Estimate the square root of 110

Question

Solve in your head. Estimate \sqrt{110} to three decimal places. Aim: under 20 seconds.

Accepted Answer

Half-integer squares are easy via the ending-in-5 trick: 10.5^2 = 110.25 (from 105^2 = 11025). That's a much closer anchor than 100 or 121. Linearize downward from 110.25: \sqrt{110} \approx 10.5 - \frac{0.25}{2 	imes 10.5} = 10.5 - \frac{0.25}{21} \approx 10.5 - 0.0119 = 10.488. True value: 10.4881. Three decimals, no calculator. Takeaway. Double your set of anchors for free: every number ending in 5 has an instantly computable square, so between consecutive perfect squares there is always a half-integer anchor (10.5^2 = 110.25, 7.5^2 = 56.25, …). Closer anchor \Rightarrow smaller correctio…

Estimate the square root of 110

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