Number Systems
Complete theory, solved examples, properties, surds, exponents & practice exercises
📋 Contents
- Classification of Numbers
- Identifying Prime Numbers
- Representing Rational Numbers on the Number Line
- Rational Numbers in Decimal Form
- Properties of Rational Numbers
- Irrational Numbers & Properties
- Rationalization Factor
- Laws of Radicals / Exponents
- Representing Irrational Numbers on the Number Line
- Magnifying Glass Method
- Conversion of Repeating Decimals to Rational Numbers
- Surds – Definition & Laws
- Operations on Surds
- Rationalization of Surds
- Exponents of Real Numbers
- Solved Illustrations
- Exercise – 1 (MCQ)
- Exercise – 2 (Subjective)
1. Classification of Numbers
N = {1, 2, 3, 4, …}
W = {0, 1, 2, 3, 4, …}
I = {…, −3, −2, −1, 0, 1, 2, 3, …}
e.g. 2/3, 37/15, −17/19.
All natural numbers, whole numbers, and integers are rational. Rational numbers include integers (no decimal part), terminating fractions (e.g. 0.75, −0.02), and non-terminating but recurring decimals (e.g. 0.666…, −2.333…).
Types of Fractions
| Type | Description | Example |
|---|---|---|
| Common fraction | Denominator is not 10 or a power of 10 | 3/7 |
| Decimal fraction | Denominator is 10 or a power of 10 | 3/10, 7/100 |
| Proper fraction | Numerator < Denominator | 3/7 |
| Improper fraction | Numerator > Denominator | 7/3 |
| Mixed fraction | Integral part + fractional part | 2⅓ |
| Compound fraction | Numerator & denominator are themselves fractions | (1/2)/(3/4) |
e.g. √2 = 1.41421356…, √3 = 1.73205…, π = 3.14159…
P = {2, 3, 5, 7, 11, 13, 17, 19, 23, …}
C = {4, 6, 8, 9, 10, 12, …}
e.g. 4 and 9 are co-prime because HCF(4, 9) = 1.
E = {…, −4, −2, 0, 2, 4, …}
O = {…, −5, −3, −1, 1, 3, 5, …}
e.g. 3i, 4i, −5i
The set of complex numbers is the superset of all other sets of numbers.
Hierarchy of Number Systems
Pictorial hierarchy (outermost = largest set):
2. Identifying Prime Numbers
Method (Two Steps)
- Step 1: Find the approximate square root of the given number.
- Step 2: Divide the given number by every prime number less than that square root. If the number is not divisible by any of them, it is prime; otherwise, it is composite.
Approximate square root of 571 ≈ 24.
Prime numbers less than 24: 2, 3, 5, 7, 11, 13, 17, 19, 23.
571 is not divisible by any of these, therefore 571 is a prime number.
1 is neither prime nor composite.
3. Representing Rational Numbers on the Number Line
To locate a fraction such as 3/7 on the number line, divide the unit interval (from 0 to 1) into 7 equal parts. The 3rd part from 0 represents 3/7.
Number Line – Locating 3/7
013/7Finding n Rational Numbers Between a and b
Compute the common difference d = (b − a) / (n + 1). The required rational numbers are:
Here a = 2, b = 3, n = 4.
d = (3 − 2) / (4 + 1) = 1/5.
The four rational numbers are: 11/5, 12/5, 13/5, 14/5 (i.e. 2.2, 2.4, 2.6, 2.8).
From a < b:
• Adding a to both sides: 2a < a + b, so (a+b)/2 > a.
• Adding b to both sides: a + b < 2b, so (a+b)/2 < b.
Thus the 1st rational number is m₁ = (a+b)/2.
2nd: m₂ = (a + m₁)/2, and 3rd: m₃ = (m₁ + b)/2.
