What is the square root of one-half?

Question

No calculator. Solve in your head. Estimate \sqrt{0.5} to three decimal places. Aim: under 10 seconds.

Accepted Answer

Rewrite as a reciprocal root: \sqrt{0.5} = \sqrt{	frac12} = \frac{1}{\sqrt2}. With \sqrt2 \approx 1.414: \frac{1}{1.414} \approx 0.707. Check: 0.707^2 = 0.4998 \approx 0.5. Takeaway. 	frac{1}{\sqrt2} = 	frac{\sqrt2}{2} \approx 0.707 is one of the most reused constants going, it is \cos 45^\circ, the RMS-to-peak ratio, and the value where a probability-weighting or volatility term is often anchored. Learn it as a partner to \sqrt2 \approx 1.414.

What is the square root of one-half?

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