Irrational Numbers

Irrational numbers include the square root, cube root, fourth root, and nth root of many numbers. Whenever numbers are preceded with a radical sign, the numbers are called radicals

Radical sign:

A radical sign is a math symbol that looks almost like a 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

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


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!


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


2 = 1.41421356237309504880168872420969807856967

Even more

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:

a. If you can quickly find a root for the radical, the radical is rational

b. 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!

c. 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:



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 a rational number

Example #2:

Is 5325 irrational?

Find the following n-root symbol on your calculator:


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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