Arithmetic (from the Greek ἀριθμός arithmos, "number") is a
branch of mathematics that consists of the study of numbers,
especially the properties of the traditional operations on
them—addition, subtraction, multiplication and division. Arithmetic
is an elementary part of number theory, and number theory is
considered to be one of the top-level divisions of modern mathematics,
along with algebra, geometry, and analysis. The terms arithmetic and
higher arithmetic were used until the beginning of the
20th century as synonyms for number theory and are sometimes
still used to refer to a wider part of number theory.
Multiplication (× or · or *)
2.4 Division (÷ or /)
4 Compound unit arithmetic
4.1 Basic arithmetic operations
4.2 Principles of compound unit arithmetic
4.3 Operations in practice
Arithmetic in education
7 See also
7.1 Related topics
10 External links
The prehistory of arithmetic is limited to a small number of artifacts
which may indicate the conception of addition and subtraction, the
best-known being the
Ishango bone from central Africa, dating from
somewhere between 20,000 and 18,000 BC, although its
interpretation is disputed.
The earliest written records indicate the Egyptians and Babylonians
used all the elementary arithmetic operations as early as
2000 BC. These artifacts do not always reveal the specific
process used for solving problems, but the characteristics of the
particular numeral system strongly influence the complexity of the
methods. The hieroglyphic system for Egyptian numerals, like the later
Roman numerals, descended from tally marks used for counting. In both
cases, this origin resulted in values that used a decimal base but did
not include positional notation. Complex calculations with Roman
numerals required the assistance of a counting board or the Roman
abacus to obtain the results.
Early number systems that included positional notation were not
decimal, including the sexagesimal (base 60) system for
Babylonian numerals and the vigesimal (base 20) system that
defined Maya numerals. Because of this place-value concept, the
ability to reuse the same digits for different values contributed to
simpler and more efficient methods of calculation.
The continuous historical development of modern arithmetic starts with
Hellenistic civilization of ancient Greece, although it originated
much later than the Babylonian and Egyptian examples. Prior to the
Euclid around 300 BC, Greek studies in mathematics
overlapped with philosophical and mystical beliefs. For example,
Nicomachus summarized the viewpoint of the earlier Pythagorean
approach to numbers, and their relationships to each other, in his
Introduction to Arithmetic.
Greek numerals were used by Archimedes,
Diophantus and others in a
positional notation not very different from ours. Because the ancient
Greeks lacked a symbol for zero (until the Hellenistic period), they
used three separate sets of symbols.
One set for the unit's place, one
for the ten's place, and one for the hundred's. Then for the
thousand's place they would reuse the symbols for the unit's place,
and so on. Their addition algorithm was identical to ours, and their
multiplication algorithm was only very slightly different. Their long
division algorithm was the same, and the square root algorithm that
was once taught in school[clarification needed] was known to
Archimedes, who may have invented it. He preferred it to Hero's method
of successive approximation because, once computed, a digit doesn't
change, and the square roots of perfect squares, such as 7485696,
terminate immediately as 2736. For numbers with a fractional part,
such as 546.934, they used negative powers of 60 instead of negative
powers of 10 for the fractional part 0.934.
The ancient Chinese had advanced arithmetic studies dating from the
Shang Dynasty and continuing through the Tang Dynasty, from basic
numbers to advanced algebra. The ancient Chinese used a positional
notation similar to that of the Greeks. Since they also lacked a
symbol for zero, they had one set of symbols for the unit's place, and
a second set for the ten's place. For the hundred's place they then
reused the symbols for the unit's place, and so on. Their symbols were
based on the ancient counting rods. It is a complicated question to
determine exactly when the Chinese started calculating with positional
representation, but it was definitely before 400 BC. The
ancient Chinese were the first to meaningfully discover, understand,
and apply negative numbers as explained in the Nine Chapters on the
Mathematical Art (Jiuzhang Suanshu), which was written by Liu Hui.
The gradual development of
Hindu–Arabic numerals independently
devised the place-value concept and positional notation, which
combined the simpler methods for computations with a decimal base and
the use of a digit representing 0. This allowed the system to
consistently represent both large and small integers. This approach
eventually replaced all other systems. In the early 6th century AD,
the Indian mathematician
Aryabhata incorporated an existing version of
this system in his work, and experimented with different notations. In
the 7th century,
Brahmagupta established the use of 0 as a
separate number and determined the results for multiplication,
division, addition and subtraction of zero and all other numbers,
except for the result of division by 0. His contemporary, the Syriac
Severus Sebokht (650 AD) said, "Indians possess a method
of calculation that no word can praise enough. Their rational system
of mathematics, or of their method of calculation. I mean the system
using nine symbols." The Arabs also learned this new method and
called it hesab.
