Instant Square Roots

This page will show you how to approximate square roots quickly.

Approximation
√x ≈ s + (x - s 2) /2s where s = ⌊√x⌉ (round √x; basically the square root of the nearest perfect square)

Examples
√10 ≈ 3 + (10 - 9)/6 where s = ⌊√10⌉ (round √10 = 3; the square root of 9, the nearest perfect square)

√23 ≈ 5 + (23 - 25)/10 where s = ⌊√23⌉ (round √23 = 5; the square root of 25, the nearest perfect square)

√43,560 ≈ 209 + (43,560 - 43,681)/418 where s = ⌊√43,560⌉ (round √43,560 = 209; the square root of 43,681, the nearest perfect square)

√163 ≈ 13 + (163 - 169)/26 where s = ⌊√163⌉ (round √163 = 13; the square root of 169, the nearest perfect square)

√79 ≈ 9 + (79 - 81)/18 where s = ⌊√79⌉ (round √79 = 9; the square root of 81, the nearest perfect square)

Square Root Method
This approximation can be expanded to get more accuracy:
 * 1) √10 ≈ 3 + (10 - 9)/6 = 3.167
 * 2) √10 ≈ 3.167 + (10 - 3.167 2) /6 = 3.162
 * 3) √10 ≈ 3.162 + (10 - 3.162 2) /6 = 3.16233