Definition of Continued Fraction Expansion for Rationals

A continued fraction of a rational number is a fraction that looks like so: $$2+\dfrac{1}{3+\dfrac{1}{4+\dfrac{1}{5+\dfrac{1}{3+\dfrac{1}{\ddots+\dfrac{1}{5}}}}}}$$ or in general, $$a_1+\dfrac{1}{a_2+\dfrac{1}{a_3+\dfrac{1}{\ddots+\dfrac{1}{a_n}}}},$$ where $a_1,a_2,\cdots,a_n$ are positive integers.

I agree. This takes up a lot of vertical space, so we will be using the notation $[a_1,a_2,\cdots,a_n]$ to represent the previous continued fraction.

Additionally, sometimes we're lazy, and we do not want to compute the whole fraction. Suppose we have $[1,1,1,2,1,21,1]$. We can compute this, and we get $\frac{250}{159}\approx 1.57233$. However, if you're computing this by hand, it would be a pain to get past that 21. So, what if we just didn't compute it?

When we compute $[1,1,1,2,1]$, we have $\frac{11}{7}\approx 1.571428$. This is extremely close! When truncating a continued fraction, the resulting fractions are called convergents.

Note that when we truncate right before a large number, the denominator will not recieve that final large increase, and the resulting fraction will be a good approximation.

This of course inspires us to come up with a "magic" trick! Ask a friend to use their calculator and compute a fraction, and then give you the first 10 or so digits of the decimal expansion. Then, using your favorite method, compute the continued fraction. Then, truncate it right before a large number.

For example, we have $\frac{1234}{237}\approx 5.206751054$ (note that this is simplified). We compute that the continued fraction is $[5,4,1,5,7,1,20887,3,\cdots]$. We have a large number, just as expected!. So, we truncate the fraction right before it, to get $[5,4,1,5,7,1]$. Computing this fraction gives $\frac{1234}{237}$, which is what we wanted.

If your friend happens to be careless and picks a fraction that is not reduced, then you will actually reduce it for them! So if they claim that you're wrong, ask them to check if their fraction is reducible.

To find out how to calculate the continued fraction expansion, go here.