A simple or regular continued fraction is a continued fraction with numerators all equal to one, and denominators built from a sequence ${\displaystyle \{a_{i}\}}$ of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fraction like

or an infinite continued fraction like

Typically, such a continued fraction is obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In the finite case, the iteration/recursion is stopped after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers ${\displaystyle a_{i}}$ are called the coefficients or terms of the continued fraction.

Simple continued fractions have a number of remarkable properties related to the Euclidean algorithm for integers or real numbers. Every rational number ⁠${\displaystyle p}$/${\displaystyle q}$⁠ has two closely related expressions as a finite continued fraction, whose coefficients a_i can be determined by applying the Euclidean algorithm to ${\displaystyle (p,q)}$. The numerical value of an infinite continued fraction is irrational; it is defined from its infinite sequence of integers as the limit of a sequence of values for finite continued fractions. Each finite continued fraction of the sequence is obtained by using a finite prefix of the infinite continued fraction's defining sequence of integers. Moreover, every irrational number ${\displaystyle \alpha }$ is the value of a unique infinite regular continued fraction, whose coefficients can be found using the non-terminating version of the Euclidean algorithm applied to the incommensurable values ${\displaystyle \alpha }$ and 1. This way of expressing real numbers (rational and irrational) is called their continued fraction representation.

Consider, for example, the rational number ⁠415/93⁠, which is around 4.4624. As a first approximation, start with 4, which is the integer part; ⁠415/93⁠ = 4 + ⁠43/93⁠. The fractional part is the reciprocal of ⁠93/43⁠ which is about 2.1628. Use the integer part, 2, as an approximation for the reciprocal to obtain a second approximation of 4 + ⁠1/2⁠ = 4.5. Now, ⁠93/43⁠ = 2 + ⁠7/43⁠; the remaining fractional part, ⁠7/43⁠, is the reciprocal of ⁠43/7⁠, and ⁠43/7⁠ is around 6.1429. Use 6 as an approximation for this to obtain 2 + ⁠1/6⁠ as an approximation for ⁠93/43⁠ and 4 + ⁠1/2 + ⁠1/6⁠⁠, about 4.4615, as the third approximation. Further, ⁠43/7⁠ = 6 + ⁠1/7⁠. Finally, the fractional part, ⁠1/7⁠, is the reciprocal of 7, so its approximation in this scheme, 7, is exact (⁠7/1⁠ = 7 + ⁠0/1⁠) and produces the exact expression $${\displaystyle 4+{\cfrac {1}{2+{\cfrac {1}{6+{\cfrac {1}{7}}}}}}}$$ for ⁠415/93⁠.

That expression is called the continued fraction representation of ⁠415/93⁠. This can be represented by the abbreviated notation ⁠415/93⁠ = [4; 2, 6, 7]. It is customary to place a semicolon after the first number to indicate that it is the whole part. Some older textbooks use all commas in the (n + 1)-tuple, for example, [4, 2, 6, 7].

If the starting number is rational, then this process exactly parallels the Euclidean algorithm applied to the numerator and denominator of the number. In particular, it must terminate and produce a finite continued fraction representation of the number. The sequence of integers that occur in this representation is the sequence of successive quotients computed by the Euclidean algorithm. If the starting number is irrational, then the process continues indefinitely. This produces a sequence of approximations, all of which are rational numbers, and these converge to the starting number as a limit. This is the (infinite) continued fraction representation of the number. Examples of continued fraction representations of irrational numbers are:

Continued fractions are, in some ways, more "mathematically natural" representations of a real number than other representations such as decimal representations, and they have several desirable properties: