Complete Math Book

Chapter 1: Number Systems

Numbers are the foundation of mathematics. Over time, mathematicians have developed different types of numbers to solve increasingly complex problems.

1.1 Types of Numbers

Type	Symbol	Description	Examples
Natural Numbers	ℕ	Counting numbers starting from 1	1, 2, 3, 4, ...
Whole Numbers	𝕎	Natural numbers including zero	0, 1, 2, 3, ...
Integers	ℤ	Positive, negative, and zero	..., -2, -1, 0, 1, 2, ...
Rational Numbers	ℚ	Numbers expressible as p/q	1/2, 3/4, -5/3, 0.75
Irrational Numbers	𝕀	Cannot be expressed as p/q	√2, π, e
Real Numbers	ℝ	All rational and irrational numbers	All of the above
Complex Numbers	ℂ	Real + imaginary part	3 + 2i, -1 + i

1.2 Properties of Real Numbers

Commutative: a + b = b + a and a × b = b × a
Associative: (a + b) + c = a + (b + c)
Distributive: a(b + c) = ab + ac
Identity: a + 0 = a and a × 1 = a
Inverse: a + (−a) = 0 and a × (1/a) = 1

1.3 Number Line

The number line is a visual representation of all real numbers arranged in order from left (negative) to right (positive), with zero in the center.

1.4 Absolute Value

Definition: |x| = x if x ≥ 0
|x| = −x if x < 0

Example:

|−5| = 5 |3| = 3 |0| = 0

Chapter 2: Arithmetic

Arithmetic is the branch of mathematics dealing with basic operations on numbers.

2.1 Four Basic Operations

Operation	Symbol	Example
Addition	+	5 + 3 = 8
Subtraction	−	9 − 4 = 5
Multiplication	× or ·	6 × 7 = 42
Division	÷ or /	15 ÷ 3 = 5

2.2 Order of Operations (PEMDAS/BODMAS)

Order: P - Parentheses / Brackets
E - Exponents / Orders
M - Multiplication
D - Division
A - Addition
S - Subtraction

Example:

2 + 3 × (8 − 2)² ÷ 3
= 2 + 3 × 6² ÷ 3
= 2 + 3 × 36 ÷ 3
= 2 + 108 ÷ 3
= 2 + 36
= 38

2.3 Factors and Multiples

Factor: A number that divides another exactly. Factors of 12: 1, 2, 3, 4, 6, 12

Multiple: Result of multiplying a number by an integer. Multiples of 5: 5, 10, 15, 20, ...

2.4 Prime Numbers

A prime number has exactly two factors: 1 and itself.

Primes up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

2.5 LCM and GCD

Greatest Common Divisor (GCD): Largest number dividing both a and b.

Least Common Multiple (LCM): LCM(a, b) = (a × b) / GCD(a, b)

Example:

GCD(12, 18) = 6 LCM(12, 18) = (12 × 18) / 6 = 36

2.6 Divisibility Rules

Divisible by	Rule
2	Last digit is even
3	Sum of digits divisible by 3
4	Last two digits divisible by 4
5	Ends in 0 or 5
6	Divisible by both 2 and 3
9	Sum of digits divisible by 9
10	Ends in 0

Chapter 3: Fractions & Decimals

3.1 Fractions

A fraction represents a part of a whole: a/b where a is the numerator and b is the denominator.

Types of Fractions

Proper: Numerator < Denominator (e.g., 3/4)
Improper: Numerator ≥ Denominator (e.g., 7/4)
Mixed Number: Whole + Fraction (e.g., 1¾)

Operations on Fractions

Addition/Subtraction (same denominator): a/c + b/c = (a+b)/c

Addition/Subtraction (different denominators): a/b + c/d = (ad + bc) / bd

Multiplication: a/b × c/d = ac/bd

Division: a/b ÷ c/d = a/b × d/c = ad/bc

Examples:

2/3 + 1/4 = 8/12 + 3/12 = 11/12
3/4 × 2/5 = 6/20 = 3/10
3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2

3.2 Decimals

Decimals are another way to represent fractions with denominators that are powers of 10.

Converting Fractions to Decimals

Divide the numerator by the denominator: 3/4 = 3 ÷ 4 = 0.75

Rounding Decimals

If the digit after the rounding place is ≥ 5, round up; otherwise, round down.

Example:

3.14159 rounded to 2 decimal places = 3.14
2.675 rounded to 2 decimal places = 2.68

Chapter 4: Percentages & Ratios

4.1 Percentages

Percent means "per hundred." It represents a ratio out of 100.

