotle appeared to him, and as a consequence al-Mamun ordered that Arabic translation be made of as many Greek works as possible, including Ptolemy's ''Almagest'' and Euclid's ''Elements''. Greek works would be given to the Muslims by the
Byzantine Empire
The Byzantine Empire, also referred to as the Eastern Roman Empire or Byzantium, was the continuation of the Roman Empire primarily in its eastern provinces during Late Antiquity and the Middle Ages, when its capital city was Constantinopl ...
in exchange for treaties, as the two empires held an uneasy peace.
Many of these Greek works were translated by
Thabit ibn Qurra Thabit ( ar, ) is an Arabic name for males that means "the imperturbable one". It is sometimes spelled Thabet.
People with the patronymic
* Ibn Thabit, Libyan hip-hop musician
* Asim ibn Thabit, companion of Muhammad
* Hassan ibn Sabit (died 674 ...
(826–901), who translated books written by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius.
[ "but al-Khwarizmi's work had a serious deficiency that had to be removed before it could serve its purpose effectively in the modern world: a symbolic notation had to be developed to replace the rhetorical form. This step the Arabs never took, except for the replacement of number words by number signs. ..Thabit was the founder of a school of translators, especially from Greek and Syriac, and to him we owe an immense debt for translations into Arabic of works by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius."]
Arabic mathematicians established algebra as an independent discipline, and gave it the name "algebra" (''al-jabr''). They were the first to teach algebra in an
elementary form and for its own sake.
There are three theories about the origins of Arabic Algebra. The first emphasizes Hindu influence, the second emphasizes Mesopotamian or Persian-Syriac influence and the third emphasizes Greek influence. Many scholars believe that it is the result of a combination of all three sources.
[ "Al-Khwarizmi continued: "We have said enough so far as numbers are concerned, about the six types of equations. Now, however, it is necessary that we should demonstrate geometrically the truth of the same problems which we have explained in numbers." The ring of this passage is obviously Greek rather than Babylonian or Indian. There are, therefore, three main schools of thought on the origin of Arabic algebra: one emphasizes Hindu influence, another stresses the Mesopotamian, or Syriac-Persian, tradition, and the third points to Greek inspiration. The truth is probably approached if we combine the three theories."]
Throughout their time in power, the Arabs used a fully rhetorical algebra, where often even the numbers were spelled out in words. The Arabs would eventually replace spelled out numbers (e.g. twenty-two) with
Arabic numerals
Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write Decimal, decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers ...
(e.g. 22), but the Arabs did not adopt or develop a syncopated or symbolic algebra
until the work of
Ibn al-Banna
Ibn al‐Bannāʾ al‐Marrākushī ( ar, ابن البناء المراكشي), full name: Abu'l-Abbas Ahmad ibn Muhammad ibn Uthman al-Azdi al-Marrakushi () (29 December 1256 – 31 July 1321), was a Moroccan polymath who was active as a math ...
, who developed a symbolic algebra in the 13th century, followed by
Abū al-Hasan ibn Alī al-Qalasādī
Abū'l-Ḥasan ibn ʿAlī ibn Muḥammad ibn ʿAlī al-Qurashī al-Qalaṣādī ( ar, أبو الحسن علي بن محمد بن علي القرشي البسطي; 1412–1486) was a Muslim Arab mathematician from Al-Andalus specializing in Is ...
in the 15th century.
''Al-jabr wa'l muqabalah''
The Muslim
Persia
Iran, officially the Islamic Republic of Iran, and also called Persia, is a country located in Western Asia. It is bordered by Iraq and Turkey to the west, by Azerbaijan and Armenia to the northwest, by the Caspian Sea and Turkmeni ...
n mathematician
Muhammad ibn Mūsā al-Khwārizmī
Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persians, Persian polymath from Khwarazm, who produced vastly influential works in Mathematics ...
was a faculty member of the "
House of Wisdom" (''Bait al-Hikma'') in Baghdad, which was established by Al-Mamun. Al-Khwarizmi, who died around 850 CE, wrote more than half a dozen mathematical and
astronomical
Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galaxi ...
works, some of which were based on the Indian ''Sindhind''.
One of al-Khwarizmi's most famous books is entitled ''Al-jabr wa'l muqabalah'' or ''
'', and it gives an exhaustive account of solving
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s up to the second
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
. The book also introduced the fundamental concept of "
reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which Al-Khwarizmi originally described as ''al-jabr''.
[ "It is not certain just what the terms ''al-jabr'' and ''muqabalah'' mean, but the usual interpretation is similar to that implied in the translation above. The word ''al-jabr'' presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation, which is evident in the treatise; the word ''muqabalah'' is said to refer to "reduction" or "balancing"—that is, the cancellation of like terms on opposite sides of the equation."] The name "algebra" comes from the "''al-jabr''" in the title of his book.
