Now we know that square root of 2 is an Irrational number.
If you use a calculator to calculate the value of Root 2
you will get something like 1.4142135 and so on.
This value that you get is actually non-terminating.
It is never ending.
Also, you will not find any pattern in the digits after the decimal.
Hence, it's also called a non-recurring decimal.
There is no pattern that occurs again and again.
These kinds of numbers are called Irrational numbers.
Consider square root of 3 for instance.
You will get its value as 1.732050 and so on.
Even this decimal is non-terminating and non-recurring.
So root 3 is also an Irrational number.
You can try calculating the values of root 5, root 6, root 7 or root 8,
you will find that all of them result
in non-terminating, non-recurring decimals.
But wait... root 9 equals 3. This is a Rational number.
Yes, root 9 is not an Irrational number.
In fact, square root of any perfect square will be a Rational number.
But the square root of any number which is not a perfect square
will always result in an Irrational number.
Here's the interesting part. If you know these two points well,
you can even form an Irrational number by yourself. Let's see how.
We have to form a number which does not end and does not repeat any pattern.
Something like 2.01001000100001 and so on.
Here, the number does not end and is also non-recurring.
This means that there is no pattern that repeats forever.
You can even have a number like 15.2349449444944449 and so on
This is also Irrational as it is non-terminating and non-recurring.
Where does this concept help us?
Say your are asked to find two Irrational numbers between 1.23 and 1.25.
Remember, 1.24 is incorrect as its terminating.
We need a non-terminating, non-recurring decimal.
Yes, but we know
that 1.24 is between these two numbers.
After 1.24 we can start writing 010010001 and so on.
That's one Irrational number between these two numbers.
Again, let's write 1.24 and this time
we write 020020002 and so on after it.
That's your second Irrational number.
So to find Irrational numbers between two numbers,
just write a Rational number between them and then add
non-terminating, non-recurring digits after it.
