How do you do continued fraction expansion?

To calculate a continued fraction representation of a number r, write down the integer part (technically the floor) of r. Subtract this integer part from r. If the difference is 0, stop; otherwise find the reciprocal of the difference and repeat.

Do continued fractions converge?

If the sequence of convergents {xn} approaches a limit the continued fraction is convergent and has a definite value. It may diverge by oscillation (for example, the odd and even convergents may approach two different limits), or it may produce an infinite number of zero denominators Bn.

What do you mean by continued fraction?

: a fraction whose numerator is an integer and whose denominator is an integer plus a fraction whose numerator is an integer and whose denominator is an integer plus a fraction and so on.

Which is a rational number with a continued fraction expansion?

2.1.1 Rational Numbers Theorem 2.1. Every rational number has a simple continued fraction expansion which is nite and every \\fnite simple continued fraction expansion is a rational number. Proof. Suppose we start with a rational number, then Euclid’s algorithm terminates in \\fnitely many steps.

When does the continued fraction of a number continue?

In particular, it must terminate and produce a finite continued fraction representation of the number. 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.

Which is a truncation of a continued fraction?

The rational value whose [finite] continued fraction expansion is a truncation of the continued fraction expansion of a given number is called a convergent of that number. (As stated above, proper truncation of a continued fraction entails adding the last two terms whenever the last one is unity.)

Why is an infinite continued fraction useful for irrational numbers?

An infinite continued fraction representation for an irrational number is useful because its initial segments provide rational approximations to the number. These rational numbers are called the convergents of the continued fraction.