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REAL NUMBERS
┌─────────────────────────┐
│ │
RATIONAL IRRATIONAL
(p/q form) (cannot be p/q)
┌────────────┐ √2, √3, π, e...
│ │
INTEGERS FRACTIONS
..-2,-1,0, 1/2, 3/4,
1,2,... -5/7...
│
NATURAL NUMBERS
1,2,3,4,...
| Property | Statement | Example | |----------|-----------|---------| | Closure (+ and ×) | Result is rational | 1/2 + 1/3 = 5/6 ✓ | | Commutative | a+b=b+a, a×b=b×a | 1/2+1/3 = 1/3+1/2 | | Associative | (a+b)+c = a+(b+c) | ✓ for + and × | | Additive identity | a + 0 = a | 3/4 + 0 = 3/4 | | Multiplicative identity | a × 1 = a | 3/4 × 1 = 3/4 | | Additive inverse | a + (−a) = 0 | 3/4 + (−3/4) = 0 | | Multiplicative inverse | a × (1/a) = 1 | 3/4 × 4/3 = 1 | | Distributive | a(b+c) = ab+ac | 2(1/2+1/3) = 2×1/2 + 2×1/3 |
Represent 3/4 on number line:
0 3/4 1
├────────●──────────┤
0 1/4 2/4 3/4 4/4
Divide unit interval into 4 equal parts
3rd mark = 3/4
EXAMPLE: 3x + 5 = 2x + 15
Method — Move variables to left, constants to right:
3x − 2x = 15 − 5 [transpose: +2x moves as −2x]
x = 10 [transpose: +5 moves as −5]
VERIFY: 3(10) + 5 = 2(10) + 15
35 = 35 ✓
5(2x − 3) = 3(3x − 2) + 1
Step 1: Expand brackets
10x − 15 = 9x − 6 + 1
Step 2: Simplify RHS
10x − 15 = 9x − 5
Step 3: Transpose
10x − 9x = −5 + 15
x = 10
VERIFY: LHS = 5(20−3) = 5×17 = 85
RHS = 3(30−2)+1 = 3×28+1 = 84+1 = 85 ✓
Problem: The sum of three consecutive integers is 51. Find them.
Let integers: n, n+1, n+2
Equation: n + (n+1) + (n+2) = 51
3n + 3 = 51
3n = 48
n = 16
Answer: 16, 17, 18
Check: 16+17+18 = 51 ✓
QUADRILATERAL
(4 sides, sum of angles = 360°)
│
┌─────────┴──────────┐
TRAPEZIUM PARALLELOGRAM
(1 pair parallel) (2 pairs parallel)
│
┌───────────────┼───────────────┐
RECTANGLE RHOMBUS SQUARE
(4 right angles) (all sides equal) (both!)
| Property | Rectangle | Rhombus | Square | Parallelogram | |----------|-----------|---------|--------|---------------| | Opposite sides equal | ✓ | ✓ | ✓ | ✓ | | All sides equal | ✗ | ✓ | ✓ | ✗ | | All angles 90° | ✓ | ✗ | ✓ | ✗ | | Diagonals equal | ✓ | ✗ | ✓ | ✗ | | Diagonals bisect each other | ✓ | ✓ | ✓ | ✓ | | Diagonals perpendicular | ✗ | ✓ | ✓ | ✗ |
Sum of interior angles of ANY quadrilateral = 360°
Proof using triangles:
A────────────B
│ △ABD │
│ ╲ │
│ ╲ │
│ ╲ │
D────────────C
△BCD
Draw diagonal BD → two triangles formed
Each triangle has angle sum 180°
So ABCD angle sum = 2 × 180° = 360°
If ∠A=110°, ∠B=70°, ∠C=80°, find ∠D:
110+70+80+∠D = 360
∠D = 360−260 = 100°
TRAPEZIUM:
a (parallel side)
┌────────────────┐
│╲ │
h │ ╲ │ h = perpendicular height
│ ╲ │
└───────────────────┘
b (parallel side, longer)
Area = ½ × (a + b) × h
Example: a=5cm, b=9cm, h=4cm
Area = ½ × (5+9) × 4 = ½ × 14 × 4 = 28 cm²
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
RHOMBUS (diagonals d₁ and d₂):
┌──────┐
/│ │╲
/ │ d₁ │ ╲
/ │ ───── │ ╲
╲ │ │ │ /
╲ │ d₂ │ /
╲ │ │/
╲└──────┘/
Area = ½ × d₁ × d₂
Example: d₁=10cm, d₂=6cm
Area = ½ × 10 × 6 = 30 cm²
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
GENERAL POLYGON — Split into triangles:
Divide polygon into triangles from one vertex
Area = Sum of all triangle areas
CUBOID:
l (length)
┌────────────┐
/│ /│
/ │ h / │ b (breadth)
/ │ / │
└───────────────┘ │
│ │ │ │
│ └────────────┤ /
│ / │/
└──────────────┘
Total Surface Area = 2(lb + bh + hl)
Volume = l × b × h
CUBE (all sides = a):
Surface Area = 6a²
Volume = a³
CYLINDER (radius r, height h):
Curved Surface Area = 2πrh
Total Surface Area = 2πr(r + h)
Volume = πr²h
Cylinder: r = 7 cm, h = 10 cm. Find TSA and Volume.