R. Rashed and Angela Armstrong write:
''Al-Jabr'' is divided into six chapters, each of which deals with a different type of formula. The first chapter of ''Al-Jabr'' deals with equations whose squares equal its roots
the second chapter deals with squares equal to number
the third chapter deals with roots equal to a number
the fourth chapter deals with squares and roots equal a number
the fifth chapter deals with squares and number equal roots
and the sixth and final chapter deals with roots and number equal to squares
[ "in six short chapters, of the six types of equations made up from the three kinds of quantities: roots, squares, and numbers (that is and numbers). Chapter I, in three short paragraphs, covers the case of squares equal to roots, expressed in modern notation as and giving the answers and respectively. (The root was not recognized.) Chapter II covers the case of squares equal to numbers, and Chapter III solves the cases of roots equal to numbers, again with three illustrations per chapter to cover the cases in which the coefficient of the variable term is equal to, more than, or less than one. Chapters IV, V, and VI are more interesting, for they cover in turn the three classical cases of three-term quadratic equations: (1) squares and roots equal to numbers, (2) squares and numbers equal to roots, and (3) roots and numbers equal to squares."]
![Bodleian MS](https://upload.wikimedia.org/wikipedia/commons/2/25/Bodleian_MS._Huntington_214_roll332_frame36.jpg)
In ''Al-Jabr'', al-Khwarizmi uses geometric proofs,
he does not recognize the root
and he only deals with positive roots. He also recognizes that the
discriminant must be positive and described the method of
completing the square
:
In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form
:ax^2 + bx + c
to the form
:a(x-h)^2 + k
for some values of ''h'' and ''k''.
In other words, completing the square places a perfe ...
, though he does not justify the procedure. The Greek influence is shown by ''Al-Jabrs geometric foundations
and by one problem taken from Heron. He makes use of lettered diagrams but all of the coefficients in all of his equations are specific numbers since he had no way of expressing with parameters what he could express geometrically; although generality of method is intended.
Al-Khwarizmi most likely did not know of Diophantus's ''Arithmetica'',
[ "the algebra of al-Khwarizmi is thoroughly rhetorical, with none of the syncopation found in the Greek ''Arithmetica'' or in Brahmagupta's work. Even numbers were written out in words rather than symbols! It is quite unlikely that al-Khwarizmi knew of the work of Diophantus, but he must have been familiar with at least the astronomical and computational portions of Brahmagupta; yet neither al-Khwarizmi nor other Arabic scholars made use of syncopation or of negative numbers."] which became known to the Arabs sometime before the 10th century.
And even though al-Khwarizmi most likely knew of Brahmagupta's work, ''Al-Jabr'' is fully rhetorical with the numbers even being spelled out in words.
So, for example, what we would write as
:
Diophantus would have written as
:
And al-Khwarizmi would have written as
:One square and ten roots of the same amount to thirty-nine ''
dirhem
The dirham, dirhem or dirhm ( ar, درهم) is a silver unit of currency historically and currently used by several Arab and Arab influenced states. The term has also been used as a related unit of mass.
Unit of mass
The dirham was a un ...
s''; that is to say, what must be the square which, when increased by ten of its own roots, amounts to thirty-nine?
''Logical Necessities in Mixed Equations''
'Abd al-Hamīd ibn Turk authored a manuscript entitled ''Logical Necessities in Mixed Equations'', which is very similar to al-Khwarzimi's ''Al-Jabr'' and was published at around the same time as, or even possibly earlier than, ''Al-Jabr''.
[ "The ''Algebra'' of al-Khwarizmi usually is regarded as the first work on the subject, but a recent publication in Turkey raises some questions about this. A manuscript of a work by 'Abd-al-Hamid ibn-Turk, entitled "Logical Necessities in Mixed Equations," was part of a book on ''Al-jabr wa'l muqabalah'' which was evidently very much the same as that by al-Khwarizmi and was published at about the same time—possibly even earlier. The surviving chapters on "Logical Necessities" give precisely the same type of geometric demonstration as al-Khwarizmi's ''Algebra'' and in one case the same illustrative example . In one respect 'Abd-al-Hamad's exposition is more thorough than that of al-Khwarizmi for he gives geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution. Similarities in the works of the two men and the systematic organization found in them seem to indicate that algebra in their day was not so recent a development as has usually been assumed. When textbooks with a conventional and well-ordered exposition appear simultaneously, a subject is likely to be considerably beyond the formative stage. ..Note the omission of Diophantus and Pappus, authors who evidently were not at first known in Arabia, although the Diophantine ''Arithmetica'' became familiar before the end of the tenth century."] The manuscript gives exactly the same geometric demonstration as is found in ''Al-Jabr'', and in one case the same example as found in ''Al-Jabr'', and even goes beyond ''Al-Jabr'' by giving a geometric proof that if the discriminant is negative then the quadratic equation has no solution.
