History of mathematics

See Timeline of mathematics for a timeline of events in mathematics. See list of mathematicians for a list of biographies of mathematicians.

The word "mathematics" comes from the Greek μάθημα (máthema) which means "science, knowledge, or learning"; μαθηματικός (mathematikós) means "fond of learning". Today, the term refers to a specific body of knowledge - the rigorous, deductive study of numbers, shapes, patterns, and change.

Every modern science depends on basic mathematics at the most fundamental level, including such operations as counting, addition and subtraction.

Contents

1 Fundamentals 2 Developing the concept of "number" through equations 3 Complex numbers 4 Indian contributions 5 Miscellaneous historical notes 6 Notes 7 External links 8 References

Fundamentals

In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. With counting established, then, the ideas of addition and subtraction naturally followed. See arithmetic.

Mathematics undoubtedly could not have developed out of simple counting and arithmetic, however, without writing. Perhaps prehistoric peoples first expressed quantity by drawing lines in the ground or scratching wood. (See Numeral system: History.)

Then mathematics developed further, out of simple writing, with the development of pigments, paint and other simple tools to record and communicate "quantity" among individuals and over periods of time. Pigments and paints served another purpose in the historical development of mathematics, though, in addition to communicating quantity. Pre-historic art and other early human inventions eventually led to

• Maps for expressing distance,

• Geometric shapes and other 3-dimensional figures for expressing basic form, and

• Graphs for expressing relations.

Developing the concept of "number" through equations

Many of the extentions of the concept of number can be seen as responses to equations that would otherwise have had no solution. In each of the extentions given below we start with an equation and then give the extention to the system which allows the equation to be solved. We start with the notion of natural numbers: positive integers and zero, although it should be noted that some ancient mathematics did not have the concept of zero. Also note that it was assumed that the normal algebraic operations <math>+\ -\ \times \ /<math> return only one value (division by zero is not defined).

• <math>X+1=0<math> requires the existence of negative numbers such as <math>-1<math> for its solution. The word negative was originally used by those who opposed the introduction of such numbers.

• <math>5 \times X=3<math> requires the existence of fractional numbers for its solution. If we allow the solution of all equations of the form <math>m \times X=n<math> then we get the rational numbers (m and n are both integers).

• <math>X \times X-2=0<math> has no rational solution. Mathematicians responded by introducing radicals and real numbers, which allowed many polynomial equations to be solved.

• <math>X \times X+1=0<math> is the equation that introduces us to the complex numbers, which are discussed below.

Complex numbers

Mathematics did not start with the concept of the complex numbers. It took many years and much discussion to get this far. Roughly speaking over time mathematicians have broadened the definition of number. Opinions differ as to how to treat the complex numbers philosophically.

Many people argued that it was just an imaginary construct to solve the cubic and shouldn't be considered 'real'. This is the origin of the terms imaginary and real. However it was found that a whole new beautiful world of complex numbers opened up if you did allow them. To represent a solution to the equation shown above (i.e., <math>X*X+1=0<math>) mathematicians chose the letter i. Even with all of these extensions of the naturals we are still not finished.

In order to construct the complex numbers we need only one more assumption: Any set of real numbers with an upper bound has a least upper bound. This fills in the real line with all of the irrational numbers that cannot be derived merely from the equations above. It is worth noting that this is an entirely different type of extension. This is because of the cardinality of complex numbers is greater than that of the rationals. Once this is done all polynomial equations can be solved (although this can be done in smaller fields than the complex numbers).

Mathematicians today rarely view the development of the complex numbers in this way (the preferred teaching method does not emphasize this stepwise development) but it demonstrates the tension in mathematics between the rigorous and the creative which is the main power behind much of modern mathematics.

Interestingly the independence of the continuum hypothesis can be seen as an inability to prove whether or not certain real numbers should be thought to exist.

Indian contributions

Between 1000 B.C. and 1000 A.D. various treatises on mathematics were authored by Indian mathematicians in which were set forth for the first time, the concept of zero, the techniques of algebra and algorithm, square root and cube root. Vedic mathematics, as it is referred to today, is a separate field of study and courses are offered even in non-Indian universities.

It was from this translation of an Indian text on mathematics that the Arab mathematicians perfected the decimal system and gave the world its current system of enumeration which we call the Hindu-Arabic numerals. The concept of zero seems to have been a contribution of ancient Indian thought. Every ancient Indian language has multiple words to refer to the concept of 'void' or 'nothing' - 'Shunya' in Sanskrit. In Brahma-Phuta-Siddhanta of Brahmagupta (7th century), zero is lucidly explained and was used in Arabic books around 770 AD. From these it was carried to Europe in the 8th century. However, the concept of zero is referred to as Shunya in the early Sanskrit texts of the 4th century BC and clearly explained in Pingala’s Sutra of the 2nd century. Aryabhata in 499 AD worked the value of Pi to the fourth decimal place as 3.1416.

Miscellaneous historical notes

The Maya calendar utilized a base-20 number system which included the 'number' zero (also see Maya numerals).

In China, Zu Chongzhi (祖冲之) of the Southern and Northern Dynasties was the first person to calculate the value of Pi to seven decimal places.

The Mesopotamian cuneiform tablet Plimpton 232 records a number of Pythagorean triplets (3,4,5) (5,12,13). ..., dated 1900 BC, possibly millennia before Pythagoras¹.

Notes

• Note 1: p.187, Paul Hoffman, The man who loved only numbers: the story of Paul Erdös and the search for mathematical truth. New York: Hyperion, 1998 ISBN 0-7868-6362-5

External links

• The MacTutor History of Mathematics Archive (http://www-groups.dcs.st-andrews.ac.uk/~history/) created by John J O'Connor and Edmund F Robertson, which contains biographies, timelines and historical articles about mathematical concepts.
• Earliest uses of various mathematical symbols (http://members.aol.com/jeff570/mathsym.html) by Jeff Miller

• Earliest known uses of some of the words of mathematics (http://members.aol.com/jeff570/mathword.html) by Jeff Miller

• History of Indian mathematics (http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/index.html) by Ian Pearce
• History of Mathematics, public domain article (http://www.jewishencyclopedia.com/view.jsp?artid=259&letter=M&search=mathematics)
• Important publications in the history of mathematics
• History of calculus (http://www.dean.usma.edu/math/people/rickey/hm/default.htm) by Fred Rickey

References

Boyer, C. B.: A history of mathematics 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7)de:Geschichte der Mathematik es:Historia de las matemáticas fr:Histoire des mathématiques it:Storia della matematica lt:Matematikos istorija pl:Historia matematyki pt:História da matemática sv:Matematikens historia uk:Історія математики zh:数学史