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?

«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)

## Negative Numbers

We can also square negative numbers.

That was interesting!

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

Just the same as squaring a positive number:

## Square Roots

A square root goes the other way:

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

What can we multiply by itself to get this?

Here are some more squares and square roots:

## Decimal Numbers

It also works for decimal numbers.

Using the sliders:

• 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

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

√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:

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 only an approximation !

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

## The Easiest Way to Calculate a Square Root

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:

• 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

### How to Guess

In that case we could think «82,163» has 5 digits, so the square root might have 3 digits (100×100=10,000), and the square root of 8 (the first digit) is about 3 (3×3=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)

309,310,315, 1082, 1083, 2040, 3156, 2041, 2042, 3154