The similarity between these two works has led some historians to conclude that Arabic algebra may have been well developed by the time of al-Khwarizmi and 'Abd al-Hamid.
Abu Kamil and al-Karkhi
Arabic mathematicians treated
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
s as algebraic objects. The
Egypt
Egypt ( ar, مصر , ), officially the Arab Republic of Egypt, is a transcontinental country spanning the northeast corner of Africa and southwest corner of Asia via a land bridge formed by the Sinai Peninsula. It is bordered by the Mediter ...
ian mathematician
Abū Kāmil Shujā ibn Aslam (c. 850–930) was the first to accept irrational numbers (often in the form of a
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
E ...
,
cube root
In mathematics, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. Fo ...
or
fourth root) as solutions to quadratic equations or as
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
s in an equation. He was also the first to solve three non-linear
simultaneous equations
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single e ...
with three unknown
variables.
Al-Karkhi
( fa, ابو بکر محمد بن الحسن الکرجی; c. 953 – c. 1029) was a 10th-century Persian mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal surviving works are ...
(953–1029), also known as Al-Karaji, was the successor of
Abū al-Wafā' al-Būzjānī
Abū al-Wafāʾ Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī or Abū al-Wafā Būzhjānī ( fa, ابوالوفا بوزجانی or بوژگانی) (10 June 940 – 15 July 998) was a Persian mathematician a ...
(940–998) and he discovered the first numerical solution to equations of the form
[ "Abu'l Wefa was a capable algebraist as well as a trionometer. ..His successor al-Karkhi evidently used this translation to become an Arabic disciple of Diophantus—but without Diophantine analysis! ..In particular, to al-Karkhi is attributed the first numerical solution of equations of the form (only equations with positive roots were considered),"] Al-Karkhi only considered positive roots.
Al-Karkhi is also regarded as the first person to free algebra from
geometrical
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
operations and replace them with the type of
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
operations which are at the core of algebra today. His work on algebra and polynomials gave the rules for arithmetic operations to manipulate polynomials. The
historian of mathematics
The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments ...
F. Woepcke, in ''Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi'' (
Paris
Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. S ...
, 1853), praised Al-Karaji for being "the first who introduced the theory of algebraic calculus". Stemming from this, Al-Karaji investigated
binomial coefficients
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
and
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although o ...
.
Omar Khayyám, Sharaf al-Dīn, and al-Kashi
Omar Khayyám (c. 1050 – 1123) wrote a book on Algebra that went beyond ''Al-Jabr'' to include equations of the third degree.
[ "Omar Khayyam (c. 1050 – 1123), the "tent-maker," wrote an ''Algebra'' that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the sixteenth century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). .. For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ..One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved.""] Omar Khayyám provided both arithmetic and geometric solutions for quadratic equations, but he only gave geometric solutions for general
cubic equations
In algebra, a cubic equation in one variable is an equation of the form
:ax^3+bx^2+cx+d=0
in which is nonzero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
since he mistakenly believed that arithmetic solutions were impossible.
His method of solving cubic equations by using intersecting conics had been used by
Menaechmus :''There is also a Menaechmus in Plautus' play, ''The Menaechmi''.''
Menaechmus ( el, Μέναιχμος, 380–320 BC) was an ancient Greek mathematician, geometer and philosopher born in Alopeconnesus or Prokonnesos in the Thracian Chersonese, w ...
,
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
, and
Ibn al-Haytham (Alhazen), but Omar Khayyám generalized the method to cover all cubic equations with positive roots.
He only considered positive roots and he did not go past the third degree.
He also saw a strong relationship between geometry and algebra.
In the 12th century,
Sharaf al-Dīn al-Tūsī
Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī ( fa, شرفالدین مظفر بن محمد بن مظفر توسی; 1135 – 1213) was an Iranian mathematician and astronomer of the Islamic Golden Age (during the ...
(1135–1213) wrote the ''Al-Mu'adalat'' (''Treatise on Equations''), which dealt with eight types of cubic equations with positive solutions and five types of cubic equations which may not have positive solutions. He used what would later be known as the "
Ruffini-
Horner
Horner is an English and German surname that derives from the Middle English word for the occupation ''horner'', meaning horn-worker or horn-maker, or even horn-blower.