Key Formulas: Percentage = (Part / Whole) × 100
Part = (Percentage / 100) × Whole
Percentage Change = ((New - Old) / Old) × 100

Examples:

What is 25% of 200? → (25/100) × 200 = 50
40 out of 160 is what %? → (40/160) × 100 = 25%
Price went from $50 to $60. % increase? → ((60-50)/50) × 100 = 20%

4.2 Ratios

A ratio compares two quantities: a : b

Equivalent Ratios: a:b = ka:kb for any non-zero k

4.3 Proportions

Direct Proportion: y = kx (y increases as x increases)

Inverse Proportion: y = k/x (y decreases as x increases)

Chapter 5: Basic Algebra

Algebra uses symbols (variables) to represent unknown quantities and express mathematical relationships.

5.1 Variables and Expressions

Variable: A symbol (usually a letter) representing an unknown value, e.g., x, y
Expression: A combination of variables, numbers, and operations, e.g., 3x + 2
Equation: Two expressions set equal, e.g., 2x + 3 = 7

5.2 Simplifying Expressions

Like Terms: Terms with the same variable and exponent.

Example:

3x + 2y + 5x − y
= (3x + 5x) + (2y − y)
= 8x + y

5.3 Expanding (Distributive Law)

Formula: a(b + c) = ab + ac

Example:

3(2x + 5) = 6x + 15
(x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6

5.4 Factoring

Common Factor: 6x + 9 = 3(2x + 3)
Difference of Squares: a² − b² = (a+b)(a−b)
Perfect Square: a² + 2ab + b² = (a+b)²
Trinomial: x² + 5x + 6 = (x+2)(x+3)

Chapter 6: Linear Equations

6.1 One-Variable Linear Equations

Standard Form: ax + b = c

Example:

2x + 3 = 11
2x = 8
x = 4

6.2 Linear Equations in Two Variables

Standard Form: ax + by = c

Slope-Intercept Form: y = mx + b (m = slope, b = y-intercept)

Point-Slope Form: y − y₁ = m(x − x₁)

6.3 Slope

Formula: m = (y₂ − y₁) / (x₂ − x₁)

Slope represents the rate of change (rise over run).

6.4 Systems of Linear Equations

Methods to solve systems: Substitution, Elimination, Graphing

Example (Elimination):

x + y = 5
x − y = 1
────────
2x = 6 → x = 3
y = 5 − 3 = 2
Solution: (3, 2)

Chapter 7: Quadratic Equations

7.1 Standard Form

Standard Form: ax² + bx + c = 0 (where a ≠ 0)

7.2 Methods of Solving

Factoring

Example:

x² + 5x + 6 = 0
(x + 2)(x + 3) = 0
x = −2 or x = −3

Quadratic Formula

Formula: x = (−b ± √(b² − 4ac)) / 2a

Example:

2x² − 4x − 6 = 0
a=2, b=−4, c=−6
x = (4 ± √(16 + 48)) / 4 = (4 ± 8) / 4
x = 3 or x = −1

Completing the Square

Example:

x² + 6x + 5 = 0
x² + 6x = −5
x² + 6x + 9 = 4
(x + 3)² = 4
x + 3 = ±2
x = −1 or x = −5

7.3 Discriminant

Discriminant: Δ = b² − 4ac

Δ > 0 → Two distinct real roots
Δ = 0 → One repeated real root
Δ < 0 → Two complex (no real) roots

7.4 Vertex Form

Vertex Form: y = a(x − h)² + k
Vertex: (h, k)

Chapter 8: Polynomials

8.1 Definition

A polynomial is an expression of the form: aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀

8.2 Degree and Classification

Degree	Name	Example
0	Constant	5
1	Linear	3x + 2
2	Quadratic	x² + 3x + 1
3	Cubic	x³ − 2x² + x
4	Quartic	x⁴ + x²
5	Quintic	x⁵ − x

8.3 Operations on Polynomials

Addition/Subtraction: Combine like terms.

Multiplication: Use distributive property (FOIL for binomials).

Division: Use polynomial long division or synthetic division.

8.4 Remainder and Factor Theorems

Remainder Theorem: If P(x) is divided by (x − a), the remainder is P(a).

Factor Theorem: (x − a) is a factor of P(x) if and only if P(a) = 0.