4. Rational Numbers in Decimal Representation
e.g. 1/2 = 0.5, 11/16 = 0.6875, 3/20 = 0.15
e.g. 1/3 = 0.333… = 0.3̄ | 1/7 = 0.142857142857… | 5/6 = 0.8333…
5. Properties of Rational Numbers
For rational numbers a, b, c:
| # | Property | Statement |
|---|---|---|
| i | Commutative – Addition | a + b = b + a |
| ii | Associative – Addition | (a + b) + c = a + (b + c) |
| iii | Additive Inverse | a + (−a) = 0 (0 is identity; −a is additive inverse) |
| iv | Commutative – Multiplication | a · b = b · a |
| v | Associative – Multiplication | (a · b) · c = a · (b · c) |
| vi | Multiplicative Inverse | a · (1/a) = 1 (1 is identity; 1/a is multiplicative inverse) |
| vii | Distributive Property | a · (b + c) = a·b + a·c |
6. Irrational Numbers & Their Properties
The square root of any prime number is also irrational: √2, √3, √5, √7, …
Decimal Values of Common Irrationals
√3 = 1.73205080… (non-recurring, non-terminating)
π = 3.14159265… (non-recurring, non-terminating)
Properties of Irrational Numbers
| S.No. | Property | Illustration |
|---|---|---|
| 1 | Sum of two irrationals need not be irrational | √2 + (−√2) = 0 (rational) |
| 2 | Difference of two irrationals need not be irrational | √3 − √3 = 0 (rational) |
| 3 | Product of two irrationals need not be irrational | √2 × √2 = 2 (rational) |
| 4 | Quotient of two irrationals need not be irrational | √8 ÷ √2 = √4 = 2 (rational) |
| 5 | Sum of a rational and an irrational is irrational | 5 + √3 is irrational |
| 6 | Difference of a rational and an irrational is irrational | 5 − √3 is irrational |
| 7 | Product of a non-zero rational & irrational is irrational | 5 × √3 = 5√3 is irrational |
| 8 | Quotient of a rational and an irrational is irrational | 2 ÷ √3 = 2/√3 is irrational |
Assume 2 + √3 is rational. Then √3 = (2 + √3) − 2 would also be rational (since difference of two rationals is rational). But √3 is known to be irrational — contradiction! ∴ 2 + √3 is irrational. ∎
Let √3 + √2 = r (rational). Squaring: 3 + 2√6 + 2 = r²,
so 5 + 2√6 = r², ∴ √6 = (r² − 5)/2.
Since r is rational, (r² − 5)/2 is rational. But √6 is irrational — contradiction! ∎
Method 1: √5 ≈ 2.236 (between 2 and 2.5) and √(21/4) = √5.25 ≈ 2.291 are both irrational.
Method 2: Non-terminating, non-repeating numbers like 2.101001000100001… and 2.201001000100001… lie between 2 and 2.5.
0.1201001000100001… and 0.1202002000200002… (non-terminating, non-recurring).
7. Rationalization Factor (RF)
For integers a, b and natural numbers x, y:
- √x and √x are RF of each other: √x · √x = x (rational).
- (a + √x) and (a − √x) are RF of each other: their product = a² − x (rational).
- (√x + √y) and (√x − √y) are RF of each other: their product = x − y (rational).
(i) 1/√5 × √5/√5 = √5/5
(ii) 1/(3+√2) × (3−√2)/(3−√2) = (3−√2)/(9−2) = (3−√2)/7
(iii) 1/(√5−√3) × (√5+√3)/(√5+√3) = (√5+√3)/(5−3) = (√5+√3)/2
8. Laws of Radicals (Exponents)
Let a > 0 be a real number and p, q be rational numbers:
| # | Law | Description |
|---|---|---|
| 1 | aᵖ · aᵍ = aᵖ⁺ᵍ | Multiply powers with same base → add exponents |
| 2 | aᵖ ÷ aᵍ = aᵖ⁻ᵍ | Divide powers with same base → subtract exponents |
| 3 | (aᵖ)ᵍ = aᵖᵍ | Power of a power → multiply exponents |
| 4 | (ab)ᵖ = aᵖ · bᵖ | Power distributes over product |
9. Representing Irrational Numbers on the Number Line
Representing √2 on the Number Line
We use the Pythagorean theorem geometrically:
- Draw a unit length OA = 1 on the number line (from origin O to point A).
- At A, draw AB perpendicular to OA with AB = 1 unit.
- By Pythagoras: OB = √(OA² + AB²) = √(1² + 1²) = √2.
- With O as centre and OB as radius, draw an arc intersecting the number line at P. Point P represents √2.
Representing √3 on the Number Line (continuing)
- On the hypotenuse OB (= √2) of right-angled △AOB, draw BC ⊥ OB with BC = 1 unit.
- Join OC. By Pythagoras: OC = √(OB² + BC²) = √(2 + 1) = √3.
- With O as centre and OC as radius, draw an arc intersecting the number line at P. Point P represents √3.
Construction diagram for √2 and √3 on the number line
OAB√2√2C√3√3Finding √x on the Number Line (General Method)
- Draw a line and mark point A. Mark point B such that AB = x units.
- Extend to point P such that BP = 1 unit.
- Find midpoint O of AP.
- Draw a semicircle with centre O and radius OP.
- Draw a perpendicular at B intersecting the semicircle at D. Then BD = √x.
- With B as centre and BD as radius, mark arc on the number line at E. BE = √x.
10. Visualising Numbers on the Number Line (Magnifying Glass Method)
- Identify the two consecutive integers between which the number lies.
- Divide that unit interval into 10 equal parts to identify the tenths digit.