People
*Alison Horner (born 1966), British businesswoman
* Arthur Horner (dis ...
method" to
numerically approximate the root of a cubic equation. He also developed the concepts of the
maxima and minima
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ra ...
of curves in order to solve cubic equations which may not have positive solutions. He understood the importance of the
discriminant of the cubic equation and used an early version of
Cardano's formula
In algebra, a cubic equation in one variable is an equation of the form
:ax^3+bx^2+cx+d=0
in which is nonzero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the ...
to find algebraic solutions to certain types of cubic equations. Some scholars, such as Roshdi Rashed, argue that Sharaf al-Din discovered the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of cubic polynomials and realized its significance, while other scholars connect his solution to the ideas of Euclid and Archimedes.
Sharaf al-Din also developed the concept of a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
. In his analysis of
the equation
for example, he begins by changing the equation's form to
. He then states that the question of whether the equation has a solution depends on whether or not the “function” on the left side reaches the value
. To determine this, he finds a maximum value for the function. He proves that the maximum value occurs when
, which gives the functional value
. Sharaf al-Din then states that if this value is less than
, there are no positive solutions; if it is equal to
, then there is one solution at
; and if it is greater than
, then there are two solutions, one between
and
and one between
and
.
In the early 15th century,
Jamshīd al-Kāshī
Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) ( fa, غیاث الدین جمشید کاشانی ''Ghiyās-ud-dīn Jamshīd Kāshānī'') (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxania) was a Persian astronomer ...
developed an early form of
Newton's method
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ...
to numerically solve the equation
to find roots of
. Al-Kāshī also developed
decimal fractions
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
and claimed to have discovered it himself. However, J. Lennart Berggrenn notes that he was mistaken, as decimal fractions were first used five centuries before him by the
Baghdad
Baghdad (; ar, بَغْدَاد , ) is the capital of Iraq and the second-largest city in the Arab world after Cairo. It is located on the Tigris near the ruins of the ancient city of Babylon and the Sassanid Persian capital of Ctesiphon ...
i mathematician
Abu'l-Hasan al-Uqlidisi Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi ( ar, أبو الحسن أحمد بن ابراهيم الإقليدسي) was a Muslim Arab mathematician, who was active in Damascus and Baghdad. He wrote the earliest surviving book on the positional use ...
as early as the 10th century.
[
]
Al-Hassār, Ibn al-Banna, and al-Qalasadi
Al-Hassār
Al-Hassar or Abu Bakr Muhammad ibn Abdallah ibn Ayyash al-Hassar ( ar, أبو بكر محمد ابن عياش الحصَار) was a 12th-century Moroccan mathematician. He is the author of two books ''Kitab al-bayan wat-tadhkar'' (Book of Demonstr ...
, a mathematician from Morocco
Morocco (),, ) officially the Kingdom of Morocco, is the westernmost country in the Maghreb region of North Africa. It overlooks the Mediterranean Sea to the north and the Atlantic Ocean to the west, and has land borders with Algeria to ...
specializing in Islamic inheritance jurisprudence
Islamic Inheritance jurisprudence is a field of Islamic jurisprudence ( ar, فقه) that deals with inheritance, a topic that is prominently dealt with in the Qur'an. It is often called ''Mīrāth'', and its branch of Islamic law is technically ...
during the 12th century, developed the modern symbolic mathematical notation
Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathem ...
for fractions
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
, where the numerator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
and denominator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
are separated by a horizontal bar. This same fractional notation appeared soon after in the work of Fibonacci
Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Wester ...
in the 13th century.
Abū al-Hasan ibn Alī al-Qalasādī
Abū'l-Ḥasan ibn ʿAlī ibn Muḥammad ibn ʿAlī al-Qurashī al-Qalaṣādī ( ar, أبو الحسن علي بن محمد بن علي القرشي البسطي; 1412–1486) was a Muslim Arab mathematician from Al-Andalus specializing in Is ...
(1412–1486) was the last major medieval Arab
The Arabs (singular: Arab; singular ar, عَرَبِيٌّ, DIN 31635: , , plural ar, عَرَب, DIN 31635: , Arabic pronunciation: ), also known as the Arab people, are an ethnic group mainly inhabiting the Arab world in Western Asia, ...
algebraist, who made the first attempt at creating an algebraic notation since Ibn al-Banna
Ibn al‐Bannāʾ al‐Marrākushī ( ar, ابن البناء المراكشي), full name: Abu'l-Abbas Ahmad ibn Muhammad ibn Uthman al-Azdi al-Marrakushi () (29 December 1256 – 31 July 1321), was a Moroccan polymath who was active as a math ...
two centuries earlier, who was himself the first to make such an attempt since Diophantus
Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...
and Brahmagupta
Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical trea ...
in ancient times. The syncopated notations of his predecessors, however, lacked symbols for mathematical operations
In mathematics, an operation is a function which takes zero or more input values (also called "''operands''" or "arguments") to a well-defined output value. The number of operands is the arity of the operation.