Chapter 9: Inequalities

9.1 Inequality Symbols

Symbol	Meaning
<	Less than
>	Greater than
≤	Less than or equal to
≥	Greater than or equal to
≠	Not equal to

9.2 Solving Linear Inequalities

⚠️ Important: When multiplying or dividing both sides by a negative number, reverse the inequality sign!

Example:

−2x + 3 > 7
−2x > 4
x < −2 (sign flipped when dividing by −2)

9.3 Compound Inequalities

AND (Intersection): a < x < b means x > a AND x < b

OR (Union): x < a OR x > b

9.4 Absolute Value Inequalities

|x| < a means: −a < x < a
|x| > a means: x < −a or x > a

Chapter 10: Functions

10.1 Definition

A function f from set A to set B assigns exactly one output f(x) ∈ B to each input x ∈ A.

A relation is a function if every x-value maps to exactly one y-value (Vertical Line Test).

10.2 Function Notation

f(x) = expression in x
e.g., f(x) = 2x + 3 → f(4) = 2(4) + 3 = 11

10.3 Types of Functions

Type	Form	Shape
Linear	f(x) = mx + b	Straight line
Quadratic	f(x) = ax² + bx + c	Parabola
Cubic	f(x) = ax³	S-curve
Exponential	f(x) = aˣ	Exponential curve
Logarithmic	f(x) = log(x)	Log curve
Absolute Value	f(x) = \|x\|	V-shape
Square Root	f(x) = √x	Half parabola

10.4 Domain and Range

Domain: All valid input (x) values
Range: All possible output (y) values

10.5 Composite and Inverse Functions

Composite: (f ∘ g)(x) = f(g(x))

Inverse: If f(a) = b, then f⁻¹(b) = a
To find: swap x and y, then solve for y

Chapter 11: Sequences & Series

11.1 Sequences

An ordered list of numbers following a pattern.

11.2 Arithmetic Sequences

General Term: aₙ = a₁ + (n − 1)d

Sum of n Terms: Sₙ = n/2 × (a₁ + aₙ) = n/2 × [2a₁ + (n−1)d]

Where d = common difference

Example:

Sequence: 3, 7, 11, 15, ... (d = 4)
a₁₀ = 3 + (10−1)×4 = 39
S₁₀ = 10/2 × (3 + 39) = 210

11.3 Geometric Sequences

General Term: aₙ = a₁ × rⁿ⁻¹

Sum of n Terms: Sₙ = a₁(1 − rⁿ) / (1 − r), r ≠ 1

Sum to Infinity (|r| < 1): S∞ = a₁ / (1 − r)

Where r = common ratio

11.4 Sigma Notation

∑(i=1 to n) f(i) = f(1) + f(2) + ... + f(n)

Chapter 12: Exponents & Logarithms

12.1 Laws of Exponents

Laws: aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ
(ab)ⁿ = aⁿbⁿ
a⁰ = 1 (a ≠ 0)
a⁻ⁿ = 1/aⁿ
a^(m/n) = ⁿ√(aᵐ)

12.2 Logarithms

Definition: log_b(x) = y ⟺ bʸ = x

Special Logs: log₁₀(x) = log(x) (common log)
logₑ(x) = ln(x) (natural log, e ≈ 2.718)

12.3 Laws of Logarithms

Product Rule: log(ab) = log(a) + log(b)

Quotient Rule: log(a/b) = log(a) − log(b)

Power Rule: log(aⁿ) = n·log(a)

Change of Base: log_b(a) = log(a) / log(b)

Chapter 13: Plane Geometry

13.1 Basic Concepts

Point: Has position but no size
Line: Extends infinitely in both directions
Ray: Starts at a point and extends in one direction
Line Segment: Part of a line with two endpoints
Angle: Formed by two rays sharing a vertex

13.2 Types of Angles

Type	Measure
Acute	0° < θ < 90°
Right	θ = 90°
Obtuse	90° < θ < 180°
Straight	θ = 180°
Reflex	180° < θ < 360°

13.3 Angle Relationships

Complementary: Two angles summing to 90°
Supplementary: Two angles summing to 180°
Vertical Angles: Opposite angles formed by intersecting lines (equal)

13.4 Polygons - Area and Perimeter

Shape	Area	Perimeter
Square (side a)	a²	4a
Rectangle (l×w)	l × w	2(l + w)
Triangle (b, h)	½bh	a + b + c
Parallelogram	b × h	2(a + b)
Trapezoid (a, b, h)	½(a+b)h	a + b + c + d
Rhombus (d₁, d₂)	½d₁d₂	4a

