Number Systems
The number system organises every number we use into neat families. This chapter sorts numbers into rational and irrational, builds up to the real numbers, looks at how their decimal expansions behave, and shows how to represent them on the number line, apply the laws of exponents, and rationalise denominators. It is the foundation of all of Class 9 algebra.
Learning objectives
- Distinguish rational and irrational numbers.
- Recognise terminating and non-terminating decimal expansions.
- Represent real numbers on the number line.
- Apply the laws of exponents for real numbers.
- Rationalise the denominator of a surd.
Key concepts
Rational and irrational numbers
A rational number can be written as p/q where p and q are integers and q โ 0 (e.g. 3/4, โ2, 0.5). An irrational number cannot (e.g. โ2, โ3, ฯ). Together they form the real numbers, which fill the entire number line.
Decimal expansions
A rational number has a decimal that either terminates (0.75) or is non-terminating but recurring (0.333โฆ). An irrational number has a decimal that is non-terminating and non-recurring.
Laws of exponents
For real numbers and rational powers: aแต ร aโฟ = aแตโบโฟ, aแต รท aโฟ = aแตโปโฟ, (aแต)โฟ = aแตโฟ, aแต ร bแต = (ab)แต, and aโฐ = 1. These also apply to roots written as fractional powers.
Rationalising the denominator
To remove a surd from a denominator, multiply the numerator and denominator by a suitable surd (the conjugate for sums). For example, 1/โ2 = โ2/2, and 1/(โ3 + 1) is multiplied by (โ3 โ 1).
Important formulas
Product of surds
โa ร โb = โ(ab)
Laws of exponents
aแตยทaโฟ = aแตโบโฟ ; (aแต)โฟ = aแตโฟ ; aโฐ = 1
Rationalising
1/โa = โa / a
Conjugate (sum)
1/(โa + b) ร (โa โ b)/(โa โ b)
Key definitions
- Rational number
- A number that can be written as p/q with integers p, q and q โ 0.
- Irrational number
- A number that cannot be written as p/q (e.g. โ2, ฯ).
- Real number
- Any rational or irrational number; together they form the number line.
- Surd
- An irrational root such as โ2 or โ5 that cannot be simplified to a rational number.
Solved examples
Q1. Is 0.333โฆ rational? If so, express it as a fraction.
Solution: Yes. Let x = 0.333โฆ, then 10x = 3.333โฆ; subtracting, 9x = 3, so x = 1/3. A recurring decimal is rational.
Q2. Rationalise the denominator of 1/โ2.
Solution: Multiply top and bottom by โ2: (1 ร โ2)/(โ2 ร โ2) = โ2/2.
Q3. Simplify 2ยณ ร 2ยฒ.
Solution: Using aแต ร aโฟ = aแตโบโฟ: 2ยณโบยฒ = 2โต = 32.
Common mistakes to avoid
- Thinking every square root is irrational โ โ4 = 2 is rational.
- Treating a non-terminating recurring decimal as irrational (it is rational).
- Adding exponents when multiplying different bases.
- Forgetting to multiply the numerator too while rationalising.
Number Systems โ MCQ Quiz
11 questions with instant feedback. Use number keys 1โ4 to answer.
Which of these is irrational?
Practice questions
Short answer
Give an example of a rational and an irrational number between 1 and 2.
Rational: 1.5 (= 3/2); irrational: โ2 โ 1.414.
Is the product of a non-zero rational and an irrational number rational or irrational?
Irrational (e.g. 2 ร โ3 = 2โ3).
Express 0.6 (recurring) as a fraction.
Let x = 0.666โฆ, 10x = 6.666โฆ, so 9x = 6, x = 2/3.
Long answer
Rationalise the denominator of 1/(โ5 โ โ2).
Multiply numerator and denominator by the conjugate (โ5 + โ2): (โ5 + โ2)/[(โ5 โ โ2)(โ5 + โ2)] = (โ5 + โ2)/(5 โ 2) = (โ5 + โ2)/3.
Show that 1.272727โฆ is a rational number.
Let x = 1.272727โฆ. Then 100x = 127.2727โฆ. Subtracting, 99x = 126, so x = 126/99 = 14/11, which is of the form p/q โ hence rational.
HOTS (Higher Order Thinking)
Insert two irrational numbers between 2 and 3.
For example, โ5 (โ 2.236) and โ7 (โ 2.645) both lie between 2 and 3 and are irrational.
Is โ2 + โ3 rational or irrational? Explain.
It is irrational. If it were rational, squaring would force โ6 to be rational, which it is not โ so โ2 + โ3 cannot be rational.
Quick revision
Revision notes
- Rational = p/q (q โ 0); irrational cannot be written that way.
- Decimals: terminating/recurring โ rational; non-terminating non-recurring โ irrational.
- Laws of exponents apply to real (and fractional) powers.
- Rationalise by multiplying with a surd or conjugate.
Key takeaways
- Not all roots are irrational โ check if they simplify.
- Recurring decimals are rational.
- Conjugates clear surds from denominators.
Frequently asked questions
What is the difference between rational and irrational numbers?
Rational numbers can be written as p/q (q โ 0); irrational numbers cannot, and their decimals are non-terminating and non-recurring.
How do I rationalise a denominator?
Multiply the numerator and denominator by a suitable surd โ the same surd for a single root, or the conjugate for a sum or difference.
Is ฯ rational?
No, ฯ is irrational; 22/7 is only an approximation, not its exact value.