Square roots

Can you calculate the area of a square? It is easy. Just multiply one side by itself or raise one side to the power of two - square it. If the side is two, the square has an area of four. But what about this square?

It's area is nine. How long are the sides? Which number will give you nine if you square it? Right. The side is three.

What you've just calculated in your head was the square root of nine. ​ This little V shaped symbol is a root sign. It's the opposite of a power expression. The root sign asks you to find a number, which times itself is the number that you see under that little roof. Nine for example is three times three. So we say that three is the square root of nine just like nine is three to the second power or three squared. ​ ​A bit more formally we can say that if X squared equals A and X is equal to or greater than zero, then the square root of A is X.

Does it look difficult? Pick a value for X. Say two. Two squared is four. Then the square root of four is two.

It's pretty easy to find the square root of 1, 4, 9, 16 and 25. These are perfect squares and you know them from the multiplication table. They're one times one, two times two, three times three, four times four, and five times five.​ But how do you calculate the square root of a number that is not a perfect square? Such as ten. There are several ways.

The simplest way is to use a calculator. First, enter the number you want to find the square root of - ten - then press the square root button. Nowadays calculators are not expensive and most cellphones have them as a function, but in the past it was not common to have your own calculator. Back then they used tables, which were often at the back of a math book, where you could look up square roots along with a few other things that are hard to calculate manually. ​ If you don't have a calculator or a table of square roots, you can calculate the number by using an algorithm. Start with a feasible guess and calculate the root of that number.

You then adapt your guess a bit at a time until you have the answer. If the answer is too large, you lower the initial value. And if it is too small, you increase it. ​ There are algorithms that help you find square roots with as few tries as possible. If you want to experiment with such an algorithm search for the Babylonian method on Wikipedia. You cannot find the square root of a negative number. Can you guess why?

The square root answers the question. Which number multiplied by itself equals this number and a number multiplied by itself is always positive. Minus times minus is always plus. Therefore a negative number can never be the product of two equal factors. The cubic root on the other hand is different.

You can find the cubic root of a negative number, but that's another story. The square root is simply the number you should multiply by itself to find the number you have in front of you. This is an operation inverse to raising a number to the power of two.