Chapter 14: Triangles

14.1 Types of Triangles

By Sides: Equilateral (all sides equal), Isosceles (two sides equal), Scalene (no sides equal)

By Angles: Acute (all angles < 90°), Right (one 90°), Obtuse (one angle > 90°)

14.2 Triangle Properties

Sum of interior angles = 180°
Exterior angle = Sum of two non-adjacent interior angles
Triangle Inequality: a + b > c for any two sides a, b and third side c

14.3 Pythagorean Theorem

For a right triangle with legs a, b and hypotenuse c: a² + b² = c²

Example:

a = 3, b = 4 → c = √(9 + 16) = √25 = 5
Common triples: (3,4,5), (5,12,13), (8,15,17)

14.4 Heron's Formula

Area of triangle with sides a, b, c: s = (a + b + c) / 2 (semi-perimeter)
Area = √(s(s−a)(s−b)(s−c))

14.5 Triangle Congruence (SSS, SAS, ASA, AAS, RHS)

Criterion	Meaning
SSS	All three sides equal
SAS	Two sides and included angle equal
ASA	Two angles and included side equal
AAS	Two angles and non-included side equal
RHS	Right angle, hypotenuse, and one side equal

14.6 Similar Triangles

Triangles are similar if their corresponding angles are equal. Corresponding sides are proportional.

Chapter 15: Circles

15.1 Basic Terminology

Radius (r): Distance from center to circumference
Diameter (d): d = 2r
Chord: Line segment with both endpoints on circle
Arc: Part of the circumference
Tangent: Line touching circle at exactly one point
Secant: Line intersecting circle at two points

15.2 Formulas

Circumference: C = 2πr = πd

Area: A = πr²

Arc Length: L = (θ/360°) × 2πr

Sector Area: A = (θ/360°) × πr²

15.3 Circle Theorems

Angle at center = 2 × angle at circumference (same arc)
Angles in the same segment are equal
Angle in a semicircle = 90°
Opposite angles in a cyclic quadrilateral sum to 180°
Tangent is perpendicular to radius at point of tangency
Two tangents from external point are equal in length

15.4 Equation of a Circle

Center (h, k), radius r: (x − h)² + (y − k)² = r²

Center at origin: x² + y² = r²

Chapter 16: Solid Geometry

Solid geometry deals with three-dimensional shapes.

16.1 Common 3D Shapes

Shape	Volume	Surface Area
Cube (a)	a³	6a²
Cuboid (l,w,h)	lwh	2(lw + lh + wh)
Sphere (r)	(4/3)πr³	4πr²
Cylinder (r,h)	πr²h	2πr(r + h)
Cone (r,h,l)	(1/3)πr²h	πr(r + l)
Pyramid (b,h)	(1/3)Bh	B + (1/2)Pl

B = base area, P = base perimeter, l = slant height

16.2 Euler's Formula for Polyhedra

V − E + F = 2

Where V = vertices, E = edges, F = faces

Chapter 17: Coordinate Geometry

17.1 Cartesian Plane

The Cartesian coordinate system uses two perpendicular number lines (x-axis and y-axis) to describe point positions as (x, y).

17.2 Key Formulas

Distance Formula: d = √((x₂−x₁)² + (y₂−y₁)²)

Midpoint Formula: M = ((x₁+x₂)/2, (y₁+y₂)/2)

Slope: m = (y₂−y₁)/(x₂−x₁)

Collinear Points: Points are collinear if slopes between pairs are equal.

17.3 Conic Sections

Conic	Equation
Circle	(x−h)² + (y−k)² = r²
Parabola	y = a(x−h)² + k
Ellipse	(x−h)²/a² + (y−k)²/b² = 1
Hyperbola	(x−h)²/a² − (y−k)²/b² = 1

Chapter 18: Trigonometric Functions

18.1 Basic Ratios (Right Triangle)

SOH CAH TOA: sin(θ) = Opposite / Hypotenuse
cos(θ) = Adjacent / Hypotenuse
tan(θ) = Opposite / Adjacent
csc(θ) = 1/sin(θ)
sec(θ) = 1/cos(θ)
cot(θ) = 1/tan(θ) = cos(θ)/sin(θ)

18.2 Standard Angle Values

Angle	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3
90°	1	0	undefined

18.3 Signs in Quadrants (ASTC)

All positive (Q1), Sin positive (Q2), Tan positive (Q3), Cos positive (Q4)

18.4 Sine and Cosine Rules

Sine Rule: a/sin(A) = b/sin(B) = c/sin(C)

