1
Properties of Numbers
Natural Numbers
Counting numbers.
N = {1,2,3,4,...}
Whole Numbers
W = {0,1,2,3,...}
Integers
Z = {...,-3,-2,-1,0,1,2,3,...}
Rational Numbers
p/q, q ≠ 0
Irrational Numbers
Numbers that cannot be expressed as p/q.
Examples: √2, √3, π
Real Numbers
Rational + irrational numbers.
R = Q ∪ Irrational
Important Properties
Commutative
a+b=b+a
a×b=b×a
Associative
(a+b)+c=a+(b+c)
(ab)c=a(bc)
Distributive
a(b+c)=ab+ac
Identity
a+0=a
a×1=a
Inverse
a + (-a)=0
a × 1/a = 1
Divisibility & Factors
- 2 → last digit is even.
- 3 → sum of digits divisible by 3.
- 5 → last digit 0 or 5.
- 9 → sum of digits divisible by 9.
- 10 → last digit 0.
- 11 → alternating digit sum is divisible by 11.
For two positive integers: HCF × LCM = Product of numbers
Indices / Exponents
aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ
a⁰ = 1
a⁻ⁿ = 1/aⁿ
a^(m/n) = ⁿ√(aᵐ)
2
Squares and Cubes
Square
n² = n × n
Examples: 1, 4, 9, 16, 25, 36...
Cube
n³ = n × n × n
Examples: 1, 8, 27, 64, 125...
Square Root
√(n²)=n
Cube Root
∛(n³)=n
Important Square Identities
(a+b)² = a² + 2ab + b²
(a-b)² = a² - 2ab + b²
a²-b² = (a-b)(a+b)
Important Cube Identities
(a+b)³ = a³+3a²b+3ab²+b³
(a-b)³ = a³-3a²b+3ab²-b³
a³+b³ = (a+b)(a²-ab+b²)
a³-b³ = (a-b)(a²+ab+b²)
Interesting Properties
- The square of an odd number is odd.
- The square of an even number is even.
- A perfect square cannot have an odd number of zeros at the end.
- Perfect cubes can be positive or negative.
- The sum of the first n odd numbers = n².
1+3+5+...+(2n-1)=n²
3
Major Algebraic Identities
Basic Identities
Identity 1
(a+b)²=a²+2ab+b²
Identity 2
(a-b)²=a²-2ab+b²
Identity 3
a²-b²=(a-b)(a+b)
Identity 4
(x+a)(x+b)=x²+(a+b)x+ab
Cubic Identities
(a+b)³=a³+b³+3ab(a+b)
(a-b)³=a³-b³-3ab(a-b)
a³+b³=(a+b)(a²-ab+b²)
a³-b³=(a-b)(a²+ab+b²)
Three-variable Identity
a³+b³+c³-3abc=(a+b+c)(a²+b²+c²-ab-bc-ca)
If a+b+c=0, then:
a³+b³+c³=3abc
Quadratic Equation
ax²+bx+c=0
x = [-b ± √(b²-4ac)] / 2a
Discriminant D = b²-4ac
- D > 0 → two distinct real roots
- D = 0 → equal roots
- D < 0 → no real roots
4
Major Geometry Properties
Lines and Angles
Straight angle = 180°
Complete angle = 360°
Complementary angles = 90°
Supplementary angles = 180°
Triangle
Angle Sum
A+B+C=180°
Exterior Angle
Exterior angle = sum of opposite interior angles
Area
A=½bh
Pythagoras
a²+b²=c²
Quadrilaterals
Sum of interior angles = 360°
| Shape | Important Property |
|---|---|
| Square | All sides equal, all angles 90° |
| Rectangle | Opposite sides equal, all angles 90° |
| Parallelogram | Opposite sides and angles equal |
| Rhombus | All sides equal |
| Trapezium | One pair of parallel sides |
Polygon
Sum of interior angles = (n-2) × 180°
Each interior angle of regular polygon = [(n-2)180°]/n
Number of diagonals = n(n-3)/2
Circle
Circumference = 2πr
Area = πr²
Diameter = 2r
- Angle in a semicircle = 90°.
- Equal chords subtend equal angles at the centre.
- The perpendicular from centre to a chord bisects the chord.
- Tangent is perpendicular to radius at point of contact.
5
Trigonometric Ratios & Properties
Six Trigonometric Ratios
| Ratio | Formula |
|---|---|
| sin θ | Perpendicular / Hypotenuse |
| cos θ | Base / Hypotenuse |
| tan θ | Perpendicular / Base |
| cosec θ | 1 / sin θ |
| sec θ | 1 / cos θ |
| cot θ | 1 / tan θ |
Fundamental Identities
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
Standard Values
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | undefined |
Compound Angles
sin(A+B)=sinA cosB + cosA sinB
sin(A-B)=sinA cosB - cosA sinB
cos(A+B)=cosA cosB - sinA sinB
cos(A-B)=cosA cosB + sinA sinB
Double Angles
sin2A=2sinA cosA
cos2A=cos²A-sin²A
cos2A=2cos²A-1
cos2A=1-2sin²A
tan2A=2tanA/(1-tan²A)
6
2D & 3D Area, Surface Area and Volume
2D Shapes
| Shape | Area | Perimeter |
|---|---|---|
| Square | a² | 4a |
| Rectangle | lb | 2(l+b) |
| Triangle | ½bh | a+b+c |
| Circle | πr² | 2πr |
| Parallelogram | bh | 2(a+b) |
| Trapezium | ½(a+b)h | a+b+c+d |
Heron's Formula
s=(a+b+c)/2
Area=√[s(s-a)(s-b)(s-c)]
3D Shapes
| Solid | Surface Area | Volume |
|---|---|---|
| Cube | 6a² | a³ |
| Cuboid | 2(lb+bh+hl) | lbh |
| Cylinder | 2πr(h+r) | πr²h |
| Cone | πr(l+r) | ⅓πr²h |
| Sphere | 4πr² | 4/3πr³ |
| Hemisphere | 3πr² | 2/3πr³ |
Cone slant height: l=√(r²+h²)
7
Statistics
Mean
Mean = Σx / n
Weighted Mean
Weighted Mean = Σwx / Σw
Median
Arrange observations in ascending or descending order.
