Irrational And Real Numbers
Class 8th Mathematics (old) MHB Solution
Exercise 7
Question 1.Classify the following decimal fractions as terminating and non-terminating recurring decimals.
(1) 0.777… (2) 0.777
(3) 4.7182 (4) 
(5) 9.165165 (6) 
(7) 0.52888 (8) 
(9) 
Answer:
(1) 0.777…
This number is non-terminating. Because the last digit 7 is repeating.
(2) 0.777
This number is terminating, because the number ends with 7, but not extending again three sevens.
(3) 4.7182
This is terminating, because the number is ending with 2 and not extending again.
(4)
This number is non-terminating, because 7182 is mentioned with a bar (‾). That says 7182 is repeating.
(5) 9.16516
This number is terminating, because the number is not extending after 9.16516.
(6)
This number is non-terminating, because 165 is mentioned with a bar (‾). That says 165 is repeating.
(7) 0.52888
This number is terminating, because the number is not extending after 0.52888.
(8)
This number is non-terminating, because 8 is mentioned with a bar (‾). That says 8 is repeating.
(9)
This number is non-terminating, because 136 is mentioned with a bar (‾). That says 136 is repeating.
Exercise 8
Question 1.Write the following numbers in the non-terminating recurring form.
(1) 0.16 (2) 7.439
(3) 10.605 (4) 0.058
(5) 1.06 (6) 0.0002
Answer:
(1) 0.16
Here 0.16 is also written as 0.1600… adding zero to the right side of the decimal number does not change the value of that number. So 0.16 can also be written as 
now this is a non-terminating number.
(2) 7.439
Here 7.439 is also written as 7.4390… adding zero to the right side of the decimal number does not change the value of that number. So 0.16 can also be written as 
now this is a non-terminating number.
(3) 10.605
Here 10.605 is also written as 10.6050… adding zero to the right side of the decimal number does not change the value of that number. So 10.605 can also be written as 
now this is a non-terminating number.
(4) 0.058
Here 0.058 is also written as 0.0580… adding zero to the right side of the decimal number does not change the value of that number. So 0.058 can also be written as 
now this is a non-terminating number.
(5) 1.06
Here 1.06 is also written as 1.060… adding zero to the right side of the decimal number does not change the value of that number. So 1.06 can also be written as 
now this is a non-terminating number.
(6) 0.0002
Here 0.0002 is also written as 0.00020… adding zero to the right side of the decimal number does not change the value of that number. So 0.0002 can also be written as 
now this is a non-terminating number.
Exercise 9
Question 1.Classify the following numbers into two groups and label each group correctly.
(1) 
(2) 0.31045693…
(3) 
(4) 
(5) 
(6) 0.1010010001…
Answer:
We can classify the above given decimals into 3 groups. Terminating and Non-Terminating numbers.
i) Recurring and Non-Recurring numbers.
ii) Rational and Irrational numbers.
First, we should know what is terminating and non-terminating.
i.Terminating Numbers : The numbers which have an end (or) which have countable digits(finite).
Eg. 10.0123789456, 2.5896374
ii.Non-Terminating Numbers: The number which doesn’t has an end (or) They consists of uncountable digits(infinity).
Eg.20.15637894387…, 78.9848678898…
(Group 2): Recurring numbers: It is a decimal which has repeated set of numbers. Eg.
20.02020202…. , 46.464646464….
Non-Recurring numbers: It is a decimal which doesn’t have repeated set of numbers
Eg. 10.9848678898…., 97.852963741…
Group 3: Rational and Irrational numbers:
A number which can be written in the form of 
where numerator and denominators are integers except zero, then that is known as rational number.
A number which cannot be written in the form of or when numerator and denominators are not integers, then that is known as irrational numbers.
Exercise 10
Question 1.Make a table with columns for rational numbers, irrational numbers and real numbers and write the following numbers in their proper places in the table.
(1) 
(2) 
(3) 4.10547194… (4) 4.8
(5) 
(6) 
(7) 
(8) 
(9) 3.819023… (10) 6.10203040
Answer:
Real Numbers:
If a number found to be any of the following number, then that number is known as real number.
• Rational number
• Irrational numbers
• Integers
• Whole number
• Natural numbers
1) 1.
This is an rational number so it s a real number too.
2) √5
Since it is not a perfect square. It is known as irrational.
3) 4.10547194…
The dot in the end of the decimals indicates they are non-repeating numbers. It is clearly an irrational.
4) 4.8
It seems clearly as a rational number.
5) 0.
Since it s a recurring number. It is known as rational number
6) √25
This is perfect square, so it is a rational number.
7) √10
This is not a perfect square, so it is an irrational number.
8) √196
Since it is perfect square. It is a rational number.
9) 3.819023…
Since there is no repeating number in this, it is known as irrational number.
10) 6.10203040
Since it has repeating numbers this is rational.
