Estimate the square root of 20

Question

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

Accepted Answer

Use the ending-in-5 trick: 4.5^2 = 20.25 (from 45^2 = 2025). This anchor is a quarter above the target, much closer than 16 or 25. Linearize downward from 20.25: \sqrt{20} \approx 4.5 - \frac{0.25}{2 	imes 4.5} = 4.5 - \frac{0.25}{9} \approx 4.5 - 0.0278 = 4.472. True value: 4.4721. Three decimals, no calculator. Takeaway. The gap 0.25 between 20 and 4.5^2 = 20.25 divided by 2 	imes 4.5 gives the correction directly. Smaller anchor gaps make the single linear step accurate to three or four decimals with no calculator.

Estimate the square root of 20

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