The most commonly studied operat ...
.[ Al-Qalasadi "took the first steps toward the introduction of algebraic symbolism by using letters in place of numbers"][ and by "using short Arabic words, or just their initial letters, as mathematical symbols."][
]
Europe and the Mediterranean region
Just as the death of Hypatia
Hypatia, Koine pronunciation (born 350–370; died 415 AD) was a neoplatonist philosopher, astronomer, and mathematician, who lived in Alexandria, Egypt, then part of the Eastern Roman Empire. She was a prominent thinker in Alexandria where ...
signals the close of the Library of Alexandria
The Great Library of Alexandria in Alexandria, Egypt, was one of the largest and most significant libraries of the ancient world. The Library was part of a larger research institution called the Mouseion, which was dedicated to the Muses, th ...
as a mathematical center, so does the death of Boethius
Anicius Manlius Severinus Boethius, commonly known as Boethius (; Latin: ''Boetius''; 480 – 524 AD), was a Roman senator, consul, ''magister officiorum'', historian, and philosopher of the Early Middle Ages. He was a central figure in the tr ...
signal the end of mathematics in the Western Roman Empire
The Western Roman Empire comprised the western provinces of the Roman Empire at any time during which they were administered by a separate independent Imperial court; in particular, this term is used in historiography to describe the period fr ...
. Although there was some work being done at Athens
Athens ( ; el, Αθήνα, Athína ; grc, Ἀθῆναι, Athênai (pl.) ) is both the capital and largest city of Greece. With a population close to four million, it is also the seventh largest city in the European Union. Athens dominates ...
, it came to a close when in 529 the Byzantine
The Byzantine Empire, also referred to as the Eastern Roman Empire or Byzantium, was the continuation of the Roman Empire primarily in its eastern provinces during Late Antiquity and the Middle Ages, when its capital city was Constantinopl ...
emperor Justinian
Justinian I (; la, Iustinianus, ; grc-gre, Ἰουστινιανός ; 48214 November 565), also known as Justinian the Great, was the Byzantine emperor from 527 to 565.
His reign is marked by the ambitious but only partly realized ''renovat ...
closed the pagan
Paganism (from classical Latin ''pāgānus'' "rural", "rustic", later "civilian") is a term first used in the fourth century by early Christians for people in the Roman Empire who practiced polytheism, or ethnic religions other than Judaism. ...
philosophical schools. The year 529 is now taken to be the beginning of the medieval period. Scholars fled the West towards the more hospitable East, particularly towards Persia
Iran, officially the Islamic Republic of Iran, and also called Persia, is a country located in Western Asia. It is bordered by Iraq and Turkey to the west, by Azerbaijan and Armenia to the northwest, by the Caspian Sea and Turkmeni ...
, where they found haven under King Chosroes and established what might be termed an "Athenian Academy in Exile".[ "The death of Boethius may be taken to mark the end of ancient mathematics in the Western Roman Empire, as the death of Hypatia had marked the close of Alexandria as a mathematical center; but work continued for a few years longer at Athens. ..When in 527 Justinian became emperor in the East, he evidently felt that the pagan learning of the Academy and other philosophical schools at Athens was a threat to orthodox Christianity; hence, in 529 the philosophical schools were closed and the scholars dispersed. Rome at the time was scarcely a very hospitable home for scholars, and Simplicius and some of the other philosophers looked to the East for haven. This they found in Persia, where under King Chosroes they established what might be called the "Athenian Academy in Exile."(Sarton 1952; p. 400)."] Under a treaty with Justinian, Chosroes would eventually return the scholars to the Eastern Empire
The Byzantine Empire, also referred to as the Eastern Roman Empire or Byzantium, was the continuation of the Roman Empire primarily in its eastern provinces during Late Antiquity and the Middle Ages, when its capital city was Constantinopl ...
. During the Dark Ages, European mathematics was at its nadir with mathematical research consisting mainly of commentaries on ancient treatises; and most of this research was centered in the Byzantine Empire
The Byzantine Empire, also referred to as the Eastern Roman Empire or Byzantium, was the continuation of the Roman Empire primarily in its eastern provinces during Late Antiquity and the Middle Ages, when its capital city was Constantinopl ...
