Types of Numbers

Numbers can be classified into sets called number systems. For different methods of expressing numbers with symbols, see numeral systems.

Natural numbers

The most familiar numbers are the natural numbers, which to some mean the non-negative integers {0, 1, 2, …} and to others mean the positive integers {1, 2, 3, …}. The positive integers are referred to as the counting numbers. In the base ten number system, in almost universal use, the symbols for natural numbers are written using ten digits, 0 through 9. An implied place value system, one that increments in powers of ten, is used for numbers greater than nine. Thus, numbers greater than nine have numerals formed with two or more digits. The symbol for the set of all natural numbers is N.

Integers

Negative numbers are numbers which are less than zero. They are used to indicate a number that is opposite to the corresponding positive number (the absolute value), but equal in magnitude. For example, if a positive number is used to indicate distance to the right of a fixed point, a negative number would indicate distance to the left. Similarly, if a positive number indicates a bank deposit, then a negative number indicates a withdrawal. When the negative whole numbers are combined with the positive whole numbers and zero, one obtains the integers Z (German Zahl, plural Zahlen).

Rational numbers

A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero natural number denominator. The fraction m/n represents the quantity arrived at when a whole is divided into n equal parts, and m of those equal parts are chosen. Two different fractions may correspond to the same rational number; for example 1/2 and 2/4 are the same. If the absolute value of m is greater than n, the absolute value of the fraction is greater than one. Fractions can be positive, negative, or zero. The set of all fractions includes the integers, since every integer can be written as a fraction with denominator 1. The symbol for the rational numbers is a bold face Q. (for quotient).

Real numbers

Decimal numbers are another way in which numbers can be expressed. In the base ten number system, they are written as a string of digits, with a period (decimal point) (in, for example, the US and UK) or a comma (in, for example, continental Europe) to the right of the ones place. A rational number expressed as a decimal must terminate or repeat. An example, the decimal 1.25 terminates, 0.3333... (unending threes) repeats, both are rational. All repeating and terminating decimals can be written as fractions; 1.25 = 5/4 and 0.3333... = 1/3. Any decimal represents an infinite series. Unlike repeating and terminating decimals, non-repeating, non-terminating decimals cannot be written as fractions, and are called irrational numbers. These include 0.1010010001... as well as constants such as π (pi), and √2, the square root of 2.

The real numbers are made up of all numbers that can be expressed as decimals, both rational and irrational. The symbol for the real numbers is R. The real numbers are used to represent measurements, and correspond to the points on the number line. As measurements are only made to a certain level of precision, there is always some error margin when using real numbers to represent them. This is often dealt with by giving an appropriate number of significant figures.

The real numbers are uniquely characterized by their mathematical properties: they are the only complete ordered field. They are not, however, an algebraically closed field.

Complex numbers

Moving to a greater level of abstraction, the real numbers can be extended to the complex numbers C. This set of numbers arose, historically, from the question of whether a negative number can have a square root. From this problem, a new number was discovered: the square root of negative one. This number is denoted by i, a symbol assigned by Leonhard Euler. The complex numbers consist of all numbers of the form a + b i, where a and b are real numbers. When a is zero, then a + b i is called imaginary. Likewise, when b is zero, then a + b i is real, since there is no imaginary component. A complex number that has a and b as integers is called a Gaussian integer. The complex numbers are an algebraically closed field, meaning that every polynomial with complex coefficients can be factored into linear factors with complex coefficients. Complex numbers correspond to points on the complex plane.

Each of the number systems mentioned above is a subset of the next number system.

Infinitary extensions

Superreal, hyperreal and surreal numbers extend the real numbers by adding infinitesimally small numbers and infinitely large numbers. While real numbers may have infinitely long expansions to the right of the decimal point, these numbers allow for infinitely long expansions to the left. The number system which results depends on what base is used for the digits: any base is possible, but a system with the best mathematical properties is obtained when the base is a prime number. This leads to the p-adic numbers.

Other types

While the natural numbers or the real numbers suffice for most everyday purposes, mathematicians have discovered other sets of numbers with specialized uses. Some are subsets of the complex numbers. For example, algebraic numbers are the roots of polynomials with rational coefficients. Real numbers that are not algebraic are called transcendental numbers. Sets of numbers that are not subsets of the complex numbers include the quaternions H , invented by Sir William Rowan Hamilton, in which multiplication is not commutative, and the octonions, in which multiplication is not associative. Elements of function fields of finite characteristic behave in some ways like numbers and are often regarded as numbers by number theorists.