Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah

The Number System

This page contains concise explanations of commonly used types of numbers.

Natural Numbers

Natural numbers are the numbers 1, 2, 3, ... .


All natural numbers are integers, but also 0, -1, -2, ... .

Rational numbers

Rational numbers are fractions p/q where q is non-zero and p and q are both integer. For example, all integers are rational (pick q =1). Other examples of rational numbers are

3/2, -5/7, -1232321/74567467.

Real numbers

Without getting technical, real numbers are all numbers that can be written as a possibly never repeating decimal fraction. For example, all rational numbers are real. Their decimal representations do repeat. Decimal fractions whose representation do not repeat are irrational. Example of irrational real numbers include e = 2.71828... (the base of the natural logarithm), pi=3.14..., the ratio of a circle's perimeter to its diameter, and the square root of 2, which is 1.4142... .

Complex numbers

Complex numbers are numbers of the form

where a and b are real numbers and i is the square root of negative 1, i.e.,


The symbol i is called the imaginary unit.

Algebraic numbers

Algebraic numbers are complex numbers that can be obtained as the root of a polynomial with integer coefficients. For example, rational numbers are algebraic, and so are roots of integers.

Transcendental numbers

Transcendental numbers are real numbers that are not algebraic. For example e and pi are transcendental (and real).

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