. The end of the medieval period is set as the fall of Constantinople
la, Constantinopolis ota, قسطنطينيه
, alternate_name = Byzantion (earlier Greek name), Nova Roma ("New Rome"), Miklagard/Miklagarth (Old Norse), Tsargrad ( Slavic), Qustantiniya (Arabic), Basileuousa ("Queen of Cities"), Megalopolis (" ...
to the Turks
Turk or Turks may refer to:
Communities and ethnic groups
* Turkic peoples, a collection of ethnic groups who speak Turkic languages
* Turkish people, or the Turks, a Turkic ethnic group and nation
* Turkish citizen, a citizen of the Republic ...
in 1453.
Late Middle Ages
The 12th century saw a flood of translations from Arabic
Arabic (, ' ; , ' or ) is a Semitic languages, Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C ...
into Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
and by the 13th century, European mathematics was beginning to rival the mathematics of other lands. In the 13th century, the solution of a cubic equation by Fibonacci
Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Wester ...
is representative of the beginning of a revival in European algebra.
As the Islamic world was declining after the 15th century, the European world was ascending. And it is here that algebra was further developed.
Symbolic algebra
Modern notation for arithmetic operations was introduced between the end of the 15th century and the beginning of the 16th century by Johannes Widmann
Johannes Widmann (c. 1460 – after 1498) was a German mathematician. The + and - symbols first appeared in print in his book ''Mercantile Arithmetic'' or ''Behende und hüpsche Rechenung auff allen Kauffmanschafft'' published in Leipzig in 1489 ...
and Michael Stifel
Michael Stifel or Styfel (1487 – April 19, 1567) was a German monk, Protestant reformer and mathematician. He was an Augustinian who became an early supporter of Martin Luther. He was later appointed professor of mathematics at Jena Universi ...
. At the end of 16th century, François Viète
François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...
introduced symbols, now called variables, for representing indeterminate or unknown numbers. This created a new algebra consisting of computing with symbolic expressions as if they were numbers.
Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equation
In mathematics, a quartic equation is one which can be expressed as a ''quartic function'' equaling zero. The general form of a quartic equation is
:ax^4+bx^3+cx^2+dx+e=0 \,
where ''a'' ≠ 0.
The quartic is the highest order polynomi ...
s, developed in the mid-16th century. The idea of a determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
was developed by Japanese mathematician Kowa Seki
, Selin, Helaine. (1997). ''Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures,'' p. 890 also known as ,Selin, was a Japanese mathematician and author of the Edo period.
Seki laid foundations for the subs ...
in the 17th century, followed by Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
ten years later, for the purpose of solving systems of simultaneous linear equations using matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
. Gabriel Cramer
Gabriel Cramer (; 31 July 1704 – 4 January 1752) was a Genevan mathematician. He was the son of physician Jean Cramer and Anne Mallet Cramer.
Biography
Cramer showed promise in mathematics from an early age. At 18 he received his doctorate ...
also did some work on matrices and determinants in the 18th century.
The symbol ''x''
By tradition, the first unknown variable
Variable may refer to:
* Variable (computer science), a symbolic name associated with a value and whose associated value may be changed
* Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
in an algebraic problem is nowadays represented by the symbol
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...
and if there is a second or a third unknown, then these are labeled and respectively. Algebraic is conventionally printed in italic type
In typography, italic type is a cursive font based on a stylised form of calligraphic handwriting. Owing to the influence from calligraphy, italics normally slant slightly to the right. Italics are a way to emphasise key points in a printed tex ...
to distinguish it from the sign of multiplication.
Mathematical historians generally agree that the use of in algebra was introduced by René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...
and was first published in his treatise ''La Géométrie
''La Géométrie'' was published in 1637 as an appendix to ''Discours de la méthode'' (''Discourse on the Method''), written by René Descartes. In the ''Discourse'', he presents his method for obtaining clarity on any subject. ''La Géométrie ...
'' (1637). In that work, he used letters from the beginning of the alphabet for known quantities, and letters from the end of the alphabet for unknowns. It has been suggested that he later settled on (in place of ) for the first unknown because of its relatively greater abundance in the French and Latin typographical fonts of the time.
Three alternative theories of the origin of algebraic were suggested in the 19th century: (1) a symbol used by German algebraists and thought to be derived from a cursive letter mistaken for ; (2) the numeral ''1'' with oblique strikethrough; and (3) an Arabic/Spanish source (see below). But the Swiss-American historian of mathematics Florian Cajori
Florian Cajori (February 28, 1859 – August 14 or 15, 1930) was a Swiss-American historian of mathematics.
Biography
Florian Cajori was born in Zillis, Switzerland, as the son of Georg Cajori and Catherine Camenisch. He attended schools first ...
examined these and found all three lacking in concrete evidence; Cajori credited Descartes as the originator, and described his and as "free from tradition and their choice purely arbitrary."
