Square Root of a Number

What does it mean to find the square root of a number?

Well, when we ask for the square root of a number, we are really asking, "what number when multiplied by itself gives the number we want the square root of".

For example, what is the square root of the number 16? The real question we must answer is, "what number, when multiplied by itself, gives us the value 16"? The obvious answer is 4, because 4 x 4 = 16. So, the square root of 16 is 4.

That’s pretty easy. But suppose we want to know the square root of 5; that is, we want to know what number, when multiplied by itself, equals 5? That’s not quite so easy. Most of us would use a calculator to find that answer. A calculator will have a key with the symbol which is the symbol for square root. Using the square root key on my calculator for the square root of 5 gives me the answer 2.236..., the approximate value of the square root of 5 or . We can check this answer by multiplying 2.236 by itself so that 2.236 x 2.236 = 4.999... or essentially 5. Additional digits need to be carried with the number to obtain exact values.

We could find square root values using a trial-and-error procedure but that takes quite a bit of computational work. Suppose we want the square root of 2, for example. We might start by trying 1.5 for the square root. 1.5 x 1.5 = 2.25, so 1.5 is too large. We next might try 1.4 which gives us 1.4 x1.4 = 1.96. So, 1.4 is too small and 1.5 is too large, so we might next try 1.45 as our trial value. 1.45 x 1.45 = 2.1025, so 1.45 is a bit too large. Let’s next try 1.41. 1.41 x 1.41 = 1.9881 which is too small. So, at this point, we know that the square root of 2 is somewhere between 1.41 and 1.45. If we continue our trial-and-error procedure, we can get our value for the square root of 2 as close to its correct value as we wish. Using a calculator would tell us that , correct to as many decimal places as we wish by using as many trials as necessary to achieve the accuracy we want.

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