Definition of Continued Fraction Expansion for Reals

Similarly to before, we define a continued fraction expansion of a real number to be just like a rational's continued fraction. However, it cannot look like this: $$[a_1,a_2,\cdots,a_n]=a_1+\dfrac{1}{a_2+\dfrac{1}{a_3+\dfrac{1}{\ddots+\dfrac{1}{a_n}}}}.$$

Right, if it looked like that, then all we would have to do is compute starting from the bottom, and then after each step, we would still have a rational number. All we are doing is taking the reciprocal and adding an integer. Both of those operations will never change a rational number to an irrational number. Since we started with a rational number, namely $a_n$, we will get a rational number in the end.

So, we have to somehow prevent this from happening. If $a_n$ exists, then we can't do anything to make it irrational, because all the other $a_i$ have to be integers. So, how about if $a_n$ doesn't exist? That would mean the continued fraction would not end , like so: $$[a_1,a_2,a_3,\cdots]=a_1+\dfrac{1}{a_2+\dfrac{1}{a_3+\ddots}}.$$

Let's look at some examples

$\sqrt{2}=[1,2,2,2,\cdots]$ $\sqrt{3}=[1,1,2,1,2,1,2,\cdots]$ $e=[2,1,2,1,1,4,1,1,6,1,1,\cdots]$ $\pi=[3,7,15,1,292,1,1,1,2\cdots]$

For $e$, it has a really nice pattern, except for the $2$ at the beginning. However, note that if we allowed zeros in continued fractions, we would have $$e=[2,1,2,1,1,4,1,1,6,1,1,\cdots]=[1,0,1,1,2,1,1,4,1,1,\cdots],$$ which is extremely nice, because it follows our pattern!

However, the continued fraction for $\pi$ seems random. There are no apparent pattern in the terms. However, there are some large values in the fraction. Like we did before, let's try to truncate it and compute the values.

Taking $[3,7]$ gives $\frac{22}{7}$, the infamous approximation for $\pi$ (some people think that $\pi=\frac{22}{7}$)! However, there is an even larger term, $292$. If we truncate before that, we have $[3,7,15,1]=\frac{355}{113}$, another well known approximation.