- Zoom into the relevant tenth-interval and divide again into 10 equal parts to identify the hundredths digit, and so on.
Step 1: 5 < 5.377… < 6 → lies between 5 and 6.
Step 2: 5.3 < 5.377… < 5.4 → lies between 5.3 and 5.4.
Step 3: 5.37 < 5.377… < 5.38 → lies between 5.37 and 5.38.
Step 4: 5.377 < 5.377… < 5.378 → lies between 5.377 and 5.378.
By successive magnification we can pinpoint the position with increasing accuracy.
11. Converting Repeating Decimals to Rational Form p/q
Let x = 0.474747… …(1)
Multiply by 10: 10x = 4.74747… …(2)
Multiply again by 10: 100x = 47.4747… …(3)
Subtract (2) from (3): 90x = 43, so x = 43/90.
Let x = 0.6666… …(1)
10x = 6.6666… …(2)
Subtracting (1) from (2): 9x = 6 → x = 6/9 = 2/3.
12. Surds – Definition & Laws
• a = radicand (must be a rational number),
• n = order (index) of the surd,
• √ is the radical sign.
ⁿ√a = a^(1/n)
Some Examples of Surds
- √2 is a surd (radicand 2 is rational and result is irrational).
- √(3 + √2) is a surd (sum of rational + irrational gives a surd).
- ∛7 is a surd.
Expressions That Are NOT Surds
- √9 = 3 (result is rational, so NOT a surd).
- ∛(2+√3) — radicand itself is irrational, so NOT a surd by definition.
Basic Laws of Surds
| # | Law | Example |
|---|---|---|
| i | (ⁿ√a)ⁿ = a | (√5)² = 5 |
| ii | ⁿ√a · ⁿ√b = ⁿ√(ab) [same order] | √2 · √3 = √6 |
| iii | ⁿ√a / ⁿ√b = ⁿ√(a/b) | √8 / √2 = √4 = 2 |
| iv | ᵐ√(ⁿ√a) = ᵐⁿ√a | ²√(³√a) = ⁶√a |
| v | ⁿ√(aᵐ) = a^(m/n) [changing order] | √3 = ⁶√(3³) = ⁶√27 |
13. Operations on Surds
Addition and Subtraction
Only surds with the same order and same radicand (like surds) can be added or subtracted.
= (15 − 6 + 4)√2 = 13√2
Multiplication and Division
LCM of orders 3 and 2 = 6.
³√4 = ⁶√(4²) = ⁶√16 | ²√3 = ⁶√(3³) = ⁶√27
Product = ⁶√(16 × 27) = ⁶√432
Comparison of Surds
If x > y > 0 and n > 1, then ⁿ√x > ⁿ√y. To compare surds of different orders, convert to the same order using LCM.
LCM of 2 and 3 = 6.
²√3 = ⁶√(3³) = ⁶√27 | ³√5 = ⁶√(5²) = ⁶√25
Since 27 > 25, ∴ ²√3 > ³√5.
14. Rationalization of Surds
| Expression | Rationalizing Factor (RF) | Product (rational) |
|---|---|---|
| √a | √a | a |
| a + √b | a − √b | a² − b |
| √a + √b | √a − √b | a − b |
| √a + √b | √a − √b (conjugate) | a − b |
| a − √b | a + √b | a² − b |
15. Exponents of Real Numbers
aⁿ = a × a × a × … (n times)
e.g. 2³ = 2 × 2 × 2 = 8
a⁻ⁿ = 1/aⁿ
a^(p/q) = (aᵖ)^(1/q) = ᵍ√(aᵖ)
Written as a^(1/n) or ⁿ√a.
Remark: If a < 0 and n is even, the principal nth root is NOT defined in the real number system. e.g. (−9)^(1/2) is not a real number.
Laws of Rational Exponents
For positive reals a, b and rational exponents m, n:
| # | Law | Description |
|---|---|---|
| i | aᵐ × aⁿ = aᵐ⁺ⁿ | Same base, multiply → add exponents |
| ii | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | Same base, divide → subtract exponents |
| iii | (aᵐ)ⁿ = aᵐⁿ | Power of a power → multiply |
| iv | (ab)ᵐ = aᵐ · bᵐ | Power distributes over product |
| v | a^(m/n) = (aᵐ)^(1/n) = (a^(1/n))ᵐ | Fractional exponent |
| vi | (a/b)ᵐ = aᵐ/bᵐ | Power distributes over quotient |
| vii | a⁻ᵐ = 1/aᵐ | Negative exponent = reciprocal |
16. Solved Illustrations
(i) 0.15 = 15/100 = 3/20
(ii) 0.675 = 675/1000 = 27/40
(iii) 0.00026 = 26/100000 = 13/50000
Let x = 0.4777… → 10x = 4.777… → 100x = 47.777…
100x − 10x = 43 → 90x = 43 → x = 43/90
Suppose √2 = p/q in lowest terms (p, q integers, no common factor, q ≠ 0).