Nevertheless, the Hispano-Arabic hypothesis continues to have a presence in popular culture
Popular culture (also called mass culture or pop culture) is generally recognized by members of a society as a set of practices, beliefs, artistic output (also known as, popular art or mass art) and objects that are dominant or prevalent in a ...
today. It is the claim that algebraic is the abbreviation of a supposed loanword
A loanword (also loan word or loan-word) is a word at least partly assimilated from one language (the donor language) into another language. This is in contrast to cognates, which are words in two or more languages that are similar because th ...
from Arabic in Old Spanish. The theory originated in 1884 with the German orientalist Paul de Lagarde
Paul Anton de Lagarde (2 November 1827 – 22 December 1891) was a German biblical scholar and orientalist, sometimes regarded as one of the greatest orientalists of the 19th century. Lagarde's strong support of anti-Semitism, vocal opposition ...
, shortly after he published his edition of a 1505 Spanish/Arabic bilingual glossary in which Spanish ''cosa'' ("thing") was paired with its Arabic equivalent, شىء (''shayʔ''), transcribed as ''xei''. (The "sh" sound in Old Spanish
Old Spanish, also known as Old Castilian ( es, castellano antiguo; osp, romance castellano ), or Medieval Spanish ( es, español medieval), was originally a dialect of Vulgar Latin spoken in the former provinces of the Roman Empire that provided ...
was routinely spelled ) Evidently Lagarde was aware that Arab mathematicians, in the "rhetorical" stage of algebra's development, often used that word to represent the unknown quantity. He surmised that "nothing could be more natural" ("Nichts war also natürlicher...") than for the initial of the Arabic word— romanized as the Old Spanish —to be adopted for use in algebra. A later reader reinterpreted Lagarde's conjecture as having "proven" the point. Lagarde was unaware that early Spanish mathematicians used, not a ''transcription'' of the Arabic word, but rather its ''translation'' in their own language, "cosa". There is no instance of ''xei'' or similar forms in several compiled historical vocabularies of Spanish.
Gottfried Leibniz
Although the mathematical notion of function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
was implicit in trigonometric
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
and logarithmic tables, which existed in his day, Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
was the first, in 1692 and 1694, to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa
In common usage, the abscissa refers to the (''x'') coordinate and the ordinate refers to the (''y'') coordinate of a standard two-dimensional graph.
The distance of a point from the y-axis, scaled with the x-axis, is called abscissa or x coo ...
, ordinate
In common usage, the abscissa refers to the (''x'') coordinate and the ordinate refers to the (''y'') coordinate of a standard two-dimensional graph.
The distance of a point from the y-axis, scaled with the x-axis, is called abscissa or x c ...
, tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
, chord, and the perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
. In the 18th century, "function" lost these geometrical associations.
Leibniz realized that the coefficients of a system of linear equations could be arranged into an array, now called a matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
, which can be manipulated to find the solution of the system, if any. This method was later called Gaussian elimination
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...
. Leibniz also discovered Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
and symbolic logic, also relevant to algebra.
Abstract algebra
The ability to do algebra is a skill cultivated in mathematics education
In contemporary education, mathematics education, known in Europe as the didactics or pedagogy of mathematics – is the practice of teaching, learning and carrying out scholarly research into the transfer of mathematical knowledge.
Although re ...
. As explained by Andrew Warwick, Cambridge University
, mottoeng = Literal: From here, light and sacred draughts.
Non literal: From this place, we gain enlightenment and precious knowledge.
, established =
, other_name = The Chancellor, Masters and Schola ...
students in the early 19th century practiced "mixed mathematics", doing exercise
Exercise is a body activity that enhances or maintains physical fitness and overall health and wellness.
It is performed for various reasons, to aid growth and improve strength, develop muscles and the cardiovascular system, hone athletic ...
s based on physical variables such as space, time, and weight. Over time the association of variable
Variable may refer to:
* Variable (computer science), a symbolic name associated with a value and whose associated value may be changed
* Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
s with physical quantities faded away as mathematical technique grew. Eventually mathematics was concerned completely with abstract polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s, complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s, hypercomplex number
In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers.
The study of hypercomplex numbers in the late 19th century forms the basis of modern group represent ...
s and other concepts. Application to physical situations was then called applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
or mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
, and the field of mathematics expanded to include abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
. For instance, the issue of constructible number
In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length , r, can be constructed with compass and straightedge in a finite number of steps. Equivalently, r is cons ...
s showed some mathematical limitations, and the field of Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
was developed.
The father of algebra
The title of "the father of algebra" is frequently credited to the Persian mathematician Al-Khwarizmi
Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astronom ...
