- See Timeline of mathematics for a timeline of events in mathematics. See list of mathematicians for a list of biographies of mathematicians.
- Also see The Nine Chapters on the Mathematical Art for information about the development of mathematics in China.
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.
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
The Lo Shu Square is one example of representative artwork yielding to mathematics. In this case the representation of patterns on a turtle's back is believed to have spawned the idea of the Magic square around 2800 BC.
Around the same time that the ancient China had developed the Lo Shu Square, ancient Egyptian mathematicians were applying the basic principles of mathematics that they knew to surveying. This knowledge culminated in their construction of several Egyptian pyramids and of the extraordinary Great Pyramid of Giza around 2600 BC. Interestingly, a closer analysis of the Great Pyramid's construct as well as that of many other ancient Egyptian structures yield some indications that the ancient Egyptians were much farther developed in their mathematical knowledge than historians have been led to believe solely by reading extant mathematical documents. See Great Pyramid of Giza and Moscow and Rhind Mathematical Papyri for details.
Similar statements hold for the peoples of the Indus Valley Civilization (c. 2600 BC), who are credited with the earliest known physical use of decimal fractions in an extraordinarily advanced system of ancient weights and measures. The prehistoric Indus Valley engineers applied their technological advances to ancient seafaring and even dentistry! As with the ancient Egyptians, there are other indications as well that the ancient Indus Valley peoples were even farther advanced than what is suggested by these already amazing accomplishments.
Whether or not the ancient Egyptians and Indus Valley peoples actually were, of course, we may never know for certain.
Disciplines
Historically, the major disciplines within mathematics arose out of the need to do calculations in commerce, to measure land and to predict astronomical events. These three needs can be roughly related to the broad subdivision of mathematics into the studies of structure, space and change.
The study of structure starts with numbers, firstly the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. The deeper properties of whole numbers are studied in number theory. The investigation of methods to solve equations leads to the field of abstract algebra, which, among other things, studies rings and fields, structures that generalize the properties possessed by the familiar numbers. The physically important concept of vector, generalized to vector spaces and studied in linear algebra, belongs to the two branches of structure and space.
The study of space originates with geometry, first the Euclidean geometry and trigonometry of familiar three-dimensional space, but later also included hyperbolic geometry which plays a central role in general relativity. Several long standing questions about ruler-and-compass constructions were finally settled by Galois theory. The modern fields of differential geometry and algebraic geometry generalize geometry in different directions: differential geometry emphasizes the concepts of coordinate system, smoothness and direction, while in algebraic geometry geometrical objects are described as solution sets of polynomial equations. Group theory investigates the concept of symmetry abstractly and provides a link between the studies of space and structure. Topology connects the study of space and the study of change by focusing on the concept of continuity.
Understanding and describing change in measurable quantities is the common theme of the natural sciences, and calculus was developed as a most useful tool for doing just that. The central concept used to describe a changing variable is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods to solve these are studied in the field of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. For several reasons, it is convenient to generalise to the complex numbers which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions, laying the groundwork for quantum mechanics among many other things. Many phenomena in nature can be described by dynamical systems and chaos theory deals with the fact that many of these systems exhibit unpredictable yet deterministic behavior.
In order to clarify and investigate the foundations of mathematics, the fields of set theory, mathematical logic and model theory were developed.
When computers were first conceived, several essential theoretical concepts were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, information theory and algorithmic information theory. Many of these questions are now investigated in theoretical computer science.
Discrete mathematics is the common name for those fields of mathematics useful in computer science.
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 
return only one value (division by zero is not defined).
- X + 1 = 0 requires the existence of negative numbers such as - 1 for its solution. The word negative was originally used by those who opposed the introduction of such numbers.
requires the existence of fractional numbers for its solution. If we allow the solution of all equations of the form 
then we get the rational numbers (m and n are both integers).
has no rational solution. Mathematicians responded by introducing radicals and real numbers, which allowed many polynomial equations to be solved.
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., X * X + 1 = 0) 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 foreign 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.
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), the Zero is lucidly explained and was rendered into 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.
Note however that though the concept of 'Zero' is documented as a contribution of ancient Indian thought, it is recognizably ludicrous for us to suppose that ancient Egyptian mathematics could have become as advanced as it was (see also Moscow and Rhind Mathematical Papyri and golden ratio [see Corinna Rossi, Architecture and Mathematics in Ancient Egypt, Cambridge University Press, 2004, pp. 23-56]) without such an idea of "nothingness." See next paragraph below.
Miscellaneous historical notes
As early as 2450 BC ancient Egyptian mathematicians derived the earliest known systematic method for the approximative calculation of the circle on the basis of the "sacred" 3-4-5 triangle. Additionally, the Rhind Mathematical Papyrus, dating to 1650 BC, preserves the first known aproximate value of π (at 3.16) as well as the earliest known attempt at squaring the circle. (The Rhind Mathematical Papyrus, written by the scribe Ahmes, is allegedly a copy of a lost scroll dating around 1850 BC.) Apparently, Egyptian mathematicians were millennia ahead of the rest of the world.
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 Northern and Southern Dynasties was the first person to calculate the value of Pi to seven decimal places.
External links
References
Boyer, C. B. (1991). A History of Mathematics. New York, John Wiley & Sons, Inc.