Estimating a square root

For a big number, you can close in on its square root from both sides, using squares you already know.

Take √1936. Step 1: 1936 is between 1600 = 40² and 2500 = 50², so the root is between 40 and 50.

Step 2: 1936 ends in 6, and squares ending in 6 come from numbers ending in 4 or 6. So the root is 44 or 46.

Step 3: test the middle, 45². Write it as (40 + 5)(40 + 5) = 40² + 2 × 40 × 5 + 5² = 1600 + 400 + 25 = 2025. (A shortcut for numbers ending in 5: multiply the front digit by the next number, 4 × 5 = 20, then write 25 after it: 2025.)

Step 4: 2025 > 1936, so the root is below 45: 40 < √1936 < 45. Of 44 and 46, only 44 fits. Check: 44² = 1936.

The same squeezing works when a number is not a perfect square. Aribam and Bijou take turns calling out a number for the other to find the root of: 25 gives 5, 81 gives 9. Then Aribam says 250, and Bijou is stuck: 250 isn't a perfect square. Can he at least get close?

He knows 100 < 250 < 400, so 10 < √250 < 20: a start, but wide. Then 15² = 225 and 16² = 256, so 15 < √250 < 16. 256 is much nearer to 250 than 225 is, so √250 is about 16, and a little less than 16.

Akhil has a square piece of cloth of area 125 cm² and wants a square handkerchief of side 15 cm. He can't: 15² = 225, far more than 125. The nearest squares to 125 are 11² = 121 and 12² = 144, and 144 is too big. So the largest handkerchief with a whole-number side is 11 cm.