A continued fraction is a mathematical expression written as a fraction whose denominator contains a sum involving another fraction, which may itself be a simple or a continued fraction. If this iteration (repetitive process) terminates with a simple fraction, the result is a finite continued fraction; if it continues indefinitely, the result is an infinite continued fraction. Any rational number can be expressed as a finite continued fraction, and any irrational number can be expressed as an infinite continued fraction. The special case in which all numerators are equal to one is referred to as a simple continued fraction.

Different areas of mathematics use different terminology and notation for continued fractions. In number theory, the unqualified term continued fraction usually refers to simple continued fractions, whereas the general case is referred to as generalized continued fractions. In complex analysis and numerical analysis, the general case is usually referred to by the unqualified term continued fraction.

The numerators and denominators of continued fractions can be sequences ${\displaystyle \{a_{i}\},\{b_{i}\}}$ of constants or functions.

A continued fraction is an expression of the form

where the a_n (n > 0) are the partial numerators, the b_n are the partial denominators, and the leading term b₀ is called the integer part of the continued fraction.

The successive convergents of the continued fraction are formed by applying the fundamental recurrence formulas:

where A_n is the numerator and B_n is the denominator, called continuants, of the nth convergent. They are given by the three-term recurrence relation

with initial values