Stepped Reckoner was the first calculator that could perform
all four arithmetic operations.
Codex Vigilanus described an early form of Arabic
numerals (omitting 0) by 976 AD, Leonardo of Pisa
(Fibonacci) was primarily responsible for spreading their use
Europe after the publication of his book
Liber Abaci in
1202. He wrote, "The method of the Indians (Latin Modus Indoram)
surpasses any known method to compute. It's a marvelous method. They
do their computations using nine figures and symbol zero".
In the Middle Ages, arithmetic was one of the seven liberal arts
taught in universities.
The flourishing of algebra in the medieval
Islamic world and in
Europe was an outgrowth of the enormous simplification of
computation through decimal notation.
Various types of tools have been invented and widely used to assist in
numeric calculations. Before Renaissance, they were various types of
abaci. More recent examples include slide rules, nomograms and
mechanical calculators, such as Pascal's calculator. At present, they
have been supplanted by electronic calculators and computers.
See also: Algebraic operation
The basic arithmetic operations are addition, subtraction,
multiplication and division, although this subject also includes more
advanced operations, such as manipulations of percentages, square
roots, exponentiation, and logarithmic functions.
performed according to an order of operations. Any set of objects upon
which all four arithmetic operations (except division by 0) can
be performed, and where these four operations obey the usual laws, is
called a field.
Main article: Addition
Addition is the basic operation of arithmetic. In its simplest form,
addition combines two numbers, the addends or terms, into a single
number, the sum of the numbers (Such as 2 + 2 = 4 or 3 + 5 = 8).
Adding more than two numbers can be viewed as repeated addition; this
procedure is known as summation and includes ways to add infinitely
many numbers in an infinite series; repeated addition of the
number 1 is the most basic form of counting.
Addition is commutative and associative so the order the terms are
added in does not matter. The identity element of addition (the
additive identity) is 0, that is, adding 0 to any number
yields that same number. Also, the inverse element of addition (the
additive inverse) is the opposite of any number, that is, adding the
opposite of any number to the number itself yields the additive
identity, 0. For example, the opposite of 7 is −7, so 7 +
(−7) = 0.
Addition can be given geometrically as in the following example:
If we have two sticks of lengths 2 and 5, then if we place the sticks
one after the other, the length of the stick thus formed is 2 + 5 = 7.
Main article: Subtraction
See also: Method of complements
Subtraction is the inverse of addition.
Subtraction finds the
difference between two numbers, the minuend minus the subtrahend. If
the minuend is larger than the subtrahend, the difference is positive;
if the minuend is smaller than the subtrahend, the difference is
negative; if they are equal, the difference is 0.
Subtraction is neither commutative nor associative. For that reason,
it is often helpful to look at subtraction as addition of the minuend
and the opposite of the subtrahend, that is a − b = a + (−b). When
written as a sum, all the properties of addition hold.
There are several methods for calculating results, some of which are
particularly advantageous to machine calculation. For example, digital
computers employ the method of two's complement. Of great importance
is the counting up method by which change is made. Suppose an amount P
is given to pay the required amount Q, with P greater than Q. Rather
than performing the subtraction P − Q and counting out that amount
in change, money is counted out starting at Q and continuing until
reaching P. Although the amount counted out must equal the result of
the subtraction P − Q, the subtraction was never really done and the
value of P − Q might still be unknown to the change-maker.
Multiplication (× or · or *)
Main article: Multiplication
Multiplication is the second basic operation of arithmetic.
Multiplication also combines two numbers into a single number, the
product. The two original numbers are called the multiplier and the
multiplicand, sometimes both simply called factors.
Multiplication may be viewed as a scaling operation. If the numbers
are imagined as lying in a line, multiplication by a number, say x,
greater than 1 is the same as stretching everything away
from 0 uniformly, in such a way that the number 1 itself is
stretched to where x was. Similarly, multiplying by a number less
than 1 can be imagined as squeezing towards 0. (Again, in
such a way that 1 goes to the multiplicand.)
Multiplication is commutative and associative; further it is
distributive over addition and subtraction. The multiplicative
identity is 1, that is, multiplying any number by 1 yields
that same number. Also, the multiplicative inverse is the reciprocal
of any number (except 0; 0 is the only number without a
multiplicative inverse), that is, multiplying the reciprocal of any
number by the number itself yields the multiplicative identity.
