Estimate the square root of 104

Question

Solve in your head. Estimate \sqrt{104} to three decimal places. Aim: under 10 seconds.

Accepted Answer

Linearize around the nearest perfect square, 100 = 10^2: \sqrt{a^2 + \varepsilon} \approx a + \frac{\varepsilon}{2a}. With a = 10 and \varepsilon = 4 (since 104 = 100 + 4): \sqrt{104} \approx 10 + \frac{4}{20} = 10 + 0.2 = 10.200. True value: \sqrt{104} \approx 10.19804. The estimate 10.200 is high by about 0.002, the usual small overshoot from the concavity of \sqrt{x} (the tangent line sits above the curve). Cross-check: \sqrt{104} = 2\sqrt{26}, and \sqrt{26} \approx 5.09902, giving 10.19804, consistent. Takeaway. With a base square of 100, the division by 2a = 20 is trivial: just take a te…

Estimate the square root of 104

Your answer

More Mental Math questions