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 = 3.167... where s = ⌊√10⌉ (round √10 = 3; the square root of 9, the nearest perfect square)

√23 ≈ 5 + (23 - 25)/10 = 4.8 where s = ⌊√23⌉ (round √23 = 5; the square root of 25, the nearest perfect square)

√43,560 ≈ 209 + (43,560 - 43,681)/418 = 208.7105... where s = ⌊√43,560⌉ (round √43,560 = 209; the square root of 43,681, the nearest perfect square)

√163 ≈ 13 + (163 - 169)/26 = 12.769... where s = ⌊√163⌉ (round √163 = 13; the square root of 169, the nearest perfect square)

√79 ≈ 9 + (79 - 81)/18 = 8.8888... 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: An even faster calculation would be to set s equal to the previous result (also known as the Bakhshali method):
 * 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
 * 4) √10 ≈ 3.16233 + (10 - 3.16233 2) /6 = 3.162275
 * 5) √10 ≈ 3.162275 + (10 - 3.162275 2) /6 = 3.1622778
 * 1) √10 ≈ 3 + (10 - 9)/6 = 3.167
 * 2) √10 ≈ 3.167 + (10 - 3.167 2) /(2 * 3.167) = 3.16228
 * 3) √10 ≈ 3.16228 + (10 - 3.16228 2) /(2 * 3.16228) = 3.1622776601692
 * 4) √10 ≈ 3.1622776601692 + (10 - 3.1622776601692 2) /(2 * 3.1622776601692) = 3.16227766016837908

Root Generalization
n√x ≈ s + (x - s n) /n(sn-1) where s = ⌊n√x⌉ (round n√x; basically the square root of the nearest perfect power of n)