Number systems
Class 9 Maths · Number systems · 34 questions with answers
The number system chapter is short, entirely rule-based, and among the most reliably scored in the year - which makes it a good place to start. Rationalising in particular is mechanical once a child sees that it is just the difference-of-squares identity from Class 8, used to remove a root from the bottom of a fraction.
- Decides whether a number is rational or irrational and gives the reason
- Rationalises a denominator using the conjugate
- Converts a recurring decimal into a fraction
- Applies the laws of exponents to fractional powers
Exercise 1 of 3 · 12 questions
Rational or irrational? Give the reason.
Write the answer · Warm-up
Here’s one done for you
√2 × √8 is rational, and that surprises most children. Combining the roots gives √16 = 4. The lesson is that a number is not irrational because it has a root sign in it - it is irrational because it cannot be written as a fraction. Simplify first, then judge.
- 1)√25
- 2)√7
- 3)0.3333...
- 4)π
- 5)2 + √3
- 6)√2 × √8
- 7)√16
- 8)√11
- 9)0.101001000100001...
- 10)22/7
- 11)√5 × √5
- 12)3 - √2
Answers
- 1) Rational - it equals 5, which can be written as 5/1.
- 2) Irrational - 7 is not a perfect square, so its root cannot be written as a fraction.
- 3) Rational - a recurring decimal, equal to 1/3.
- 4) Irrational - its decimal expansion is non-terminating and non-recurring. Note that 22/7 is an approximation, not π itself.
- 5) Irrational - a rational number added to an irrational one is always irrational.
- 6) Rational - it equals √16, which is 4.
- 7) Rational. √16 = 4, which can be written as 4/1.
- 8) Irrational. 11 is not a perfect square, so its root is non-terminating and non-repeating.
- 9) Irrational. The decimal never terminates and never settles into a repeating block.
- 10) Rational. It is a ratio of two integers, and it is an approximation of π rather than π itself.
- 11) Rational. √5 × √5 = 5.
- 12) Irrational. A rational number minus an irrational number is always irrational.
Exercise 2 of 3 · 10 questions
Rationalise the denominator.
Write the answer · Guided
Here’s one done for you
For 1/(2 + √3), multiply top and bottom by (2 - √3). The denominator becomes (2 + √3)(2 - √3) = 4 - 3 = 1, so the answer is simply 2 - √3. This is the Class 8 identity (a + b)(a - b) = a² - b² doing all the work. Once a child recognises that, rationalising stops being a separate technique and becomes an application of something they already know.
- 1)1/√5
- 2)1/(√7 - √6)
- 3)5/(√3 - 1)
- 4)1/(2 + √3)
- 5)7/(3 + 2√2)
- 6)1/√3
- 7)1/(√5 + √2)
- 8)3/(2 - √3)
- 9)1/(√7 + 2)
- 10)4/(3 - √5)
Answers
- 1) √5/5. Multiply top and bottom by √5.
- 2) √7 + √6. Multiply by the conjugate (√7 + √6); the denominator becomes 7 - 6 = 1.
- 3) 5(√3 + 1)/2. Multiply by (√3 + 1); the denominator becomes 3 - 1 = 2.
- 4) 2 - √3. Multiply by (2 - √3); the denominator becomes 4 - 3 = 1.
- 5) 21 - 14√2. Multiply by (3 - 2√2); the denominator becomes 9 - 8 = 1.
- 6) √3/3. Multiply top and bottom by √3.
- 7) (√5 - √2)/3. Multiply by the conjugate: the denominator becomes 5 - 2 = 3.
- 8) 3(2 + √3) = 6 + 3√3. The denominator becomes 4 - 3 = 1.
- 9) (√7 - 2)/3. The denominator becomes 7 - 4 = 3.
- 10) (3 + √5). The denominator becomes 9 - 5 = 4, and 4(3 + √5)/4 = 3 + √5.
Exercise 3 of 3 · 12 questions
Simplify, or convert as asked.
Write the answer · Practice
Here’s one done for you
To convert 0.474747... multiply by 100 rather than by 10, because two digits repeat. Then 100x - x = 47, so x = 47/99. The rule follows from that rather than needing to be memorised: multiply by 10 for one repeating digit, 100 for two, 1000 for three - enough to line the repeating part up so it cancels.
- 1)2^(2/3) × 2^(1/3)
- 2)(3^(1/5))^5
- 3)64^(1/2) × 64^(1/3)
- 4)Express 0.666... as a fraction.
- 5)Express 0.474747... as a fraction.
- 6)Simplify 1/(3 - √2) + 1/(3 + √2).
- 7)5^(1/2) × 5^(1/2)
- 8)(2^3)^(1/3)
- 9)27^(2/3)
- 10)Express 0.777... as a fraction.
- 11)Express 0.363636... as a fraction.
- 12)Simplify 1/(2 - √3) + 1/(2 + √3).
Answers
- 1) 2. Adding the powers gives 2^1.
- 2) 3. Multiplying the powers gives 3^1.
- 3) 32. That is 8 × 4, or equivalently 64^(5/6).
- 4) 2/3. Let x = 0.666...; then 10x = 6.666...; subtracting gives 9x = 6, so x = 2/3.
- 5) 47/99. Let x = 0.474747...; then 100x = 47.4747...; subtracting gives 99x = 47.
- 6) 6/7. The two fractions combine to give ((3 + √2) + (3 - √2))/(9 - 2) = 6/7.
- 7) 5. The powers add: 1/2 + 1/2 = 1.
- 8) 2. The powers multiply: 3 × 1/3 = 1.
- 9) 9. 27^(1/3) = 3, and 3² = 9.
- 10) 7/9. Let x = 0.777..., then 10x = 7.777..., so 9x = 7.
- 11) 36/99 = 4/11. Let x = 0.3636..., then 100x = 36.3636..., so 99x = 36.
- 12) 4. The two denominators are conjugates, so the terms become (2 + √3) + (2 - √3).
While your child works
- Simplify before judging whether a number is irrational. √2 × √8 looks irrational and is 4.
- Rationalising is just (a + b)(a - b) = a² - b² from Class 8. Say that once and the technique stops feeling new.
- This chapter is short and reliably scored. It is a good place to start the year, and a good place to regain confidence mid-year.