Odd n → Median = (n+1)/2 th observation
Even n → Median = average of n/2 and (n/2+1) observations
Mode
The value occurring most frequently is the mode.
Grouped Data
Mean = Σfᵢxᵢ / Σfᵢ
Median = l + [(n/2 - cf)/f]h
Mode = l + [(f₁-f₀)/(2f₁-f₀-f₂)]h
Range
Range = Maximum - Minimum
Variance
Variance = Σ(x-x̄)²/n
Standard Deviation
σ = √[Σ(x-x̄)²/n]
Coefficient of Variation
CV = (σ/Mean) × 100
Mean Deviation
MD = Σ|x-A|/n
8
Major Graphs & Curves
Linear
y=mx+c
Quadratic
y=ax²+bx+c
Cubic
y=x³
Reciprocal
y=1/x
Important Functions
Linear → y=x
Quadratic → y=x²
Cubic → y=x³
Modulus → y=|x|
Reciprocal → y=1/x
Exponential → y=aˣ
Logarithmic → y=logₐx
Sine → y=sin x
Cosine → y=cos x
Tangent → y=tan x
Coordinate Geometry
Distance = √[(x₂-x₁)²+(y₂-y₁)²]
Midpoint = ((x₁+x₂)/2,(y₁+y₂)/2)
Slope m=(y₂-y₁)/(x₂-x₁)
Equation of line: y-y₁=m(x-x₁)
9
Differentiation & Integration
Differentiation
d/dx(c)=0
d/dx(xⁿ)=nxⁿ⁻¹
d/dx(eˣ)=eˣ
d/dx(aˣ)=aˣ ln(a)
d/dx(ln x)=1/x
d/dx(sin x)=cos x
d/dx(cos x)=-sin x
d/dx(tan x)=sec²x
d/dx(cot x)=-cosec²x
d/dx(sec x)=sec x tan x
d/dx(cosec x)=-cosec x cot x
Rules of Differentiation
Product: (uv)' = u'v + uv'
Quotient: (u/v)' = (vu'-uv')/v²
Chain Rule: d[f(g(x))]/dx = f'(g(x))g'(x)
Integration
∫xⁿ dx = xⁿ⁺¹/(n+1)+C, n≠-1
∫1/x dx = ln|x|+C
∫eˣ dx=eˣ+C
∫sin x dx=-cos x+C
∫cos x dx=sin x+C
∫sec²x dx=tan x+C
∫cosec²x dx=-cot x+C
Integration by Parts
∫u dv = uv - ∫v du
Fundamental Idea
Differentiation measures the rate of change.
Integration can represent accumulation and area under a curve.
Integration can represent accumulation and area under a curve.
10
Probability
Basic Probability
P(E)=Favourable outcomes / Total outcomes
0 ≤ P(E) ≤ 1
Complementary Event
P(E')=1-P(E)
Addition Rule
P(A∪B)=P(A)+P(B)-P(A∩B)
For mutually exclusive events:
P(A∪B)=P(A)+P(B)
Multiplication Rule
P(A∩B)=P(A)P(B|A)
For independent events:
P(A∩B)=P(A)P(B)
Conditional Probability
P(A|B)=P(A∩B)/P(B)
Bayes' Theorem
P(Aᵢ|B)=P(Aᵢ)P(B|Aᵢ) / ΣP(Aⱼ)P(B|Aⱼ)
Expected Value
E(X)=ΣxP(x)
11
Factorial, Permutation, Combination & Binomial
Factorial
n! = n(n-1)(n-2)...3×2×1
0! = 1
Permutation
ⁿPᵣ = n!/(n-r)!
Permutation is concerned with arrangements where order matters.
Combination
ⁿCᵣ = n!/[r!(n-r)!]
Combination is concerned with selection where order does not matter.
Important Properties
ⁿC₀ = ⁿCₙ = 1
ⁿCᵣ = ⁿCₙ₋ᵣ
ⁿPᵣ = ⁿCᵣ × r!
Binomial Theorem
(a+b)ⁿ = Σ [ⁿCᵣ aⁿ⁻ʳ bʳ]
General Term
Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ
Important Binomial Results
(a+b)²=a²+2ab+b²
(a+b)³=a³+3a²b+3ab²+b³
Sum of coefficients = 2ⁿ
Alternating sum of coefficients = 0
Pascal's Triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
★
Quick Mathematics Formula Reference
Pythagoras
a²+b²=c²
Circle Area
πr²
Circle Circumference
2πr
Quadratic Formula
(-b±√D)/2a
Distance
√[(x₂-x₁)²+(y₂-y₁)²]
Probability
Favourable/Total
Derivative
d(xⁿ)/dx=nxⁿ⁻¹
Integral
∫xⁿdx=xⁿ⁺¹/(n+1)+C
Combination
n!/[r!(n-r)!]
Permutation
n!/(n-r)!