The product of a and b is written as a × b or a·b. When a or b are
expressions not written simply with digits, it is also written by
simple juxtaposition: ab. In computer programming languages and
software packages in which one can only use characters normally found
on a keyboard, it is often written with an asterisk: a * b.
Division (÷ or /)
Main article: Division (mathematics)
Division is essentially the inverse of multiplication. Division finds
the quotient of two numbers, the dividend divided by the divisor. Any
dividend divided by 0 is undefined. For distinct positive
numbers, if the dividend is larger than the divisor, the quotient is
greater than 1, otherwise it is less than 1 (a similar rule
applies for negative numbers). The quotient multiplied by the divisor
always yields the dividend.
Division is neither commutative nor associative. As it is helpful to
look at subtraction as addition, it is helpful to look at division as
multiplication of the dividend times the reciprocal of the divisor,
that is a ÷ b = a × 1/b. When written as a product, it obeys all the
properties of multiplication.
Decimal representation refers exclusively, in common use, to the
written numeral system employing arabic numerals as the digits for a
radix 10 ("decimal") positional notation; however, any numeral
system based on powers of 10, e.g., Greek, Cyrillic, Roman, or
Chinese numerals may conceptually be described as "decimal notation"
or "decimal representation".
Modern methods for four fundamental operations (addition, subtraction,
multiplication and division) were first devised by
India. This was known during medieval
Europe as "Modus Indoram" or
Method of the Indians.
Positional notation (also known as "place-value
notation") refers to the representation or encoding of numbers using
the same symbol for the different orders of magnitude (e.g., the "ones
place", "tens place", "hundreds place") and, with a radix point, using
those same symbols to represent fractions (e.g., the "tenths place",
"hundredths place"). For example, 507.36 denotes 5 hundreds
(102), plus 0 tens (101), plus 7 units (100), plus
3 tenths (10−1) plus 6 hundredths (10−2).
The concept of 0 as a number comparable to the other basic digits is
essential to this notation, as is the concept of 0's use as a
placeholder, and as is the definition of multiplication and addition
with 0. The use of 0 as a placeholder and, therefore, the
use of a positional notation is first attested to in the Jain text
India entitled the Lokavibhâga, dated 458 AD and it was
only in the early 13th century that these concepts, transmitted
via the scholarship of the Arabic world, were introduced into Europe
by Fibonacci using the Hindu–Arabic numeral system.
Algorism comprises all of the rules for performing arithmetic
computations using this type of written numeral. For example, addition
produces the sum of two arbitrary numbers. The result is calculated by
the repeated addition of single digits from each number that occupies
the same position, proceeding from right to left. An addition table
with ten rows and ten columns displays all possible values for each
sum. If an individual sum exceeds the value 9, the result is
represented with two digits. The rightmost digit is the value for the
current position, and the result for the subsequent addition of the
digits to the left increases by the value of the second (leftmost)
digit, which is always one. This adjustment is termed a carry of the
The process for multiplying two arbitrary numbers is similar to the
process for addition. A multiplication table with ten rows and ten
columns lists the results for each pair of digits. If an individual
product of a pair of digits exceeds 9, the carry adjustment
increases the result of any subsequent multiplication from digits to
the left by a value equal to the second (leftmost) digit, which is any
value from 1 to 8 (9 × 9 = 81). Additional steps define the final
Similar techniques exist for subtraction and division.
The creation of a correct process for multiplication relies on the
relationship between values of adjacent digits. The value for any
single digit in a numeral depends on its position. Also, each position
to the left represents a value ten times larger than the position to
the right. In mathematical terms, the exponent for the radix (base)
of 10 increases by 1 (to the left) or decreases by 1
(to the right). Therefore, the value for any arbitrary digit is
multiplied by a value of the form 10n with integer n. The
list of values corresponding to all possible positions for a single
digit is written as ..., 102, 10, 1, 10−1, 10−2, ... .
Repeated multiplication of any value in this list by 10 produces
another value in the list. In mathematical terminology, this
characteristic is defined as closure, and the previous list is
described as closed under multiplication. It is the basis for
correctly finding the results of multiplication using the previous
technique. This outcome is one example of the uses of number theory.
Compound unit arithmetic
Compound unit arithmetic is the application of arithmetic
operations to mixed radix quantities such as feet and inches, gallons
and pints, pounds shillings and pence, and so on. Prior to the use of
decimal-based systems of money and units of measure, the use of
compound unit arithmetic formed a significant part of commerce and
Basic arithmetic operations
The techniques used for compound unit arithmetic were developed over
many centuries and are well-documented in many textbooks in many
different languages. In addition to the basic
arithmetic functions encountered in decimal arithmetic, compound unit
arithmetic employs three more functions:
Reduction where a compound quantity is reduced to a single quantity,
for example conversion of a distance expressed in yards, feet and
inches to one expressed in inches.
