A continued fraction is a special complex fraction. If a0, a1, a2, ..., an, ... are all integers, they are called infinite continued fractions and finite continued fractions respectively. They can be abbreviated as a0, a1, a2, ..., an, ... and a0, a1, a2, ..., an. Generally, a finite continued fraction represents a rational number, and an infinite continued fraction represents an irrational number. If a0, a1, a2, ..., an, ... are all real numbers, the above-mentioned continued fractions can be called infinite continued fractions and finite continued fractions respectively. For the calculation needs of modern mathematics, a0, a1, a2, ..., an, ... in the continued fraction can also be taken as a polynomial with x as the variable. In modern computational mathematics, it is often related to certain differential equations and difference equations, and is related to the application of function construction related to certain recursive relations.

Continued fraction representation avoids these two problems of real number representation. Let's consider how to describe a number such as 415/93, which is approximately 4.4624. It is approximately 4, but in fact it is a little more than 4, approximately 4 + 1/2. However, the 2 in the denominator is inaccurate; a more accurate denominator is a little more than 2, approximately 2 + 1/6, so 415/93 is approximately 4 + 1/(2 + 1/6). However, the 6 in the denominator is inaccurate; a more accurate denominator is a little more than 6, actually 6+1/7. So 415/93 is actually 4+1/(2+1/(6+1/7)). This is accurate.

Removing the redundancy in the expression 4 + 1/(2 + 1/(6 + 1/7)) yields the abbreviated notation [4; 2, 6, 7].