Short Answers - Number System Class 9 Notes | EduRev

Ques 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.

Ques 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.

Ques 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
⇒
⇒
⇒
⇒
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Ques 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.
 

Ques 5. Express  as a rational number.
 Solution: Let x = = 0.3333              ... ...(1)

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


Ques 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

Ques 7. Simplify: (√3 +√2)²
Solution: (√3 +)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

Ques 8. Rationalise the denominator of
Solution: Since RF of

Thus,  

Ques 9. Find 
Solution: Since 64 = 4 x 4 x 4 = 4³

Ques 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:  

Ques 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.

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

Ques 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.

Ques 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...

⇒ 990x = 243

⇒  

Thus  

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