Methods of computing square roots

There are a numbers of ways to calculate square roots of numbers, and even more ways to estimate them.

The mathematical operation of finding a root is the opposite operation of exponentiation, and therefore involves a similar but reverse thought process.

Firstly, one needs to know how precise the result is expected to be. This is because often square roots are irrational. For example, square root of a nice round whole number 28 is a fraction which in its decimal notation has infinite length, and therefore it is impossible to express it exactly:[math]\displaystyle{ \sqrt{28} \approx 5.291502622129181.... }[/math]

Moreover, for some real numbers the square root is a complex number. For example, square root of -4 is a complex number 2i :[math]\displaystyle{ \sqrt{-4} = 2 i }[/math]

In many cases there may be multiple valid answers. For example, square root of 4 is 2, but -2 is also a valid answer. One can verify that they are both valid answers by squaring each candidate answer and checking if you obtain 4 as the result of verification:[math]\displaystyle{ 2^2 = 2 \times 2 = 4 }[/math]

[math]\displaystyle{ (-2)^2=(-2) \times (-2) = 4 }[/math]

Please note that calculating a square root is a special case of the problem of calculating N^th root.

Calculating

Most calculators provide a function for calculation of a square root.

	General Steps	Example
How to calculate a square root using a simple calculator.	First, make sure the operating space is clear. This is usually accomplished by clicking the C button a couple of times. Then type the number whose root you are trying to calculate. Then press the square root button ([math]\displaystyle{ \sqrt{} }[/math]). The number you see on the screen is one of the answers. Remember, that often there are multiple valid answers, as explained above.	Press C button a couple of times. Type 16 Press [math]\displaystyle{ \sqrt{} }[/math] button. The answer is 4. Keep in mind that -4 is also a valid answer.

Estimating

If the result does not have to be very precise, the following estimation techniques could be helpful:

Methodology

Example

Suppose you need to find square root of some number [math]\displaystyle{ N }[/math]. Find some number [math]\displaystyle{ A }[/math] such that [math]\displaystyle{ A^2 }[/math] (that is [math]\displaystyle{ A }[/math] squared, or [math]\displaystyle{ A }[/math] times [math]\displaystyle{ A }[/math]) is approximately equal to [math]\displaystyle{ N }[/math] (but how close? This needs to be expanded). Then we can think of [math]\displaystyle{ A }[/math] as being approximately a square root of [math]\displaystyle{ N }[/math].

Suppose we need to estimate the square root of 2. We know that [math]\displaystyle{ 1^2 = 1 }[/math], and [math]\displaystyle{ 2^2 = 4 }[/math]. Therefore, one of the answers to [math]\displaystyle{ \sqrt{2} }[/math] is somewhere between 1 and 2.

References

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