Making sense of irrational numbers

Hello mathematicians!

Why do I call you a mathematician? Yes, you all are excellent. Since junior and senior KG, you have met with the numbers. White pencil and black slate both were your friend to draw every number and loudly say, “it’s a one, two, …”. Later we learned tables which were again game of numbers. Still, we are playing with the numbers. However, as each standard has changed, new concepts have added to the study.

Let’s take an example of irrational numbers. Remember? Quickly recall it.


What Is an Irrational Number?

An irrational number is a real number that is can not be expressed as a ratio of integers means fraction.

Rational numbers are the opposite and can be expressed as a fraction.

Who discover the irrational numbers?

Hippasus of Metapontum, in the 5th century, B.C.He was the ancient Greek philosopher and mathematician. He proved the existence of irrational numbers first.

He found the problems with Root 2.

Later, he decided to go with the help of a square shape where he used Pythagoras theorem.

a^2 + b^2 = c^2.

Now, the problem has arrived.

The diagonal length would be the square root of 2 that means c^2 =2 and c=√2

However, he could not prove that as the ratio of two integers. Hippasus did not get the expected result. Without wasting time and never give up thinking he started again in a different way.

Step 1

He assumed that Root 2 is a rational number. As you know, every rational number can be expressed as a ratio or ratio of two integers.

He supposed the two integers namely p and q. We write a ratio in the form of p/q, it could not have common factors that mean example, 2 can be expressed as 8/4 or 10/5, we get the answer 2 after division. So, 2 can be represented as 2/1, 1 is invisible.

Step 2

Hippasus assumed that p/q can not exist. Multiplied by q on both sides and squared both sides.

The original picture is, 

√2 = p / q

After multiplying by q, you get

√2 q = p

Square of both sides give the equation,

2 (q)^2 = p^2

Step 3

Square of an odd number is always odd 

3^2 = 9

and the square of an even number is even only.

4^2 = 16

Step 4

So, p would have to be an even integer and expressed or replaced by 2x where x is an integer.

The equation has changed,

2 (q)^2 = (2x)^2

2 (q)^2 = 4x^2

This implies that q is also even.

Initially, p and q had no common factors. However, it seems both are even now and 2 is the common factor for both. The initial assumption of Root 2  being a rational number goes false. Contradiction had found. This is how making sense of irrational numbers.


Alike scientist and their experiment, mathematician Hippasus experimented and solved the contradictory examples. We never think while learning how each statement and concept arise with strong proof.

” Assumption helped to solve the confusion“!!!


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