Estimate the square root of 1000

Question

Solve in your head. Estimate \sqrt{1000} to three decimal places. Aim: under 15 seconds.

Accepted Answer

First find the nearest perfect square. 31^2 = 961 and 32^2 = 1024; since 1000 is closer to 1024 (a gap of 24) than to 961 (a gap of 39), anchor at 32. Then use \sqrt{a^2 + \varepsilon} \approx a + \frac{\varepsilon}{2a} with a = 32 and \varepsilon = -24: \sqrt{1000} \approx 32 - \frac{24}{64} = 32 - 0.375 = 31.625. True value: \sqrt{1000} \approx 31.62278. The estimate is high by about 0.002. Choosing the closer square matters: had you anchored at 31 (\varepsilon = 39), you would get 31 + 39/62 = 31.629, slightly worse, because a larger \varepsilon means a larger linearization error. The tigh…

Estimate the square root of 1000

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