a continued fraction is a special complex fraction. if a0, a1, a2,...an,... are all integers, they will be called infinite continued fractions and finite continued fractions respectively. it 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 continued fractions in the above forms 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 fractions can also be taken as polynomials with x as the variable. in modern computational mathematics, it is often related to certain differential equations and difference equations, and to the application of function construction related to certain recursive relationships.

the continued fraction representation avoids these two problems of real number representation. let us consider how to describe a number such as 415/93, which is approximately 4.4624. approximately 4, but actually a little more than 4, about 4 + 1/2. but the 2 in the denominator is inaccurate; a more accurate denominator would be a little more than 2, about 2 + 1/6, so 415/93 is approximately 4 + 1/(2 + 1/6). but the 6 in the denominator is inaccurate; it would be more accurate to have a little more than 6 in the denominator, which is actually 6+1/7. so 415/93 is actually 4+1/(2+1/(6+1/7)). that's how accurate it is.

the abbreviated notation [4; 2, 6, 7] can be obtained by removing the redundant part in the expression 4 + 1/(2 + 1/(6 + 1/7)).