## To find a rational with low denominator near a given real, use continued fractions

### Quick description

Suppose that you are given a real number, to many significant figures, told that it is close to a rational with small denominator, and asked to find that rational. Can this be done efficiently (where "efficient" means that you can identify the rational in a time that depends polynomially on the number of digits of )? Yes it can, with the help of continued fractions.

### Prerequisites

Elementary number theory, continued fractions.

### Example 1

Suppose you are told that the number is very close to a fraction such that . How can you find this fraction? For numbers this small, trial and error would work fine. (For each less than , one could find the that makes closest to and wait until you hit it almost exactly.) But for larger numbers, trial and error would take a hopelessly long time.

There is a simple method for solving this computational problem. First, let us imagine that the number in question actually equals . Then we can express as a continued fraction, which takes a time that is at worst proportional to the number of digits of (assuming that and are of roughly comparable size). For example, if the number were (but we didn't know that) then we would write it as . But then we note that if our number is merely very close to , then we will write it as , where is very close to . At that point, we recognise that we are very close to the rational .

In our example, we see that at which point we recognise that the original fraction was very close to .

### General discussion

This method plays an important role in Shor's quantum algorithm for factorizing, since the procedure outputs a real number that is close to a rational and one needs an efficient way of finding the denominator of that rational.