Short Answers - Number System Class 9 Notes | EduRev

Q.1. Find a rational number between 1 and 2.
Solution.
Let x = 1 and y = 2, then

Thus,  3/2 is a rational number between 1 and 2.

Q.2. Write a rational number equivalent to 5/9 such that its numerator is 25.
Solution. 

Thus, 25/45 is the required rational number whose numerator is 25.

Q.3. Find two rational numbers between 0.1 and 0.3.
Solution.
 Let x = 0.1, y = 0.3 and n = 2

∴ Two rational numbers between 0.1 and 0.3 are: x + d and x + 2d

Q.4. Express  in the form of decimal.

Solution. 
We have ,

Now, dividing 25 by 8, we have:
Since, the remainder is 0.
∴ The process of division terminates.
 

Q.5. Express  as a rational number.
Solution.

Multiplying by 100, we have 100x = 100 x 0.3333…
⇒ 100x = 33.3333                    … ...(2)
Subtracting (2) from (1), we have
100x – x = 33.3333… – 0.3333…
⇒ 99x = 33


Q.6. Simplify: (4+ √3) (4 −√3)
Solution. (4 +√3) (4 −√3) = (4)² – ( √3)²      [∵ (a + b)(a – b) = a² – b²]
= 16 – 3 = 13
Thus, (4 +√3) (4 −√3) = 13

Q.7. Simplify: (√3 +√2)²
Solution. 
(√3 +√2)² = (√3)² + √2 ( √3 ×2) + (√2)²      [∵ (a + b)² = a² + 2ab + b²]
= 3 + 2 √6 + 2 = 5 + 2 √6
Thus, (√3 +(√2)² = 5 + 2√6

Q.8. Rationalise the denominator of
Solution. 
Since RF of (√x - √y ) is (√x +√y )
∴ RF of (√3 - √2 ) is (√3 +√2 )
Now, we have

Thus,  


Q.9. Find 
Solution. 
Since
64 = 4 x 4 x 4 = 4³

Q.10. Define a non-terminating decimal and repeating decimals ?
Solution. The decimal expansion of some rational numbers do not have finite number of decimal places in their decimal parts, rather they have a repeating block of digits in decimal parts. Such decimal expansion is called non-terminating and repeating decimal.
Example:  

Q.11. What is the difference between "pure recurring decimals" and "mixed-recurring decimals."?
Solution. A decimal in which all the digits after the decimal point are repeated, is called a pure recurring decimal. A decimal in which at least one of the digits after the decimal point is not repeated and then some digit(s) repeated is called a mixed recurring decimal.
Example:  are pure recurring decimals.
 are mixed recurring decimals.

Q.12. What type of decimal expansion does an irrational number have?
Solution. The decimal expansion of an irrational number is "non-terminating non-recurring."

Q.13. Find a rational number lying between 
Solution.

Obviously x < y
A rational number lying between x and y

Hence, 7/20  is a rational number lying between

∴
∴  

∴ Three required numbers between 0 and 0.1 are: (x + d), (x + 2d) and (x + 3d)
Now

Thus, three rational numbers between 0 and 0.1 are: 0.025, 0.050 and 0.075.

Q.14. Express  as a fraction in the simplest form.
Solution. Let x =  = 0.24545         ... ...(1)
Then, multiplying (1) by 10,
we have 10x = 10 x 0.24545...
⇒ 10x = 2.4545                      ... ...(2)
Again multiplying (1) by 1000,
we get 1000 x x = 0.24545... x 1000
⇒ 1000x = 245.4545                ... ...(3)
Subtracting (2) from (3),
we have 1000x – 10x = 245.4545... – 2.4545...

Thus  


Q.15. If x = (2 +√5) , find the value of 
Solution. We have x = 2 + √5
∴
∴
Now