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I'm searching to find how to get the sqaure root of a number having decimal places. How to find the Square root of say $0.4$?

Shobhit
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user100520
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    Hello, welcome to Math.SE. Please read http://meta.math.stackexchange.com/questions/9959/how-to-ask-a-good-question/9960#9960 and the others there for information on writing a good question for this site. In particular, people will be more willing to help if you edit your question to include some motivation, and an explanation of your own attempts.Take a look here to learn how to write in LATEX http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – Shobhit Oct 13 '13 at 05:18
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    Do you know how to take the square root of an integer? Because you can easily write $\sqrt{\frac{2}{5}} = \sqrt{\frac{1}{25}} \cdot \sqrt{10} = \frac{1}{5} \cdot \sqrt{10}$. – Arthur Oct 13 '13 at 05:27
  • You mean by hand? With a calculator using only basic operations? It would be helpful to know what is your goal if you told us how you want to calculate the square root. – Umberto Oct 13 '13 at 06:23

2 Answers2

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Wikipedia has an excellent page on the topic.

One method is Newton's method, so for $x=0.4$, you would start with an initial guess, say $x_0=0.7$ and calculate:

$$x_{n+1}=\frac{1}{2}\left(x_n + \frac{0.4}{x_n}\right)$$ $$x_1 = 0.6357$$ $$x_2 = 0.6325$$

Which is pretty close already..

Nathaniel Bubis
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One way you could solve for it is by doing a binary search. You define an interval where you believe the answer is and then narrow it down until it is smaller and smaller.

For instance $\sqrt{.4}$ should be between $.4$ and $1$. We will pick our favorite number in that range as a guess and then based on if it is high or low we will use it to narrow the search.

$$ \sqrt{.4} \text{ is in }[.4,1]$$

Guess that it is equal to $.6$. Squaring our guess give $.6^2=.36$ which is a bit too low. So we will look for an answer between $.6$ and $1$.

$$ \text{ Now } \sqrt{.4} \text{ is in } [.6,1] $$

Guess that it is $.7$. Checking shows that $.7^2=.49$ which is too high. So look between $.7$ and $.6$.

$$ \text{ Now } \sqrt{.4} \text{ is in } [.6,.7] $$

etc.

If you continue this indefinitely you will be able to get the answer to any number of desired decimal places. This method works for any kind of root not just square roots.

Spencer
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