A number is a mathematical object used to count, label, and measure. In mathematics, the definition of number has been extended over the years to include such numbers as 0, negative numbers, rational numbers, irrational numbers, and complex numbers.
Mathematical operations are certain procedures that take one or more numbers as input and produce a number as output. Unary operations take a single input number and produce a single output number. For example, the successor operation adds 1 to an integer, thus the successor of 4 is 5. Binary operations take two input numbers and produce a single output number. Examples of binary operations include addition, subtraction, multiplication, division, and exponentiation. The study of numerical operations is called arithmetic.
A repeating or recurring decimal is a way of representing rational numbers in arithmetic. The decimal representation of a number is said to be repeating if it becomes periodic (repeating its values at regular intervals) and the infinitely-repeated digit is not zero. The decimal representation of ⅓ becomes periodic just after the decimal point, repeating the single-digit sequence "3" forever. A more complicated example is 3227/555, whose decimal becomes periodic after the second digit following the decimal point and then repeats the sequence "144" forever. At present, there is no universally-accepted notation or phrasing for repeating decimals.
If the repeated digit is a zero, the rational number is called a terminating decimal, since the number is said to "terminate" before these zeros. Instead of taking any note of the repeated zeroes, they are simply omitted. All terminating decimals can be written as a decimal fraction whose divisor is a power of 10 (1.585 = 1585/1000); they may also be written as a ratio of the form k/2n5m (1.585 = 317/2352). However, every terminating decimal also has a second representation as a repeating decimal. This is obtained by decreasing the final non-zero digit by one and appending an infinitely-repeating sequence of nines, a non-obvious phenomenon that many find puzzling. 1 = 0.999… and 1.585 = 1.584999… are two examples of this. (This type of repeating decimal can be obtained by long division if one uses a modified form of the usual division algorithm.)
In algebra, which is a broad division of mathematics, abstract algebra is a common name for the sub-area that studies algebraic structures in their own right. Such structures include groups, rings, fields, modules, vector spaces, and algebras. The specific term abstract algebra was coined at the beginning of the 20th century to distinguish this area from the other parts of algebra. The term modern algebra has also been used to denote abstract algebra.
Two mathematical subject areas that study the properties of algebraic structures viewed as a whole are universal algebra and category theory. Algebraic structures, together with the associated homomorphisms, form categories. Category theory is a powerful formalism for studying and comparing different algebraic structures.
Elementary arithmetic is the simplified portion of arithmetic which includes the operations of addition, subtraction, multiplication, and division.
Elementary arithmetic starts with the natural numbers and the written symbols (digits) which represent them. The process for combining a pair of these numbers with the four basic operations traditionally relies on memorized results for small values of numbers, including the contents of a multiplication table to assist with multiplication and division.