# Irrational Numbers

Irrational numbers include the square root, cube root, fourth root, and nth root of many numbers. Whenever a number is preceded with a radical sign, the number is called a radical. A radical sign is a math symbol that looks almost like the letter v and is placed in front of a number to indicate that the root should be taken:
Not all radicals are irrational. For example, 4 is not an irrational number.

It is because 4 = 2 and 2 is a whole number.

## More examples of irrational numbers

2 = 1.4142136

7 = 2.64575131

35 = 5.9160797831

8 = 2.82842712475

Why are the radicals above irrational? They are irrational because the decimal expansion is neither terminating nor repeating.

Non-repeating:

Take a close look at the decimal expansion of every radical above, you will notice that no single number or group of numbers repeat themselves as in the following examples.

1.222222222222 (The 2 repeats itself, so it is not irrational)

4.3636363636 (36 repeats itself, so it is not irrational)

There are basically no patterns!

Non-terminating:

The radicals above are not terminating. This means that the decimal expansion has an infinite amount of numbers.

For example 2 = 1.4142135

We wrote only 7 digits after the decimal point. However, you can get more numbers.

For example 2 = 1.41421356237309504880

More digits

2 = 1.41421356237309504880168872420969807856967

Even more digits

2 = 1.4142135623730950488016887242096980785696718753769480731766797379907

It never ends...

Whenever a number is irrational, all we can get is an approximation. We can never write the number completely.

## Difference between rational and irrational numbers.

Although rational numbers can go on and on with an infinite amount of numbers, they nonetheless have a pattern. Irrational numbers don't have a pattern.

How to find out if a radical is irrational

There are a couple of ways to check if a number is rational:
• If you can quickly find a root for the radical, the radical is rational.
• If you are only looking for the square-root, you could use the square root algorithm. Of course, the method above is lengthy and time-consuming!
• You could use a calculator. This may be the best way to check.

Example #1:

Is 3125 irrational?

Find the following cube-root symbol on your calculator:

3

3x

Depending on your calculator, you will either enter 125 and then hit the symbol. Or hit the symbol first and then enter 125.

You should get 5

5 is terminating, so it is not an irrational number.

Example #2:

Is 5325 irrational?

Find the following n-root symbol on your calculator:

xy

Play around with the calculator to get 5325 = 3.179630631616273

3.179630631616273 is non terminating and non repeating, so it is irrational.

Proof that square root of 5 is irrational

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