Class 9 Maths Question Answers - Number System

Q1: Find five rational numbers between 1 and 2.
Sol:
We have to find five rational numbers between 1 and 2.
So, let us write the numbers with denominator 5 + 1 = 6
Thus, 6/6 = 1, 12/6 = 2
From this, we can write the five rational numbers between 6/6 and 12/6 as:
7/6, 8/6, 9/6, 10/6, 11/6

Q2: Locate √3 on the number line.
Sol:
Construct BD of unit length perpendicular to OB (here, OA = AB = 1 unit) as shown in the figure.
By Pythagoras theorem,
OD = √(2 + 1) = √3
Taking O as the centre and OD as radius, draw an arc which intersects the number line at the point Q using a compass.
Therefore, Q corresponds to the value of √3 on the number line.

Q3: Find the decimal expansions of 10/3, 7/8 and 1/7.
Sol:
Therefore, 10/3 = 3.3333…
7/8 = 0.875
1/7 = 0.1428571…

Q4: What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 1/17? Perform the division to check your answer.
Sol:
Thus, 1/17 = 0.0588235294117647….
Therefore, 1/17 has 16 digits in the repeating block of digits in the decimal expansion.

Q5: Visualise 3.765 on the number line, using successive magnification.
Sol:
Visualisation of 3.765 on the number line, using successive magnification is given below:

Q6: Simplify: (√3+√7) (√3-√7).
Sol:
(√3 + √7)(√3 – √7)
Using the identity (a + b)(a – b) = a² – b²,
(√3 + √7)(√3 – √7) = (√3)² – (√7)²
= 3 – 7
= -4

Q7: Find five rational numbers between 3/5 and 4/5.
Sol:
We have to find five rational numbers between 3/5 and 4/5.
So, let us write the given numbers by multiplying with 6/6, (here 6 = 5 + 1)
Now,
3/5 = (3/5) × (6/6) = 18/30
4/5 = (4/5) × (6/6) = 24/30
Thus, the required five rational numbers will be: 19/30, 20/30, 21/30, 22/30, 23/30

Q8: Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.
Sol:
No, since the square root of a positive integer 16 is equal to 4. Here, 4 is a rational number.

Q9: Find three different irrational numbers between the rational numbers 5/7 and 9/11.
Sol:
The given two rational numbers are 5/7 and 9/11.
5/7 = 0.714285714…..
9/11 = 0.81818181……
Hence, the three irrational numbers between 5/7 and 9/11 can be:
0.720720072000…
0.730730073000…
0.808008000…

Q10: Add 2√2+ 5√3 and √2 – 3√3.
Sol:
(2√2 + 5√3) + (√2 – 3√3)
= 2√2 + 5√3 + √2 – 3√3
= (2 + 1)√2 + (5 – 3)√3
= 3√2 + 2√3