Cosine Rule: c² = a² + b² − 2ab·cos(C)

Chapter 19: Trigonometric Identities

19.1 Pythagorean Identities

sin²(θ) + cos²(θ) = 1
1 + tan²(θ) = sec²(θ)
1 + cot²(θ) = csc²(θ)

19.2 Sum and Difference Formulas

sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)
cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B)
tan(A ± B) = (tan(A) ± tan(B)) / (1 ∓ tan(A)tan(B))

19.3 Double Angle Formulas

sin(2θ) = 2sin(θ)cos(θ)
cos(2θ) = cos²(θ) − sin²(θ) = 2cos²(θ)−1 = 1−2sin²(θ)
tan(2θ) = 2tan(θ) / (1 − tan²(θ))

19.4 Half Angle Formulas

sin(θ/2) = ±√((1 − cos θ)/2)
cos(θ/2) = ±√((1 + cos θ)/2)
tan(θ/2) = sin θ / (1 + cos θ) = (1 − cos θ) / sin θ

Chapter 20: Inverse Trigonometric Functions

20.1 Definitions

arcsin(x) = sin⁻¹(x): gives angle whose sine is x
arccos(x) = cos⁻¹(x): gives angle whose cosine is x
arctan(x) = tan⁻¹(x): gives angle whose tangent is x

20.2 Domains and Ranges

Function	Domain	Range
arcsin(x)	[−1, 1]	[−π/2, π/2]
arccos(x)	[−1, 1]	[0, π]
arctan(x)	(−∞, ∞)	(−π/2, π/2)

Chapter 21: Limits

The limit describes the value a function approaches as the input approaches some value.

21.1 Notation

lim[x→a] f(x) = L means f(x) approaches L as x approaches a

21.2 Limit Laws

lim[x→a] [f(x) + g(x)] = lim f(x) + lim g(x)
lim[x→a] [f(x) · g(x)] = lim f(x) · lim g(x)
lim[x→a] [f(x)/g(x)] = lim f(x) / lim g(x), if lim g(x) ≠ 0
lim[x→a] [c · f(x)] = c · lim f(x)

21.3 Special Limits

lim[x→0] sin(x)/x = 1
lim[x→0] (1 − cos x)/x = 0
lim[x→∞] (1 + 1/x)ˣ = e

21.4 L'Hôpital's Rule

If lim f(x)/g(x) is 0/0 or ∞/∞, then:
lim[x→a] f(x)/g(x) = lim[x→a] f'(x)/g'(x)

21.5 Continuity

A function is continuous at x = a if:

f(a) is defined
lim[x→a] f(x) exists
lim[x→a] f(x) = f(a)

Chapter 22: Differentiation

Differentiation finds the rate of change of a function (its derivative).

22.1 Definition

f'(x) = lim[h→0] (f(x+h) − f(x)) / h

22.2 Basic Differentiation Rules

d/dx [c] = 0 (constant)
d/dx [xⁿ] = nxⁿ⁻¹ (power rule)
d/dx [cf(x)] = c·f'(x)
d/dx [f+g] = f' + g'
d/dx [fg] = f'g + fg' (product rule)
d/dx [f/g] = (f'g − fg') / g² (quotient rule)
d/dx [f(g(x))] = f'(g(x))·g'(x) (chain rule)

22.3 Derivatives of Common Functions

Function	Derivative
sin(x)	cos(x)
cos(x)	−sin(x)
tan(x)	sec²(x)
eˣ	eˣ
ln(x)	1/x
aˣ	aˣ ln(a)
arcsin(x)	1/√(1−x²)
arctan(x)	1/(1+x²)

22.4 Applications

Tangent Line: y − y₁ = f'(x₁)(x − x₁)
Critical Points: f'(x) = 0
Increasing/Decreasing: f'(x) > 0 or f'(x) < 0
Concavity: f''(x) > 0 (concave up), f''(x) < 0 (concave down)
Inflection Points: f''(x) = 0 (sign changes)

Chapter 23: Integration

Integration is the reverse of differentiation and finds areas under curves.