Expansion, the inverse function to reduction, is the conversion of a
quantity that is expressed as a single unit of measure to a compound
unit, such as expanding 24 oz to 1 lb, 8 oz.
Normalization is the conversion of a set of compound units to a
standard form – for example rewriting "1 ft 13 in" as "2 ft 1
Knowledge of the relationship between the various units of measure,
their multiples and their submultiples forms an essential part of
compound unit arithmetic.
Principles of compound unit arithmetic
There are two basic approaches to compound unit arithmetic:
Reduction–expansion method where all the compound unit variables are
reduced to single unit variables, the calculation performed and the
result expanded back to compound units. This approach is suited for
automated calculations. A typical example is the handling of time by
Microsoft Excel where all time intervals are processed internally as
days and decimal fractions of a day.
On-going normalization method in which each unit is treated separately
and the problem is continuously normalized as the solution develops.
This approach, which is widely described in classical texts, is best
suited for manual calculations. An example of the ongoing
normalization method as applied to addition is shown below.
UK pre-decimal currency
4 farthings (f) = 1 penny
12 pennies (d) = 1 shilling
20 shillings (s) = 1 pound (£)
The addition operation is carried out from right to left; in this
case, pence are processed first, then shillings followed by pounds.
The numbers below the "answer line" are intermediate results.
The total in the pence column is 25. Since there are 12 pennies in a
shilling, 25 is divided by 12 to give 2 with a remainder
of 1. The value "1" is then written to the answer row and
the value "2" carried forward to the shillings column. This
operation is repeated using the values in the shillings column, with
the additional step of adding the value that was carried forward from
the pennies column. The intermediate total is divided by 20 as
there are 20 shillings in a pound. The pound column is then
processed, but as pounds are the largest unit that is being
considered, no values are carried forward from the pounds column.
For the sake of simplicity, the example chosen did not have farthings.
Operations in practice
A scale calibrated in imperial units with an associated cost display.
During the 19th and 20th centuries various aids were developed to aid
the manipulation of compound units, particularly in commercial
applications. The most common aids were mechanical tills which were
adapted in countries such as the United Kingdom to accommodate pounds,
shillings, pennies and farthings and "Ready Reckoners" – books
aimed at traders that catalogued the results of various routine
calculations such as the percentages or multiples of various sums of
One typical booklet that ran to 150 pages tabulated
multiples "from one to ten thousand at the various prices from one
farthing to one pound".
The cumbersome nature of compound unit arithmetic has been recognized
for many years – in 1586, the Flemish mathematician Simon
Stevin published a small pamphlet called
De Thiende ("the tenth")
in which he declared that the universal introduction of decimal
coinage, measures, and weights to be merely a question of time while
in the modern era, many conversion programs, such as that embedded in
the calculator supplied as a standard part of the Microsoft
Windows 7 operating system display compound units in a reduced
decimal format rather than using an expanded format (i.e.
"2.5 ft" is displayed rather than "2 ft 6 in").
Until the 19th century, number theory was a synonym of "arithmetic".
The addressed problems were directly related to the basic operations
and concerned primality, divisibility, and the solution of equations
in integers, such as Fermat's last theorem. It appeared that most of
these problems, although very elementary to state, are very difficult
and may not be solved without very deep mathematics involving concepts
and methods from many other branches of mathematics. This led to new
branches of number theory such as analytic number theory, algebraic
Diophantine geometry and arithmetic algebraic geometry.
Wiles' proof of Fermat's Last Theorem
Wiles' proof of Fermat's Last Theorem is a typical example of the
necessity of sophisticated methods, which go far beyond the classical
methods of arithmetic, for solving problems that can be stated in
Arithmetic in education
Primary education in mathematics often places a strong focus on
algorithms for the arithmetic of natural numbers, integers, fractions,
and decimals (using the decimal place-value system). This study is
sometimes known as algorism.
The difficulty and unmotivated appearance of these algorithms has long
led educators to question this curriculum, advocating the early
teaching of more central and intuitive mathematical ideas.
movement in this direction was the
New Math of the 1960s and 1970s,
which attempted to teach arithmetic in the spirit of axiomatic
development from set theory, an echo of the prevailing trend in higher
Also, arithmetic was used by
Islamic Scholars in order to teach
application of the rulings related to
Zakat and Irth. This was done in
a book entitled The Best of
The book begins with the foundations of mathematics and proceeds to
its application in the later chapters.