,[ "Diophantus sometimes is called "the father of algebra," but this title more appropriately belongs to Abu Abdullah bin mirsmi al-Khwarizmi. It is true that in two respects the work of al-Khwarizmi represented a retrogression from that of Diophantus. First, it is on a far more elementary level than that found in the Diophantine problems and, second, the algebra of al-Khwarizmi is thoroughly rhetorical, with none of the syncopation found in the Greek ''Arithmetica'' or in Brahmagupta's work. Even numbers were written out in words rather than symbols! It is quite unlikely that al-Khwarizmi knew of the work of Diophantus, but he must have been familiar with at least the astronomical and computational portions of Brahmagupta; yet neither al-Khwarizmi nor other Arabic scholars made use of syncopation or of negative numbers."] supported by historians of mathematics
The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments ...
, such as Carl Benjamin Boyer
Carl Benjamin Boyer (November 3, 1906 – April 26, 1976) was an American historian of sciences, and especially mathematics. Novelist David Foster Wallace called him the "Gibbon of math history". It has been written that he was one of few histori ...
, Solomon Gandz and Bartel Leendert van der Waerden
Bartel Leendert van der Waerden (; 2 February 1903 – 12 January 1996) was a Dutch mathematician and historian of mathematics.
Biography
Education and early career
Van der Waerden learned advanced mathematics at the University of Amster ...
. However, the point is debatable and the title is sometimes credited to the Hellenistic
In Classical antiquity, the Hellenistic period covers the time in Mediterranean history after Classical Greece, between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire, as signified by the Battle of Actium in ...
mathematician Diophantus
Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...
.[ "Diophantus, the father of algebra, in whose honor I have named this chapter, lived in Alexandria, in Roman Egypt, in either the 1st, the 2nd, or the 3rd century CE."] Those who support Diophantus point to the algebra found in ''Al-Jabr
''The Compendious Book on Calculation by Completion and Balancing'' ( ar, كتاب المختصر في حساب الجبر والمقابلة, ; la, Liber Algebræ et Almucabola), also known as ''Al-Jabr'' (), is an Arabic mathematical treati ...
'' being more elementary
Elementary may refer to:
Arts, entertainment, and media Music
* ''Elementary'' (Cindy Morgan album), 2001
* ''Elementary'' (The End album), 2007
* ''Elementary'', a Melvin "Wah-Wah Watson" Ragin album, 1977
Other uses in arts, entertainment, a ...
than the algebra found in '' Arithmetica'', and ''Arithmetica'' being syncopated while ''Al-Jabr'' is fully rhetorical. However, the mathematics historian Kurt Vogel argues against Diophantus holding this title, as his mathematics was not much more algebraic than that of the ancient Babylonians
Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c. 1 ...
.
Those who support Al-Khwarizmi point to the fact that he gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,[ "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwarizmi's exposition that his readers must have had little difficulty in mastering the solutions."] and was the first to teach algebra in an elementary form and for its own sake, whereas Diophantus was primarily concerned with the theory of numbers
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
.[Gandz and Saloman (1936), ''The sources of al-Khwarizmi's algebra'', Osiris i, p. 263–277: "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".] Al-Khwarizmi also introduced the fundamental concept of "reduction" and "balancing" (which he originally used the term ''al-jabr'' to refer to), referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation.[ Other supporters of Al-Khwarizmi point to his algebra no longer being concerned "with a series of problems to be resolved, but an ]exposition
Exposition (also the French for exhibition) may refer to:
*Universal exposition or World's Fair
* Expository writing
** Exposition (narrative)
* Exposition (music)
*Trade fair
A trade fair, also known as trade show, trade exhibition, or trade e ...
which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." They also point to his treatment of an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."[ Victor J. Katz regards ''Al-Jabr'' as the first true algebra text that is still extant.]
See also
*
*
References
Sources
*
*
*
* Bashmakova, I, and Smirnova, G. (2000) ''The Beginnings and Evolution of Algebra'', Dolciani Mathematical Expositions 23. Translated by Abe Shenitzer. The Mathematical Association of America.
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
External links
"Commentary by Islam's Sheikh Zakariyya al-Ansari on Ibn al-Hā’im's Poem on the Science of Algebra and Balancing Called the Creator's Epiphany in Explaining the Cogent"
featuring the basic concepts of algebra dating back to the 15th century, from the World Digital Library
The World Digital Library (WDL) is an international digital library operated by UNESCO and the United States Library of Congress.
The WDL has stated that its mission is to promote international and intercultural understanding, expand the volume ...
.
{{DEFAULTSORT:Algebra, History of