Contents 1 History 1.1 Etymology 2 Definitions of mathematics 2.1
3 Inspiration, pure and applied mathematics, and aesthetics 4 Notation, language, and rigor 5 Fields of mathematics 5.1 Foundations and philosophy 5.2 Pure mathematics 5.2.1 Quantity 5.2.2 Structure 5.2.3 Space 5.2.4 Change 5.3 Applied mathematics 5.3.1
6 Mathematical awards 7 See also 8 Notes 9 Footnotes 10 References 11 Further reading 12 External links History Main article: History of mathematics The history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, which is shared by many animals,[17] was probably that of numbers: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely quantity of their members. Greek mathematician
Mayan numerals As evidenced by tallies found on bone, in addition to recognizing how
to count physical objects, prehistoric peoples may have also
recognized how to count abstract quantities, like time – days,
seasons, years.[18]
Evidence for more complex mathematics does not appear until around
3000 BC, when the Babylonians and Egyptians began using
arithmetic, algebra and geometry for taxation and other financial
calculations, for building and construction, and for astronomy.[19]
The earliest uses of mathematics were in trading, land measurement,
painting and weaving patterns and the recording of time.
In Babylonian mathematics, elementary arithmetic (addition,
subtraction, multiplication and division) first appears in the
archaeological record.
Persian mathematician Al-Khwarizmi (c. 780 – c. 850), the inventor of algebra. During the Golden Age of Islam, especially during the 9th and
10th centuries, mathematics saw many important innovations
building on Greek mathematics: most of them include the contributions
from Persian mathematicians such as Al-Khwarismi,
Leonardo Fibonacci, the Italian mathematician who introduced the
Carl Friedrich Gauss, known as the prince of mathematicians The German mathematician
Leonhard Euler, who created and popularized much of the mathematical notation used today Most of the mathematical notation in use today was not invented until
the 16th century.[45] Before that, mathematics was written out in
words, limiting mathematical discovery.[46] Euler (1707–1783) was
responsible for many of the notations in use today. Modern notation
makes mathematics much easier for the professional, but beginners
often find it daunting. According to Barbara Oakley, this can be
attributed to the fact that mathematical ideas are both more abstract
and more encrypted than those of natural language.[47] Unlike natural
language, where people can often equate a word (such as cow) with the
physical object it corresponds to, mathematical symbols are abstract,
lacking any physical analog.[48] Mathematical symbols are also more
highly encrypted than regular words, meaning a single symbol can
encode a number of different operations or ideas.[49]
Mathematical language can be difficult to understand for beginners
because even common terms, such as or and only, have a more precise
meaning than they have in everyday speech, and other terms such as
open and field refer to specific mathematical ideas, not covered by
their laymen's meanings. Mathematical language also includes many
technical terms such as homeomorphism and integrable that have no
meaning outside of mathematics. Additionally, shorthand phrases such
as iff for "if and only if" belong to mathematical jargon. There is a
reason for special notation and technical vocabulary: mathematics
requires more precision than everyday speech. Mathematicians refer to
this precision of language and logic as "rigor".
An abacus, a simple calculating tool used since ancient times
p ⇒ q displaystyle pRightarrow q Mathematical logic
Set theory
Pure mathematics
Quantity
Main article: Arithmetic
The study of quantity starts with numbers, first the familiar natural
numbers and integers ("whole numbers") and arithmetical operations on
them, which are characterized in arithmetic. The deeper properties of
integers are studied in number theory, from which come such popular
results as Fermat's Last Theorem. The twin prime conjecture and
( 0 ) , 1 , 2 , 3 , … displaystyle (0),1,2,3,ldots … , − 2 , − 1 , 0 , 1 , 2 … displaystyle ldots ,-2,-1,0,1,2,ldots − 2 , 2 3 , 1.21 displaystyle -2, frac 2 3 ,1.21 − e , 2 , 3 , π displaystyle -e, sqrt 2 ,3,pi 2 , i , − 2 + 3 i , 2 e i 4 π 3 displaystyle 2,i,-2+3i,2e^ i frac 4pi 3 Natural numbers Integers Rational numbers Real numbers Complex numbers Structure
Main article: Algebra
Many mathematical objects, such as sets of numbers and functions,
exhibit internal structure as a consequence of operations or relations
that are defined on the set.
