实数，是有理数和无理数的总称，前者如0、-4、81/7；后者如√2、π等。实数可以直观地看作小数（有限或无限的），它们能把数轴「填满」。但仅仅以枚举的方式不能描述实数的全体。实数和虚数共同构成复数。

根据日常经验，有理数集在数轴上似乎是「稠密」的，于是古人一直认为用有理数即能满足测量上的实际需要。以边长为1公分的正方形为例，其对角线有多长？在规定的精度下（比如误差小于0.001公分），总可以用有理数来表示足够精确的测量结果（比如1.414公分）。但是，古希腊毕达哥拉斯学派的数学家发现，只使用有理数无法完全精确地表示这条对角线的长度，这彻底地打击了他们的数学理念；他们原以为：

In mathematics, a real number is a value that represents a quantity along a continuous line. The adjective real in this context was introduced in the 17th century by Descartes, who distinguished between real and imaginary roots of polynomials.

The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2 (1.41421356…, the square root of 2, an irrational algebraic number). Included within the irrationals are the transcendental numbers, such as π (3.14159265…). Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one. The real line can be thought of as a part of the complex plane, and complex numbers include real numbers.