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:
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:
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. Each time, stop one layer later and simplify. Watch what happens to the fractions that appear.
The fractions form a little staircase:
Notice that the bottom row is the same as the top row, just shifted one step to the right. That is why this feels a bit like a stairs puzzle.
Suppose one stopping point gives the fraction a/b. The next stopping point adds one more layer, so you get
1 + 1/(a/b)
Can you simplify that and explain why the next fraction becomes
(a + b)/a
Use that to justify why the numerators and denominators march through consecutive Fibonacci numbers.
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Reveal explanation
If one stopping point is a/b, then the next one is
1 + 1/(a/b) = 1 + b/a = (a + b)/a
The old numerator a becomes the new denominator.
The new numerator is a + b, which is the sum of the previous numerator and denominator.
So if you start with 1/1, then each new step is formed by “add the two numbers, then shift”. That is exactly the Fibonacci-style pattern, which is why the convergents become 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, …
Use the rule
new value = 1 + 1/(old value)
Start with 13/8. What fraction comes next?
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Reveal answer
1 + 1/(13/8) = 1 + 8/13 = 21/13.
Again, the numerator and denominator are consecutive Fibonacci numbers.
Compare these two fractions with √2 ≈ 1.414214:
7/5 and 10/7
Which one is closer?
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Reveal answer
7/5 = 1.4, which is about 0.014214 away.
10/7 ≈ 1.428571, which is about 0.014357 away.
So 7/5 is slightly better. Continued fractions often produce unusually good approximations with small denominators.
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:
Its approximations begin:
1, 3/2, 7/5, 17/12, 41/29, …
These fractions are exceptionally good approximations to √2.
This uses algebra that younger learners can skip. 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.
Lesson 5: Infinite Fractions
Download the Lesson 5 printable handout to explore continued fractions, recursive structure and increasingly accurate approximations to irrational numbers.
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