Blog: Find square root of 110.6 and 10080 without using a calculator.

Thursday, November 4, 2010

Let N be the number you want to get the square root. Assume E1 as the initial estimate for $\displaystyle\sqrt{{N}}$ .

So the next estimate E2 will be;

$\displaystyle{E}{2}=\frac{{\frac{{N}}{{{E}{1}}}+{E}{1}}}{{2}}$

Similarly E3 will be;

$\displaystyle{E}{3}=\frac{{\frac{{N}}{{{E}{2}}}+{E}{2}}}{{2}}$

For 110.6;

We know that $\displaystyle\sqrt{{100}}={10}$ and $\displaystyle\sqrt{{121}}={11}$

So let us estimate E1 = 10 for ease of calculation.

$\displaystyle{E}{2}=\frac{{\frac{{110.6}}{{10}}+{10}}}{{2}}={10.53}$

If you need further accuracy you can estimate E3 as well.

$\displaystyle{E}{3}=\frac{{\frac{{110.6}}{{10.53}}+{10.53}}}{{2}}={10.517}$

For 10080;

We know that $\displaystyle\sqrt{{10000}}={100}$

So let us say E1 = 100.

$\displaystyle{E}{2}=\frac{{\frac{{10080}}{{100}}+{100}}}{{2}}={100.4}$

Similarly;

$\displaystyle{E}{3}=\frac{{\frac{{10080}}{{100.4}}+{100.4}}}{{2}}={100.399}$

So the answers are;

$\displaystyle\sqrt{{110.6}}={10.517}$

$\displaystyle\sqrt{{10080}}={100.399}$

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