A number is an abstract object, tokens of which are symbols used in counting and measuring. A symbol which represents a number is called a numeral, but in common usage the word number is used for both the abstract object and the symbol. In addition to their use in counting and measuring, numerals are often used for labels (telephone numbers), for ordering (serial numbers), and for codes (ISBNs). In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers. As a result, there is no one encompassing definition of number and the concept of number is open for further development.

Certain procedures which input one or more numbers and output a number are called numerical operations. Unary operations input a single number and output a single number. For example, the successor operation adds one to an integer: the successor of 4 is 5. More common are binary operations which input two numbers and output a single number. Examples of binary operations include addition, subtraction, multiplication, division, and exponentiation. The study of numerical operations is called arithmetic.

The branch of mathematics that studies structures of number systems such as groups, rings and fields is called abstract algebra.

Types of numbers

Numbers can be classified into sets, called number systems. (For different methods of expressing numbers with symbols, such as the Roman numerals, see numeral systems.)

Natural numbers

The most familiar numbers are the natural numbers or counting numbers: one, two, three, ... .

In the base ten number system, in almost universal use today for arithmetic operations, the symbols for natural numbers are written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In this base ten system, the rightmost digit of a natural number has a place value of one, and every other digit has a place value ten times that of the place value of the digit to its right. The symbol for the set of all natural numbers is N, also written $\backslash mathbb\{N\}$.

In set theory, which is capable of acting as an axiomatic foundation for modern mathematics, natural numbers can be represented by classes of equivalent sets. For instance, the number 3 can be represented as the class of all sets that have exactly three elements. Alternatively, in Peano Arithmetic, the number 3 is represented as sss0, where s is the "successor" function. Many different representations are possible; all that is needed to formally represent 3 is to inscribe a certain symbol or pattern of symbols 3 times.