Square Root Calculator

What this calculator does

So, you want the square root of a number. This tool takes a number that is zero or positive and returns its square root, worked out to ten decimal places.

There is one input, the number, and the answer appears with the familiar root sign. The square root is the most common root of all, which is why it has a tool of its own.

How to use it

1. Enter your number, which must be zero or positive.
2. Press Calculate.

The ten decimal places give plenty of precision for roots that do not come out even, such as the square root of 2.

What a square root is

The square root of a number is the value that, multiplied by itself, gives that number. The square root of 9 is 3, because 3 times 3 is 9. The square root of 25 is 5, because 5 times 5 is 25. It is called a square root because of area: if a square has an area of 9, then each side is 3 long, so the square root turns an area back into a side length. In the language of roots, the square root is the nth root with degree 2, the most basic case of all.

Two roots, and the one it gives

Strictly speaking, positive numbers have two square roots, one positive and one negative, because a negative times a negative is also positive. Both 3 and minus 3, squared, give 9. But by convention "the" square root means the positive one, called the principal square root, and that is what this tool returns. So for 9 it gives 3, not minus 3, which matches what the root sign is understood to mean. If you need the negative root as well, it is simply the same value with a minus sign.

Why negatives are not allowed

The tool accepts only numbers that are zero or greater. This is because no ordinary number, multiplied by itself, can produce a negative result: a positive times a positive is positive, and a negative times a negative is also positive. So a negative number has no square root among the real numbers, and the tool asks for a non-negative input rather than return something meaningless. Taking square roots of negatives leads into imaginary numbers, which are a separate topic.

A worked example

Enter 9 and the tool returns 3, since 3 times 3 is 9. Enter 2 and it returns about 1.4142135624, a root that never settles into a neat fraction. Enter 0 and it returns 0, since 0 times 0 is 0.

Beyond the square root

The square root is degree 2, but the same idea runs to higher degrees. The cube root, degree 3, asks which number cubed gives your number, and has its own cube root calculator. For any degree at all, the general root calculator handles it, and the radical calculator shows these roots in proper radical notation. The square root is simply the first and most familiar member of that whole family.

Questions people ask

What is a square root?

The number that, multiplied by itself, gives your number. The square root of 9 is 3 because 3 times 3 is 9.

Why not minus 3 for the square root of 9?

Because "the" square root means the positive, or principal, one by convention. Minus 3 is also a square root of 9, but the tool returns the positive value.

Why can't I take the square root of a negative?

Because no real number squared is negative. A negative number has no real square root, so the tool asks for a non-negative input.

Why is it called a square root?

Because it turns the area of a square back into its side length. A square of area 9 has sides of length 3.

What about cube roots and higher?

Those use the cube root calculator or the general root calculator. The square root is the degree 2 case of the same idea.

References

On the square root. The square root of a number is the value whose square is that number, and the principal square root is the non-negative one.

1. Eric W. Weisstein, "Square Root," from MathWorld, a Wolfram resource, on the square root and the principal square root.
2. Eric W. Weisstein, "nth Root," from MathWorld, a Wolfram resource, on the square root as the nth root with degree 2.