## How to Square A Number

To square a number, just multiply it by itself ...

### Example: What is 3 squared?

3 Squared | = | = | 3 × 3 | = | 9 |

Note: we write down "3 Squared" as 3^{2}

(the little "^{2}" means the number appears twice in multiplying)

## Some More Squares

4 Squared | = | 4^{2} | = | 4 × 4 | = | 16 |

5 Squared | = | 5^{2} | = | 5 × 5 | = | 25 |

6 Squared | = | 6^{2} | = | 6 × 6 | = | 36 |

## Square Root

A **square root** goes the other direction:

3 squared is 9, so the **square root of 9 is 3**

3 | 9 |

The square root of a number is ...

... that special value that **when multiplied by itself** gives the original number.

The square root of **9** is ...

... **3**, because **when 3 is multiplied by itself** you get **9**.

Note: When you see "root" think
In this case the tree is "9", and the root is "3". |

Here are some more squares and square roots:

4 | 16 | |

5 | 25 | |

6 | 36 |

### Example: What is the square root of 25?

Well, we just happen to know that 25 = 5 × 5, so if you multiply 5 by itself (5 × 5) you will get 25.

**So the answer is 5**

## The Square Root Symbol

This is the special symbol that means "square root", it is sort of like a tick, and actually started hundreds of years ago as a dot with a flick upwards. It is called the , and always makes math look important!radical |

You can use it like this: (you would say "the square root of 9 equals 3")

*More Advanced Topics Follow*

## You Can Also Square Negative Numbers

Have a look at this:

If you square 5 you get 25: | 5 × 5 = 25 | |

But you could also square -5 to get 25: | -5 × -5 = 25 | |

(because a negative times a negative gives a positive) |

So the **square root** of 25 can be 5 **or** -5

There can be a positive or negative answer to a square root!

*But when people talk about "the" square root they usually mean just the positive one.*

And when you use the radical symbol √ it **always** means just the positive one.

Example: √36 = 6 (not -6)

## Perfect Squares

The perfect squares are the squares of the whole numbers:

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | etc | |

Perfect Squares: | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 | 225 | ... |

It is easy to work out the square root of a perfect square, but it is **really hard** to work out other square roots.

### Example: what is the square root of 10?

Well, 3 × 3 = 9 and 4 × 4 = 16, so we can guess the answer is between 3 and 4.

- Let's try 3.5:
*3.5 × 3.5 = 12.25* - Let's try 3.2:
*3.2 × 3.2 = 10.24* - Let's try 3.1:
*3.1 × 3.1 = 9.61*

Very slow ... at this point, I get out my calculator and it says:

*3.1622776601683793319988935444327*

... but the digits just go on and on, without any pattern. So even the calculator's answer is **only an approximation ! **

(Further reading: these kind of numbers are called surds which are a special type of irrational number)

(Further reading: these kind of numbers are called surds which are a special type of irrational number)

## A Special Method for Calculating a Square Root

There are many ways to calculate a square root, but my favorite method is an easy one which gets more and more accurate depending on how many times you use it:

a) start with a guess (let's guess 4 is the square root of 10)

b) divide by the guess (10/4 = 2.5)

c) add that to the guess (2.5+4=6.5)

d) then dividethatresult by 2, in other words halve it. (6.5/2 = 3.25)

e) now, set that as thenew guess, and start at b) again

... so, our first attempt got us from 4 to 3.25

Going again (*b to e*) gets us: 3.163

Going again (*b to e*) gets us: 3.1623

And so, after 3 times around the answer is 3.1623, which is pretty good, because:

3.1623 x 3.1623 = 10.00014

This is fun to try - why not use it to try calculating the square root of 2?