# Squares and Square Roots

First learn about Squares, then Square Roots are easy.

## How to Square A Number

To square a number: **multiply it by itself**.

### Example: What is 3 squared?

3 Squared | = | = 3 × 3 = 9 |

"Squared" is often written as a little 2 like this:

This says **"4 Squared equals 16"**

(the little 2 says the number appears twice in multiplying)

## Squares From 0^{2} to 6^{2}

0 Squared | = | 0^{2} | = | 0 × 0 | = | 0 |

1 Squared | = | 1^{2} | = | 1 × 1 | = | 1 |

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

3 Squared | = | 3^{2} | = | 3 × 3 | = | 9 |

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

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

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

The squares are also on the Multiplication Table: |

## Negative Numbers

We can also square **negative numbers**.

### Example: What happens when we square (−5) ?

Answer:

(−5) × (−5) = **25 **

(because a negative times a negative gives a positive)

That was interesting!

When we square a **negative** number we get a **positive** result.

Just the same as squaring a positive number:

(For more detail read Squares and Square Roots in Algebra)

## Square Roots

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

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

A square root of a number is ...

**multiplied by itself**to give the original number.

A square root of **9** is ...

**3**, because

**when 3 is multiplied by itself**we get

**9**.

It is like asking:

What can we multiply by itself to get this?

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 |

## Decimal Numbers

It also works for decimal numbers.

Try the sliders below. *Note: the numbers here are only shown to 2 decimal places.*

Using the sliders (remembering it is only accurate to 2 decimal places):

- What is the square root of
**8**? - What is the square root of
**9**? - What is the square root of
**10**? - What is
**1**squared? - What is
**1.1**squared? - What is
**2.6**squared?

## Negatives

We discovered earlier that we can square negative numbers:

### Example: (−3) squared

(−3) × (−3) = **9**

And of course 3 × 3 = **9** also.

So the square root of 9 could be **−3** or **+3**

### Example: What are the square roots of 25?

(−5) × (−5) = 25

5 × 5 = 25

So the square roots of 25 are **−5** and **+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 mathematics look important!radical |

We use it like this:

and we say **"square root of 9 equals 3"**

### Example: What is √25?

25 = 5 × 5, in other words when we multiply 5 by itself (5 × 5) we get 25

**So the answer is:**

√25 = 5

But wait a minute! Can't the square root **also be −5**? Because (−5) × (−5) = **25 **too.

- Well the
**square root of 25**could be −5 or +5. - But when we use the
**radical symbol √**we only give the**positive (or zero) result**.

### Example: What is √36 ?

Answer: 6 × 6 = 36, so **√36 = 6**

## Perfect Squares

The Perfect Squares (also called "Square Numbers") are the squares of the integers:

PerfectSquares | |

0 | 0 |

1 | 1 |

2 | 4 |

3 | 9 |

4 | 16 |

5 | 25 |

6 | 36 |

7 | 49 |

8 | 64 |

9 | 81 |

10 | 100 |

11 | 121 |

12 | 144 |

13 | 169 |

14 | 196 |

15 | 225 |

etc... |

Try to remember them up to 12.

## Calculating Square Roots

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 √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* - ...

Getting closer to 10, but it will take a long time to get a good answer!

At this point, I get out my calculator and it says:
But the digits just go on and on, without any pattern. So even the calculator's answer is |

*Note: numbers like that are called Irrational Numbers, if you want to know more.*

## The Easiest Way to Calculate a Square Root

Use your calculator's square root button! |

And also use your common sense to make sure you have the right answer.

## A Fun Way to Calculate a Square Root

There is a fun method for calculating a square root that gets more and more accurate each time around:

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 (4 + 2.5 = 6.5) d) then divide that result by 2, in other words halve it. (6.5/2 = 3.25)e) now, set that as the new guess, and start at b) again |

- 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**

Now ... why don't **you** try calculating the square root of 2 this way?

### How to Guess

What if we have to guess the square root for a difficult number such as "82,163" ... ?

In that case we could think "82,163" has 5 digits, so the square root might have 3 digits (100x100=10,000), and the square root of 8 (the first digit) is about 3 (3x3=9), so 300 is a good start.

### Square Root Day

The 4th of April 2016 is a Square Root Day, because the date looks like **4/4/16**

The next after that is the 5th of May 2025 (5/5/25)