Decimal Expansion of Real Numbers | CBSE Class 9 | Number System Episode-3 | Genius: Maths

In this Genius Teacher video you will learn about "Decimal expansion of rational and irrational numbers" and much more about Maths Chapter 1, Number System for CBSE Class 9 explanation made easy and interesting to learn for kids in Hindi using examples of Michael Phelps!

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This interesting genius video is based on the decimal expansion of real numbers and how we can use this expansion to distinguish between rational numbers and irrational numbers.
Michael Phelps is one of the most successful and most decorated Olympian of all time.
He completed this 100 metre butterfly race in 51.21 {fifty one point two one} seconds.
51.21 can be expressed as 5121 upon 100.
"5121 upon 100 is a rational number, and if we divide 5121 with 100 then after a certain stage, the remainder becomes 0.

Such cases, where the decimal expansion terminates after a finite number of steps, is called terminating decimals."
Let's consider another rational number 9 upon 11
If we divide 9 by 11, then the remainder starts repeating itself.
"Observe this 2, 9, 2, 9.

Therefore, this case where a digit or a group of digits keeps repeating itself is known as a non-terminating recurring decimal.

The quotient in this case is written as zero point eight one with a bar on top of eight one like this."
However, if we express an irrational number like pi using decimals then we will get a non-terminating non-recurring decimal.
So, how will you distinguish between rational numbers and irrational numbers based on their decimal expansion?
Rational numbers either have a terminating decimal or a non-terminating recurring decimal; whereas irrational numbers have a non-terminating non-recurring decimal.
And, what if we want to find an irrational number between two rational numbers like 20 upon 9 and 30 upon 9?
"20 upon 9 is 2.222 and 30 upon 9 is 3.333

So, the answer is any number like 2.64575 which lies between these two numbers and has a non-terminating non-recurring decimal."
Moving on, suppose we know the decimal expansion of a rational number, then how will we express it as p upon q?
For example if the rational number is 0.181818 then how will you express it as p upon q?
"Let x equal to this number which can be expressed as zero point one eight because over here, the block one eight keeps on repeating.

Now, a block of two digits keep on repeating, therefore, we will multiply both the sides with 100 to get 100x is equal to 18.18"
"And now if we subtract both the sides with x then we will get
100x minus x is equal to 18.18 minus 0.18

So 99x is equal to 18."
"Therefore, x is equal to 18 upon 99, which can be further reduced to get 2 upon 11.

So, the rational number 0.18 can be expressed as 2 upon 11."
Now that you know how to differentiate between rational numbers and irrational numbers based on their decimal expansions, quickly go to the genius app and solve some exciting examples based on this