Decimals are one way to write irrational numbers. Continued fractions are another — and some irrational numbers become astonishingly simple.
Fractions inside fractions
A continued fraction looks like this:
1 + 1/(1 + 1/(1 + 1/(1 + …)))
At first it looks impossibly complicated. But notice that after the first “1 + 1/”, the whole expression starts again.
The golden ratio returns
Call the infinite expression x. Because the tail is another copy of x:
x = 1 + 1/x
We have seen that equation before. Its positive solution is φ. Therefore:
φ = 1 + 1/(1 + 1/(1 + 1/(1 + …)))
When an infinite expression contains a complete copy of itself, we can name the whole expression and solve an ordinary algebraic equation.
Stop early and get fractions
Cut the golden continued fraction off after a few layers:
1 = 1/1
1 + 1/1 = 2/1
1 + 1/(1 + 1/1) = 3/2
next: 5/3
then: 8/5
then: 13/8
The Fibonacci numbers have appeared yet again.
After 13/8, what is the next approximation?
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Reveal answer
1 + 8/13 = 21/13. The next approximation is 21/13.
√2 has its own pattern
The square root of 2 has the continued fraction:
√2 = 1 + 1/(2 + 1/(2 + 1/(2 + …)))
Its approximations begin:
1, 3/2, 7/5, 17/12, 41/29, …
These fractions are exceptionally good approximations to √2.
Let y = 2 + 1/(2 + 1/(2 + …)). Find y, then use it to recover √2 = 1 + 1/y.
Reveal answer
Since y = 2 + 1/y, we get y² = 2y + 1, so y = 1 + √2 (taking the positive solution).
Then 1 + 1/y = 1 + 1/(1 + √2) = √2.
What have we learned?
- Continued fractions are fractions nested inside fractions.
- The golden ratio has the simplest possible infinite continued fraction.
- Truncating it produces ratios of consecutive Fibonacci numbers.
- √2 has the repeating pattern [1; 2, 2, 2, …].
For the final lesson, we leave the main thread and explore several other numbers with strange personalities.