Number Systems
CBSE Class 9 · Mathematics · Notes, formulas and practice questions
Revise rational and irrational numbers, real numbers, decimal expansions, rationalising denominators and laws of exponents for real numbers — the central ideas of CBSE Class 9 Number Systems.
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A number can be rational or irrational. A rational number can always be written as p/q with q ≠ 0, while irrational numbers such as √2 and √3 cannot. Together they form the real number system. The chapter begins by sorting numbers into these sets and by showing that every real number has a definite place on the number line. This classification matters because the same number can be represented as a fraction, as a decimal or as a point on the line, and you need to recognise when different forms actually refer to the same number.
Decimal expansions provide the clearest test of whether a number is rational. Rational numbers always end in a terminating decimal such as 2.125 or a recurring decimal such as 0.333... . Irrational numbers give a non-terminating, non-recurring expansion, like 1.414213... for √2. A recurring decimal can be changed into p/q by multiplying it by an appropriate power of 10 and subtracting, and a terminating decimal is easy to convert. The reverse direction is also useful: if a decimal never repeats, the number represented cannot be rational.
Once the numbers are understood, the chapter moves to operations. Surds behave like algebraic terms: like surds can be added or subtracted by their coefficients, and products like √a × √b = √(ab) help simplify radicals. An important operation is rationalising the denominator, where the conjugate of a surd expression is used to remove the root from the denominator. This is often needed before substituting approximate values or comparing expressions. Mastering this also builds algebraic fluency that will be used in later chapters.
The chapter closes by extending the laws of exponents, which students know for integers, to all real numbers. The rules for multiplying, dividing and raising a power to another power remain the same when the exponents are rational as long as the base is positive. For example, a square root is the same as the power one-half. Practice with simplified surds and exponent expressions builds speed; most errors come from forgetting to reduce a fraction before checking its decimal form or from applying an exponent law to the base when the bases are different. Careful revision should link definitions, decimal forms and algebraic rules.
Key terms
- Rational number
- A number that can be written as p/q, where p and q are integers and q ≠ 0. Every rational number has a decimal expansion that either terminates or becomes recurring; examples include 7, −3/4 and 0.333... .
- Irrational number
- A number that cannot be expressed in the form p/q with q ≠ 0. Its decimal expansion never terminates and never settles into a repeating pattern. Examples are √2 and π.
- Real number
- Any number that is rational or irrational. Together these fill the entire number line, so every real number has exactly one position on the line and each point of the line represents one real number.
- Terminating decimal expansion
- A decimal that ends after a finite number of digits, such as 2.125. A reduced rational number has a terminating decimal only when the denominator has no prime factors other than 2 and 5.
- Non-terminating recurring decimal expansion
- A decimal that continues forever with a digit or block of digits repeating, for example 0.6363... or 1.2727... . Such decimals always represent rational numbers and can be converted to p/q by subtraction.
- Surd
- An irrational number that is a root of a positive rational number, for example √2. A surd cannot be simplified to a rational number. However, not every irrational number is a surd — π is irrational but not a surd.
- Rationalising the denominator
- The process of rewriting a fraction such as 1/(√7 − √3) so that no radical remains in the denominator. You multiply numerator and denominator by a suitable conjugate, which changes the denominator to a whole number.
- Conjugate surd
- For a sum like √a + √b, the conjugate is √a − √b. Their product is a − b, a number without radicals, and this property makes conjugates useful in rationalising.
- Like surds
- Surds that contain the same irrational factor, for example 3√5 and 7√5. They can be combined by adding or subtracting their coefficients, exactly like terms in algebra.
Formula sheet
| What | Formula | Notes |
|---|---|---|
| First law of exponents | aᵐ × aⁿ = aᵐ⁺ⁿ | a is a positive real number; m and n are rational exponents. When multiplying the same base, add the exponents. |
| Second law of exponents | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | a is a positive real number; m and n are rational exponents. When dividing with the same base, subtract the exponents. |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | a is a positive real number; m and n are rational exponents. Raising a power to another power means multiplying the exponents. |
| Zero exponent | a⁰ = 1 | a is any nonzero real number. The expression 0⁰ is not defined in this chapter. |
| Multiplication of square roots | √a × √b = √(ab) | a and b are non-negative real numbers. This rule is used to simplify surds such as √18 = √(9 × 2). |
| Product of conjugate surds | (√a + √b)(√a − √b) = a − b | a and b are non-negative real numbers. Since the result has no radical sign, multiplying by the conjugate rationalises a surd denominator. |
Practice questions with answers
1. Which of these numbers are irrational: √9, π, 0.333..., √8? Give a reason for each.
√9 = 3, so it is rational. 0.333... = 1/3, so it is rational. π has a non-terminating, non-recurring decimal expansion, so it is irrational. √8 = 2√2, and since 8 is not a perfect square, √8 is irrational. Therefore π and √8 are the irrational numbers.
2. Without dividing, state whether the decimal expansion of 17/8 terminates, and write the expansion.
The denominator of the reduced fraction 17/8 is 8 = 2³, whose only prime factor is 2, so the decimal terminates. 17/8 = 2.125.
3. Express 1.272727... in the form p/q, where p and q are integers, q ≠ 0.
Let x = 1.272727... . Then 100x = 127.272727... . Subtracting the first equation gives 99x = 126, so x = 126/99 = 14/11.
4. Describe how to locate √3 on the number line.
Let O = 0 and A = 1 on the number line. At A draw a perpendicular AB of length 1 unit; then OB = √2 units. Mark C at distance OB from O, so C represents √2. At C draw a perpendicular CD of length 1 unit; then OD = √3 units. Draw an arc with centre O and radius OD to meet the number line at E; E represents √3.
5. Rationalise the denominator and simplify: 1/(√7 − √3).
Multiply the numerator and denominator by the conjugate √7 + √3. The denominator becomes (√7)² − (√3)² = 7 − 3 = 4. Hence the simplified value is (√7 + √3)/4.
6. Simplify using the laws of exponents: 5⁷ × 5³ ÷ 5⁸.
When multiplying powers of the same base, add the exponents, and when dividing, subtract them. The overall exponent is 7 + 3 − 8 = 2. Therefore the value is 5² = 25.
7. Give one irrational number lying between √2 and √3.
A number x lies between √2 and √3 exactly when 2 < x² < 3, so take x = √(5/2) = √10/2 ≈ 1.581. It is irrational because 10 is not a perfect square, which makes √10 irrational, and dividing an irrational number by the non-zero rational number 2 leaves it irrational. It lies between √2 ≈ 1.414 and √3 ≈ 1.732 as required. The decimal 1.5010010001… is another valid answer, since it is non-terminating and non-recurring.
8. Is the product of two irrational numbers always irrational? Give an example.
No, it is not always irrational. For instance, √2 and √8 are both irrational, but their product √2 × √8 = √16 = 4, which is rational. So the product depends on the particular numbers chosen.
9. Simplify: √18 + √50 − √8.
Simplify each surd first: √18 = 3√2, √50 = 5√2 and √8 = 2√2. Adding the like surds gives (3 + 5 − 2)√2 = 6√2.
10. If x = 2 + √3, find the value of x + 1/x.
Rationalise the reciprocal: 1/x = 1/(2 + √3). Multiply numerator and denominator by 2 − √3 to get (2 − √3)/(4 − 3) = 2 − √3. Then x + 1/x = (2 + √3) + (2 − √3) = 4.
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