( 1 , 2 , 3 ) ( 1 , 3 , 2 ) ( 2 , 1 , 3 ) ( 2 , 3 , 1 ) ( 3 , 1 , 2 ) ( 3 , 2 , 1 ) displaystyle begin matrix (1,2,3)&(1,3,2)\(2,1,3)&(2,3,1)\(3,1,2)&(3,2,1)end matrix Combinatorics Number theory Group theory Graph theory Order theory Algebra Space
Main article: Geometry
The study of space originates with geometry – in particular,
Euclidean geometry, which combines space and numbers, and encompasses
the well-known Pythagorean theorem.
Geometry
Trigonometry
Differential geometry
Topology
Change
Main article: Calculus
Understanding and describing change is a common theme in the natural
sciences, and calculus was developed as a powerful tool to investigate
it. Functions arise here, as a central concept describing a changing
quantity. The rigorous study of real numbers and functions of a real
variable is known as real analysis, with complex analysis the
equivalent field for the complex numbers.
Calculus Vector calculus Differential equations Dynamical systems Chaos theory Complex analysis Applied mathematics
Main article: Applied mathematics
Game theory Fluid dynamics Numerical analysis Optimization Probability theory Statistics Cryptography Mathematical finance Mathematical physics Mathematical chemistry Mathematical biology Mathematical economics Control theory Mathematical awards
Arguably the most prestigious award in mathematics is the Fields
Medal,[58][59] established in 1936 and awarded every four years
(except around World War II) to as many as four individuals. The
Philosophy of mathematics
Lists of mathematics topics
Notes ^ No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art depends on the artist's imagination (see Euclid). ^ See false proof for simple examples of what can go wrong in a formal proof. ^ Like other mathematical sciences such as physics and computer science, statistics is an autonomous discipline rather than a branch of applied mathematics. Like research physicists and computer scientists, research statisticians are mathematical scientists. Many statisticians have a degree in mathematics, and some statisticians are also mathematicians. Footnotes ^ a b "mathematics, n.". Oxford English Dictionary. Oxford University
Press. 2012. Retrieved June 16, 2012. The science of space, number,
quantity, and arrangement, whose methods involve logical reasoning and
usually the use of symbolic notation, and which includes geometry,
arithmetic, algebra, and analysis.
^ Kneebone, G.T. (1963). Mathematical
References Courant, Richard and H. Robbins, What Is Mathematics? : An
Elementary Approach to Ideas and Methods, Oxford University Press,
USA; 2 edition (July 18, 1996). ISBN 0-19-510519-2.
du Sautoy, Marcus (25 June 2010). "Nicolas Bourbaki". A Brief History
of Mathematics.
Further reading Benson, Donald C., The Moment of Proof: Mathematical Epiphanies,
Oxford University Press, USA; New Ed edition (December 14, 2000).
ISBN 0-19-513919-4.
Boyer, Carl B., A History of Mathematics, Wiley; 2nd edition, revised
by Uta C. Merzbach, (March 6, 1991). ISBN 0-471-54397-7. – A
concise history of mathematics from the Concept of Number to
contemporary Mathematics.
Davis, Philip J. and Hersh, Reuben, The Mathematical Experience.
Mariner Books; Reprint edition (January 14, 1999).
ISBN 0-395-92968-7.
Gullberg, Jan, Mathematics – From the Birth of Numbers. W. W.
Norton & Company; 1st edition (October 1997).
ISBN 0-393-04002-X.
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Philosophy of mathematics
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