TSA = 2πr(r + h) = 2 × (22/7) × 7 × (7+10)
= 2 × 22 × 17 = 748 cm²
Volume = πr²h = (22/7) × 7 × 7 × 10
= 22 × 7 × 10 = 1540 cm³
Base a, exponents m and n:
Law 1: aᵐ × aⁿ = aᵐ⁺ⁿ (same base, ADD powers)
Example: 2³ × 2⁴ = 2⁷ = 128
Law 2: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (same base, SUBTRACT powers)
Example: 3⁵ ÷ 3² = 3³ = 27
Law 3: (aᵐ)ⁿ = aᵐˣⁿ (MULTIPLY powers)
Example: (2³)⁴ = 2¹²
Law 4: aᵐ × bᵐ = (ab)ᵐ (same power, combine bases)
Example: 2³ × 3³ = 6³ = 216
Law 5: a⁰ = 1 (any base, power 0 = 1)
Example: 5⁰ = 1, (−3)⁰ = 1
Law 6: a⁻ⁿ = 1/aⁿ (negative power = reciprocal)
Example: 2⁻³ = 1/2³ = 1/8
Standard Form: a × 10ⁿ, where 1 ≤ a < 10
Large numbers:
6,500,000 = 6.5 × 10⁶
Distance sun to earth: 1.5 × 10⁸ km
Small numbers:
0.000065 = 6.5 × 10⁻⁵
Size of bacteria: ~2 × 10⁻⁶ m
Converting:
Move decimal right → negative power
Move decimal left → positive power
DIRECT PROPORTION: x/y = k (constant)
If x doubles, y doubles
If x halves, y halves
Graph: Straight line through origin
y
│ /
│ /
│ /
│ /
│/
└──────── x
Example:
Speed(x): 20 40 60 80
Time(y): 5 10 15 20
Ratio x/y = 4 (constant) → DIRECT proportion
INVERSE PROPORTION: x × y = k (constant)
If x doubles, y halves
If x triples, y becomes one-third
Graph: Hyperbola (curved)
y
│╲
│ ╲
│ ╲
│ ╲────
└──────── x
Example:
Workers(x): 2 4 8 16
Days(y): 16 8 4 2
Product: 32 32 32 32 (constant) → INVERSE proportion
Direct Proportion Problem: If 15 workers can build a wall in 12 days, how many days will 20 workers take?
Workers ↑ → Days ↓ (INVERSE proportion)
15 × 12 = 20 × x
180 = 20x
x = 9 days
METHOD 1 — Common Factor:
6x²y + 9xy² + 12xy
= 3xy(2x + 3y + 4) [HCF of 6,9,12 = 3; common = xy]
METHOD 2 — Regrouping:
ac + ad + bc + bd
= a(c+d) + b(c+d)
= (a+b)(c+d)
METHOD 3 — Standard Identities:
a² − b² = (a+b)(a−b) [Difference of squares]
a² + 2ab + b² = (a+b)² [Perfect square]
a² − 2ab + b² = (a−b)² [Perfect square]
Find: (x+3)²
= x² + 2(x)(3) + 3²
= x² + 6x + 9
Find: (2a−5b)²
= (2a)² − 2(2a)(5b) + (5b)²
= 4a² − 20ab + 25b²
Find: (3x+4)(3x−4)
= (3x)² − 4² = 9x² − 16
Y-axis
│
II │ I
(−,+) │ (+,+)
│
──────────┼────────── X-axis
│
III │ IV
(−,−) │ (+,−)
│
Point (3, 4):
Start at origin (0,0)
Move 3 units RIGHT (x=3)
Move 4 units UP (y=4)
Mark the point ●
Quadrant Signs:
I: (+, +) II: (−, +)
III: (−, −) IV: (+, −)
LINE GRAPH — shows change over time:
Temperature vs Time
Temp │ ╱╲
30 │ ╱ ╲
25 │ ╱ ╲──────
20 │╱
└─────────────────► Time
Mon Tue Wed Thu Fri
PIE CHART — shows proportion:
Favourite Sports:
Cricket: 40% → (40/100)×360° = 144°
Football: 30% → 108°
Others: 30% → 108°
BAR GRAPH — comparing quantities
HISTOGRAM — bar graph for continuous data (no gaps)
Class 8 Maths NCERT complete notes — rational numbers, linear equations, quadrilaterals, practical geometry, mensuration, algebraic expressions, exponents, direct/inverse proportion, factorisation, graphs, introduction to graphs.
42 pages · 1.5 MB · Updated 2026-03-11
A rational number is any number that can be expressed as p/q where p and q are integers and q ≠ 0. Examples: 3/4, −5/7, 0, 3 (=3/1), −2. They can be positive, negative, or zero. Between any two rational numbers, infinitely many rational numbers exist.
Area of trapezium = ½ × (sum of parallel sides) × height = ½ × (a + b) × h, where a and b are the two parallel sides and h is the perpendicular height between them.
Direct proportion: when one quantity increases, other also increases in same ratio. Example: more goods → more cost. Inverse proportion: when one quantity increases, other decreases. Example: more workers → less time to finish work. In direct: x/y = k (constant). In inverse: xy = k (constant).
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