Squaring: 2 = p²/q² → 2q² = p² → p² is even → p is even (say p = 2m).
Then 2q² = 4m² → q² = 2m² → q is even.
Both p and q are even — contradicts "no common factor". ∴ √2 is irrational. ∎
Rational: (2+3)/2 = 2.5 = 5/2.
Irrational: √(2 × 3) = √6 ≈ 2.449 (lies between 2 and 3).
5² × 5⁴ = 5²⁺⁴ = 5⁶ = 15625
5⁸ ÷ 5³ = 5⁸⁻³ = 5⁵ = 3125
√2 · √3 = √6. Then √6 · √6 = 6. ∎
Step I: Draw line, mark A. Mark B with AB = 9.3 cm.
Step II: Extend to C with BC = 1 unit.
Step III: Find midpoint O of AC.
Step IV: Semicircle with centre O and radius OC.
Step V: Perpendicular at B meets semicircle at D. BD = √(9.3).
Step VI: Arc with centre B and radius BD meets number line at E. BE = √(9.3). ✓
Instant evaluation, detailed solutions & performance analytics
Q1. Which of the following is an irrational number?
Q2. Every rational number is also:
Q3. The value of 1.999… in the form p/q is:
Q4. Between any two distinct rational numbers:
Q5. √2 is:
Q6. The product of a non-zero rational and an irrational number is:
Q7. Which of the following is a terminating decimal?
Q8. The value of 5x−3 · 32x−8 = 225, x is:
Q9. The rationalizing factor of (2 + √3) is:
Q10. Which is the largest in {√2, ∛3, ⁴√5}?
Q1 → (c) | Q2 → (c) | Q3 → (c) | Q4 → (c) | Q5 → (c) | Q6 → (b) | Q7 → (c) | Q8 → (d) | Q9 → (b) | Q10 → (c)
(i) The sum of two rational numbers is rational. [True]
(ii) The sum of two irrational numbers is always irrational. [False]
(iii) The product of two rational numbers is rational. [True]
(iv) The product of two irrational numbers is always irrational. [False]
(v) The sum of a rational and an irrational is irrational. [True]
(vi) The product of a non-zero rational and an irrational is rational. [False]
(vii) Every real number is rational. [False]
(viii) Every real number is either rational or irrational. [True]
Q7 → When we measure c or d with physical tools, we get only approximate rational values; the actual c and d include irrational lengths, so there's no contradiction.
Q8 → (i) irrational (ii) rational (=2) (iii) irrational
Q10 → p = 7/2, q = 1/2
Q11 → (2√3)² − (√5)² = 12 − 5 = 7
Q12 → (i) √7 + √6 (ii) (√5 − √2)/3
FAQs on CBSE Class 9 Maths Notes – Number Systems
The Number System chapter in CBSE Class 9 Maths introduces students to different types of numbers used in mathematics. It covers natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. This chapter is important because it forms the foundation for algebra, geometry, and higher mathematical concepts. Students frequently search for Number Systems notes, real numbers, irrational numbers, rational numbers examples, and Class 9 Maths important questions. The chapter explains how numbers are classified and represented on a number line. It also helps students understand decimal expansions and the relationship between different sets of numbers. A strong understanding of Number Systems improves mathematical reasoning and problem-solving abilities, making it easier to learn advanced topics in future classes and perform well in examinations.
Students can prepare effectively for Number Systems by first understanding the classification of numbers and their properties. They should learn the definitions of natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers thoroughly. Frequently searched keywords include Number Systems formulas, laws of exponents, NCERT solutions, important questions, square roots, and chapter revision notes. Students should practice questions involving decimal expansions, representation of numbers on the number line, and operations on real numbers. Special attention should be given to the laws of exponents because they are commonly asked in examinations. Making short notes of important definitions and properties can help during revision. Regular practice of NCERT exercises and sample questions strengthens conceptual understanding and helps students score high marks in the Number Systems chapter.
One of the most searched topics in Class 9 Number Systems is the difference between rational and irrational numbers. Rational numbers are numbers that can be expressed in the form p/q, where q is not equal to zero and p and q are integers. Their decimal expansions are either terminating or non-terminating recurring. Irrational numbers cannot be expressed in the form p/q and have non-terminating, non-recurring decimal expansions. Examples include √2, √3, and π. Popular search keywords include rational numbers definition, irrational numbers examples, decimal expansion, real numbers, and number line representation. Understanding the distinction between these numbers is essential because together they form the set of real numbers. This concept is frequently tested in school examinations and serves as a foundation for higher mathematical studies.