Mathematics, study of relationships among quantities, magnitudes, and properties and of logical operations by which unknown quantities, magnitudes, and properties may be deduced. In the past, mathematics was regarded as the science of quantity, whether of magnitudes, as in geometry, or of numbers, as in arithmetic, or of the generalization of these two fields, as in algebra. Toward the middle of the 19th century, however, mathematics came to be regarded increasingly as the science of relations, or as the science that draws necessary conclusions. This latter view encompasses mathematical or symbolic logic, the science of using symbols to provide an exact theory of logical deduction and inference based on definitions, axioms, postulates, and rules for combining and transforming primitive elements into more complex relations and theorems.

This brief survey of the history of mathematics traces the evolution of mathematical ideas and concepts, beginning in prehistory. Indeed, mathematics is nearly as old as humanity itself; evidence of a sense of geometry and interest in geometric pattern has been found in the designs of prehistoric pottery and textiles and in cave paintings. Primitive counting systems were almost certainly based on using the fingers of one or both hands, as evidenced by the predominance of the numbers 5 and 10 as the bases for most number systems today.

Arithmetic, literally, the art of counting. The word, in its oldest Greek form, comes from the word arithmçtikç, which combines the ideas of two words in Greek, arithmos, meaning "number," and technç, referring to an art or skill. For the history of arithmetic and mathematics, see Mathematics.

The numbers used for counting are called the positive integers. They are generated by adding 1 to each number in an unending series, so that each number in the sequence is one more than its immediate predecessor. Different civilizations throughout history have developed different kinds of number systems. One of the most common is the one used in all modern cultures, in which objects are counted in groups of ten. Called the base 10, or decimal, system, it is the one that is used in this article.

In the base 10 system, integers are represented by digits expressing various powers of 10. For example, take the number 1534. Every digit in this number has its own place value, and the place values increase by another power of 10 as they move to the left. The first place value is a unit value (here, 4); the second place value is 10 (here, 3 × 10, or 30); the third place value is 10 × 10, or 100 (here, 5 × 100, or 500); and the fourth place value is 10 × 10 × 10, or 1000 (here, 1 × 1000, or 1000).

Algebra, branch of mathematics in which letters are used to represent basic arithmetic relations. As in arithmetic, the basic operations of algebra are addition, subtraction, multiplication, division, and the extraction of roots. Arithmetic, however, cannot generalize mathematical relations such as the Pythagorean theorem, which states that the sum of the squares of the sides of any right triangle is also a square. Arithmetic can only produce specific instances of these relations (for example, 3, 4, and 5, where 32 + 42 = 52). But algebra can make a purely general statement that fulfills the conditions of the theorem: a2 + b2 = c2. Any number multiplied by itself is termed squared and is indicated by a superscript number 2. For example, 3 × 3 is notated 32; similarly, a × a is equivalent to a2 (see Exponent; Power; Root).

Classical algebra, which is concerned with solving equations, uses symbols instead of specific numbers and uses arithmetic operations to establish ways of handling symbols (see Equation; Equations, Theory of). Modern algebra has evolved from classical algebra by increasing its attention to the structures within mathematics. Mathematicians consider modern algebra to be a set of objects with rules for connecting or relating them. As such, in its most general form, algebra may fairly be described as the language of mathematics.

Trigonometry, branch of mathematics that deals with the relationships between the sides and angles of triangles and with the properties and applications of the trigonometric functions of angles. The two branches of trigonometry are plane trigonometry, which deals with figures lying wholly in a single plane, and spherical trigonometry, which deals with triangles that are sections of the surface of a sphere.

The earliest applications of trigonometry were in the fields of navigation, surveying, and astronomy, in which the main problem generally was to determine an inaccessible distance, such as the distance between the earth and the moon, or of a distance that could not be measured directly, such as the distance across a large lake. Other applications of trigonometry are found in physics, chemistry, and almost all branches of engineering, particularly in the study of periodic phenomena, such as vibration studies of sound, a bridge, or a building, or the flow of alternating current.