Lists of mathematics topics
Outline of arithmetic
Addition of natural numbers
Finite field arithmetic
List of important publications in mathematics
^ Davenport, Harold, The Higher Arithmetic: An Introduction to the
Theory of Numbers (7th ed.), Cambridge University Press, Cambridge,
1999, ISBN 0-521-63446-6.
^ Rudman, Peter Strom (2007). How
Mathematics Happened: The First
50,000 Years. Prometheus Books. p. 64.
^ The Works of Archimedes, Chapter IV,
Arithmetic in Archimedes,
edited by T.L. Heath, Dover Publications Inc, New York, 2002.
^ Joseph Needham, Science and Civilization in China, Vol. 3, p. 9,
Cambridge University Press, 1959.
^ Reference: Revue de l'Orient Chretien by François Nau pp.327-338.
^ Reference: Sigler, L., "Fibonacci's Liber Abaci", Springer, 2003.
^ Tapson, Frank (1996). The Oxford
Mathematics Study Dictionary.
Oxford University Press. ISBN 0 19 914551 2.
^ Leonardo Pisano – p. 3: "Contributions to number theory".
Encyclopædia Britannica Online, 2006. Retrieved 18 September 2006.
^ Walkingame, Francis (1860). "The Tutor's Companion; or, Complete
Practical Arithmetic" (PDF). Webb, Millington & Co.
pp. 24–39. Archived from the original (PDF) on
^ Palaiseau, JFG (October 1816). Métrologie universelle, ancienne et
moderne: ou rapport des poids et mesures des empires, royaumes,
duchés et principautés des quatre parties du monde [Universal,
ancient and modern metrology: or report of weights and measurements of
empires, kingdoms, duchies and principalities of all parts of the
world] (in French). Bordeaux. Retrieved October 30, 2011.
^ Jacob de Gelder (1824). Allereerste Gronden der Cijferkunst
[Introduction to Numeracy] (in Dutch). 's-Gravenhage and Amsterdam: de
Gebroeders van Cleef. pp. 163–176. Retrieved March 2,
^ Malaisé, Ferdinand (1842). Theoretisch-Praktischer Unterricht im
Rechnen für die niederen Classen der Regimentsschulen der Königl.
Bayer. Infantrie und Cavalerie [Theoretical and practical instruction
in arithmetic for the lower classes of the Royal Bavarian Infantry and
Cavalry School] (in German). Munich. Retrieved 20 March 2012.
^ Encyclopædia Britannica, Vol I, Edinburgh, 1772, Arithmetick
^ Walkingame, Francis (1860). "The Tutor's Companion; or, Complete
Practical Arithmetic" (PDF). Webb, Millington & Co.
pp. 43–50. Archived from the original (PDF) on
^ Thomson, J (1824). The Ready Reckoner in miniature containing
accurate table from one to the thousand at the various prices from one
farthing to one pound. Montreal. Retrieved 25 March 2012.
^ O'Connor, John J.; Robertson, Edmund F. (January 2004),
"Arithmetic", MacTutor History of
Mathematics archive, University of
St Andrews .
^ Mathematically Correct: Glossary of Terms
^ al-Dumyati, Abd-al-Fattah Bin Abd-al-Rahman al-Banna (1887). "The
Best of Arithmetic".
World Digital Library
World Digital Library (in Arabic). Retrieved 30
Cunnington, Susan, The Story of Arithmetic: A Short History of Its
Origin and Development, Swan Sonnenschein, London, 1904
Dickson, Leonard Eugene,
History of the Theory of Numbers
History of the Theory of Numbers (3 volumes),
reprints: Carnegie Institute of Washington, Washington, 1932; Chelsea,
New York, 1952, 1966
Euler, Leonhard, Elements of Algebra, Tarquin Press, 2007
Fine, Henry Burchard (1858–1928), The
Number System of Algebra
Treated Theoretically and Historically, Leach, Shewell & Sanborn,
Karpinski, Louis Charles (1878–1956), The History of Arithmetic,
Rand McNally, Chicago, 1925; reprint: Russell & Russell, New York,
Number Theory and Its History, McGraw–Hill, New York,
Number Theory: An Approach through History, Birkhauser,
Boston, 1984; reviewed:
Mathematical Reviews 85c:01004
Look up arithmetic in Wiktionary, the free dictionary.
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MathWorld article about arithmetic
The New Student's Reference Work/
The Great Calculation According to the Indians, of Maximus Planudes
– an early Western work on arithmetic at Convergence
Weyde, P. H. Vander (1879). "Arithmetic". The American
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