Square Root of 21

The square root of 21 is about 4.5826. Square roots reverse squaring and are essential in geometry and the quadratic formula.

√21 is irrational — 21 sits between the squares 16 and 25, and no whole number between their roots can multiply by itself to give 21, so the decimal never ends nor repeats. The nearer fence sits 4 away, against 5 on the other side. Rounded to two places, 4.58 misses the true value by less than one hundredth. Slanted right-triangle sides and the quadratic formula both land on roots of exactly this kind. No fraction of integers nails it either: if a ratio a/b squared gave 21, then 21 would itself be a perfect square — which the ladder above shows it is not. That property has a name, irrationality, and it is a fact about the number, not a limit of this table.