Content Summary: 600 words, 6 min read
And now, an unprovoked foray into number theory!
Simple Continued Fractions (SCFs)
Have you run into simple continued fractions in your mathematical adventures? They look like this:
Let represent the coefficients and . If you fix you can uniquely represent with . For example:
Let us call the leading coefficients of n. Here we have represented the rational with four coefficients. It turns out that every rational number can be expressed with a finite number of leading coefficients.
Life gets interesting when you look at the leading coefficients of irrational numbers. Consider the following:
First note that these irrational numbers have an infinite number of leading coefficients.
What do you notice about ? It repeats, of course! What is the repeating sequence for ? The sequence .
How about ? Well, after the first two digits, we notice an interesting pattern then then . The value of this triplet is non-periodic, but easy enough to compute. The situation looks even more bleak when you consider the …
Thus (golden ratio) and feature repeating coefficients, but and (Euler’s number) do not. What differentiates these groups?
Of these numbers, only the transcendental numbers fail to exhibit a period. Can this pattern be generalized? Probably. 🙂 There exists an unproved conjecture in number theory, that all infinite, non-periodic leading coefficients with bounded terms are transcendental.
Real Approximation As Coefficient Trimming
Stare the digits of . Can you come up with a fraction that approximates it?
Perhaps you have picked up the trick that is surprisingly close:
But could you come up with from first principles? More to the point, could you construct a fraction that comes yet closer to ‘s position on the number line?
Decomposing these numbers into continued fractions should betray the answer:
We can approximate any irrational number by truncating . Want a more accurate approximation of ? Keep more digits:
I’ll note in passing that this style of approximation resembles how algorithms approximate the frequency of signals by discarding smaller eigenvalues.
Much ink has been spilled on the number . For example, does it contain roughly equal frequencies of 3s and 7s? When you generalize this question to any base (not just base 10), the question becomes whether is a normal number. Most mathematicians suspect the answer is Yes, but this remains pure conjecture to-date.
Let’s return to the digits of . Here is a graph of the first two hundred:
Do you see a pattern? I don’t.
Let’s zoom out. This encyclopedia displays the first 20,000 coefficients of :
So affords no obvious pattern. Is there another way to generate the digits of such that a pattern emerges?
Let quadratic continued fraction represent a number expressed as:
Set . Here only is allowed to vary. Astonishingly, the following fact is true:
Thus, continued fractions allow us to make sense out of important transcendental numbers like .
I’ll close with a quote:
Continued fractions are, in some ways, more “mathematically natural” representations of a real number than other representations such as decimal representations.