Class 9 · Maths

Number System

The number system organises all the numbers we use — from natural numbers to real numbers. This chapter covers rational and irrational numbers, their decimal expansions, representing numbers on the number line, rationalising denominators and the laws of exponents for real numbers.

Solutions

Exercise 1.1

Q1. Exercise 1.1 — Question 1

Is every whole number a rational number? Justify your answer with an example.

A rational number is any number that can be written as p/q where p and q are integers and q ≠ 0. Take any whole number, say 7. It can be written as 7/1, where p = 7 and q = 1 (q ≠ 0). The same works for every whole number, including 0 = 0/1. So every whole number satisfies the definition of a rational number.

Formula: Rational number = p/q, where p, q are integers and q ≠ 0

Final Answer: Yes — every whole number n can be written as n/1, so it is rational.

Common Mistake: Writing q = 0 in the form p/q. The denominator can never be zero.

Exam Tip: Always justify by actually writing the number in p/q form — one line earns the mark.

Exercise 1.2

Q1. Exercise 1.2 — Question 1

Explain how √2 can be located on the number line using a right-angled triangle.

Draw a number line and mark O at 0 and A at 1, so OA = 1 unit. At A, draw AB perpendicular to the number line with AB = 1 unit. Join OB. By Pythagoras' theorem, OB² = OA² + AB² = 1² + 1² = 2, so OB = √2. Now place the compass at O with radius OB and draw an arc that cuts the number line at P. The point P represents √2 on the number line.

Formula: Pythagoras: hypotenuse² = base² + height²

Final Answer: Construct a right triangle with both legs 1 unit; its hypotenuse √2 is transferred to the line with a compass arc.

Common Mistake: Measuring √2 with a ruler instead of transferring the exact length with a compass.

Exam Tip: Label O, A, B and P clearly — construction marks carry marks in the board exam.

Exercise 1.3

Q1. Exercise 1.3 — Question 1

Convert the recurring decimal 0.3333… into the form p/q.

Let x = 0.3333… … (i) Only one digit repeats, so multiply both sides by 10: 10x = 3.3333… … (ii) Subtract (i) from (ii): 10x − x = 3.3333… − 0.3333… 9x = 3 x = 3/9 = 1/3

Final Answer: 0.3333… = 1/3

Common Mistake: Multiplying by 100 when only one digit repeats — the repeating parts then do not cancel.

Exam Tip: Multiply by 10 for one repeating digit, 100 for two, and so on.

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