Question

The Bakhshali Manuscript shows that another method for calculating nonperfect squares was being used in India by 400 CE. Use this method to find the approximate value of .

The nearest perfect square that is less than 22 is
, whose square root is
.
Add the square root from step 1 to to get 4.75.
Calculate the quantity one-half times the square of divided by the value found in step 2, or 4.75.

Subtract the value found in step 3 from the value found in step 2, or 4.75.
The approximate value of is
.

Answer 1

The nearest perfect square less than 22 is 16, whose square root is 4.

22 − 16 = 6, and 2·4 = 8, so 6/8 = 0.75. Add this to 4 to get 4.75.

Calculate one-half times the square of 6/8 divided by 4.75:
(1/2)·(6/8)^2 / 4.75 = (1/2)·0.5625 / 4.75 = 0.28125 / 4.75 = 0.0592105263.

Subtract this from 4.75: 4.75 − 0.0592105263 = 4.6907894737.

The approximate value of sqrt(22) is 4.6907894737 (≈ 4.69079).