Mental Arithmetic – Finding Square Roots to 1 Decimal Point

mental-arithmetic

I have 2 questions here.

  • What is the most effective and easy way of calculating square roots in your head to an accuracy of 1 decimal point? This would need to work with at least two digit, non-perfect squares and would have to be doable mentally.

    • How would the method work?

  • Is one decimal place accurate enough for all intents and purposes that you may come across in average math? When would you need more precision?

I am curious because often times in school, I would spend time with paper and pencil, working out approximations for non-perfect squares, it would help if I could get an approximation quicker.

Thanks!

Edit: Please ensure that the method is easy to understand, and to do mentally for a student in grades 8 and above. I don't want to be just memorizing formulas without understanding how they work, I think it would be beneficial for anyone browsing this question.

Best Answer

I'd just do good old Taylor series expansion up to linear term. $$ f(x+\Delta x) \approx f(x) + f'(x) \Delta x $$ so, in case of square roots $$ \sqrt{x + \Delta x} \approx \sqrt x + \frac {\Delta x}{2\sqrt x} $$ where $x$ - is the closest perfect square. Obviously, error might be huge if $\Delta x$ is big.

For example, $$ \sqrt{66} = \sqrt{64 + 2} \approx \sqrt{64} + \frac 2{2 \sqrt{64}} = 8 + \frac 18 = 8.125 $$ whereas $\sqrt{66} \approx 8.12403840463596 \ldots$

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