Divide each term of the Fibonacci sequence by the one before it and you get the fractions 2/1, 3/2, 5/3, 8/5, 13/8…, all of them closing in on the golden ratio φ ≈ 1.618034. They approach it by alternating: one lands above φ and the next below it. Subtracting two consecutive ratios shows this with an exact fraction.
The ratios of the Fibonacci sequence
The sequence starts 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and each term is the sum of the two before it. The MacTutor biography of Fibonacci places his rabbit problem, which generates these numbers, in the Liber abaci, published in 1202. The quotient of one term by the previous one is a ratio, and written as a fraction it keeps its exact value.
| Ratio | Decimal value (≈) | Ratio − φ (≈) |
|---|---|---|
| 2/1 | 2.000000 | +0.381966 |
| 3/2 | 1.500000 | −0.118034 |
| 5/3 | 1.666667 | +0.048633 |
| 8/5 | 1.600000 | −0.018034 |
| 13/8 | 1.625000 | +0.006966 |
| 21/13 | 1.615385 | −0.002649 |
| 34/21 | 1.619048 | +0.001014 |
| 55/34 | 1.617647 | −0.000387 |
The sign in the last column flips on every row, and its size shrinks at each step to a little more than a third of the previous one (≈ 0.38 of it). The decimals are approximations rounded to six places. Since 21/13 is 1.615384… with the block 615384 repeating, writing 21/13 = 1.615385 would be a false equality. The guide on exact and repeating decimals explains that difference.
Why the ratios alternate: subtracting two consecutive ratios
The alternation comes from subtracting consecutive ratios over a common denominator. One pair is enough to see the mechanism.
Worked example: 8/5 − 13/8
Subtract two consecutive Fibonacci ratios and decide which of the two is larger.
- Find the common denominator. The least common multiple (LCM) of 5 and 8 is 40.
- Convert. 8/5 = 64/40 and 13/8 = 65/40.
- Subtract the numerators. 64 − 65 = −1, so the difference is −1/40.
- Read the sign. It is negative, so the second ratio is the larger one.
The difference is −1/40 = −0.025.
Check. 1.6 − 1.625 = −0.025.
The two numerators differ by one, and that is no accident. Writing F(n) for term n of the sequence, Cassini’s identity says F(n+1)² − F(n) · F(n+2) = (−1)ⁿ. From that,
F(n+1)/F(n) − F(n+2)/F(n+1) = (−1)ⁿ / (F(n) · F(n+1)).
In the example, n = 5, F(5) = 5 and F(6) = 8, so the denominator is 40 and the sign is negative. For 21/13 and 34/21 the result is −1/273, and for 34/21 and 55/34 it is +1/714. The numerator is always ±1 and its sign flips at every step, which is why the ratios jump from one side of φ to the other. As the denominator grows, the jumps shrink and φ is trapped between two neighboring ratios.
Where φ = (1 + √5)/2 comes from
Each ratio comes from the previous one: F(n+2)/F(n+1) = 1 + 1/(F(n+1)/F(n)). If the ratios approach a number L, then L = 1 + 1/L, that is, L² = L + 1. That equation has one positive solution, φ = (1 + √5)/2 ≈ 1.618034, and it is the value the ratios aim at. The same equality φ = 1 + 1/φ explains why φ is irrational with a very regular expansion, as the next section shows.
The continued fraction of φ is made of ones
If you substitute φ into its own denominator again and again, you get φ = [1; 1, 1, 1, …], with an integer part of 1 and an endless chain of ones. The convergents of this continued fraction are 1, 2, 3/2 and the other ratios in the table, exactly the Fibonacci ratios. You can expand any of them in the continued fractions tool.
English Wikipedia describes this as the simplest continued fraction and attributes its slow convergence to the small terms. That makes φ the number hardest to approximate by rationals. The contrast with π comes down to the size of the coefficients: in π = [3; 7, 15, 1, 292, …] the coefficient 292 makes 355/113 an excellent approximation, as the story of the fractions that approximate π tells. In φ every coefficient is 1, and no ratio gives a jump like that. To find the simple fraction closest to a given decimal, use the approximate a decimal as a fraction calculator.
No ratio equals φ exactly, because φ is irrational and every ratio is a quotient of integers. Approximations that close in from opposite sides of a limit also appear in the series 1/2 + 1/4 + 1/8 + …, although there all the partial sums stay below the limit.
Practice
- Subtract 3/2 − 5/3 and say which is larger. Then check the result with Cassini’s formula.
- Calculate 89/55 − 144/89 without converting to decimals.
- Use the compare fractions calculator with 144/89 and 233/144. Which one lies above φ?
Show the answers
- 3/2 − 5/3 = 9/6 − 10/6 = −1/6, so 5/3 is larger. With n = 3, F(3) = 2 and F(4) = 3, the formula gives (−1)³/(2 · 3) = −1/6.
- 89² = 7921 and 55 · 144 = 7920, so the numerator is 1 and the denominator is 55 · 89 = 4895. The difference is 1/4895.
- 144/89 ≈ 1.617978 lies below φ and 233/144 ≈ 1.618056 lies above. That is the pattern in the table, with the subtraction 144/89 − 233/144 = −1/12816.
Sources
- English Wikipedia, article Golden ratio: the value of φ, the Fibonacci ratios alternately lower and higher than φ, φ = 1 + 1/φ, and the continued fraction of ones as the simplest, with slow convergence (accessed October 17, 2026).
- J. J. O’Connor and E. F. Robertson, Fibonacci, MacTutor History of Mathematics, University of St Andrews: the 1202 Liber abaci and the rabbit problem (accessed October 17, 2026).