# Irrational Numbers

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We saw, in the previous section, that there may be numbers on the number line that are not **rationals**. In this section, we are going to investigate these numbers. So far, all the numbers you have come across are of the form

The Pythagoreans in Greece, followers of the famous mathematician and philosopher Pythagoras, were the first to discover the numbers which were not rationals, around 400 BC. These numbers are called **irrational numbers** (irrationals), because they cannot be written in the form of a ratio of integers. There are many myths surrounding the discovery of irrational numbers by the Pythagorean, Hippacus of Croton. In all the myths, Hippacus has an unfortunate end, either for discovering that √2 is irrational or for disclosing the secret about √2 to people outside the secret Pythagorean sect!

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Let us formally define these numbers.

A number ‘s’ is called

You already know that there are infinitely many rationals. It turns out that there are infinitely many irrational numbers too. Some examples are:

**Remark :** Recall that when we use the symbol

Some of the irrational numbers listed above are familiar to you. For example, you have already come across many of the square roots listed above and the number π.

The Pythagoreans proved that

In the next section, we will discuss why 0.10110111011110... and π are irrational.

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Let us return to the questions raised at the end of the previous section.

Remember the bag of rational numbers. If we now put all irrational numbers into the bag, will there be any number left on the number line? The answer is no! It turns out that the collection of all rational numbers and irrational numbers together make up what we call the collection of

Therefore, a real number is either *real number line.*

In the 1870s two German mathematicians, Cantor and Dedekind, showed that :

*Corresponding to every real number, there is a point on the real number line, and corresponding to every point on the number line, there exists a unique real number.*

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**Example 3**

Locate √2 on the number line.

It is easy to see how the Greeks might have discovered √2.

**Example 4**

Locate √3 on the number line.

**Example 5**

Locate √5 on the number line.