Dividing consecutive Fibonacci numbers gives values that approach the golden ratio more and more closely.

The longer version

Let the ratio of neighbouring terms settle to some value r. Because each term equals its predecessor plus the one before that, r must equal one plus one over r, and that equation rearranges into r squared equals r plus one. The positive root is exactly the golden ratio.

The approach is not one sided: successive ratios overshoot and undershoot, bracketing the target ever more tightly, so eight over five is too high while thirteen over eight is too low. Converting the golden ratio into a continued fraction produces an endless chain of ones, the simplest possible, which is why it resists close rational approximation and why plant spirals tend to favour it.

Where this entry sits

This section collects odd corners of mathematics, from stubborn prime numbers to counting systems older than modern schools. Expect famous constants, probability puzzles, shapes that break everyday intuition, and numbers far too large to picture. The full list sits on the Numbers and Math page, where every entry is listed in order.