23.1 Indefinite Integrals

∫f(x)dx = F(x) + C, where F'(x) = f(x)

23.2 Basic Integration Rules

∫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
∫sec²(x) dx = tan(x) + C
∫cf(x) dx = c∫f(x) dx
∫[f(x)+g(x)] dx = ∫f(x)dx + ∫g(x)dx

23.3 Definite Integrals

∫[a to b] f(x) dx = F(b) − F(a)

23.4 Integration Techniques

Substitution: Let u = g(x), du = g'(x)dx
Integration by Parts: ∫u dv = uv − ∫v du
Partial Fractions: Decompose rational functions
Trigonometric Substitution

23.5 Applications

Area: A = ∫[a to b] f(x) dx
Area between curves: A = ∫[a to b] [f(x) − g(x)] dx
Volume (disk method): V = π∫[a to b] [f(x)]² dx

Chapter 24: Differential Equations

A differential equation relates a function with its derivatives.

24.1 Order and Degree

Order: Highest derivative present. Degree: Power of the highest derivative.

24.2 Separable Equations

Form: dy/dx = f(x)g(y)

Separate: (1/g(y))dy = f(x)dx
Then integrate both sides.

24.3 Linear First-Order ODE

Form: dy/dx + P(x)y = Q(x)

Integrating Factor: μ = e^(∫P(x)dx)
Solution: y = (1/μ)∫μQ(x)dx

24.4 Second-Order Linear ODE

Form: ay'' + by' + cy = 0

Characteristic equation: ar² + br + c = 0
Two real roots r₁, r₂: y = C₁e^(r₁x) + C₂e^(r₂x)
Repeated root r: y = (C₁ + C₂x)e^(rx)
Complex roots α ± βi: y = e^(αx)(C₁cos(βx) + C₂sin(βx))

Chapter 25: Statistics

25.1 Measures of Central Tendency

Mean: x̄ = (Σxᵢ) / n

Median: Middle value when data is sorted.

Mode: Most frequently occurring value.

25.2 Measures of Spread

Range: R = Max − Min

Variance: σ² = Σ(xᵢ − x̄)² / n (population)
s² = Σ(xᵢ − x̄)² / (n−1) (sample)

Standard Deviation: σ = √(variance)

25.3 Normal Distribution

Bell-shaped probability distribution defined by mean (μ) and standard deviation (σ).

f(x) = (1 / σ√(2π)) × e^(−(x−μ)²/(2σ²))

68-95-99.7 Rule:
68% within 1σ, 95% within 2σ, 99.7% within 3σ

25.4 Z-Score

z = (x − μ) / σ

25.5 Correlation and Regression

Correlation Coefficient (r): −1 ≤ r ≤ 1
r close to 1: strong positive correlation
r close to −1: strong negative correlation

Linear Regression: ŷ = a + bx
b = r × (sᵧ/sₓ), a = ȳ − b·x̄

Chapter 26: Probability

26.1 Basic Probability

P(A) = Number of favorable outcomes / Total outcomes
0 ≤ P(A) ≤ 1
P(A) + P(A') = 1 where A' is the complement

26.2 Probability Rules

Addition Rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Mutually Exclusive: P(A ∪ B) = P(A) + P(B)

Multiplication Rule: P(A ∩ B) = P(A) × P(B|A)

Independent Events: P(A ∩ B) = P(A) × P(B)

26.3 Conditional Probability

P(A|B) = P(A ∩ B) / P(B)

26.4 Bayes' Theorem

P(A|B) = P(B|A) × P(A) / P(B)

26.5 Counting Principles

Permutations (order matters): P(n,r) = n! / (n−r)!

Combinations (order doesn't matter): C(n,r) = n! / (r!(n−r)!)

Fundamental Counting Principle: If event A has m outcomes and B has n outcomes, then A and B together have m×n outcomes.

26.6 Probability Distributions

Distribution	Use Case	Formula
Binomial	n trials, p success probability	P(X=k) = C(n,k)pᵏ(1−p)ⁿ⁻ᵏ
Poisson	Events in time/space	P(X=k) = (λᵏe⁻λ)/k!
Normal	Continuous, symmetric	Bell curve with μ, σ

Chapter 27: Matrices & Linear Algebra

27.1 Matrix Basics

A matrix is a rectangular array of numbers arranged in rows and columns.

A = [aᵢⱼ] is an m×n matrix with m rows and n columns

27.2 Matrix Operations

Addition: (A+B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ (same dimensions)
Scalar Multiplication: (cA)ᵢⱼ = c·Aᵢⱼ
Matrix Multiplication: (AB)ᵢⱼ = Σₖ AᵢₖBₖⱼ (A is m×n, B is n×p)
Transpose: (Aᵀ)ᵢⱼ = Aⱼᵢ

27.3 Determinant

2×2 Matrix: |A| = ad − bc for [[a,b],[c,d]]

3×3 Matrix: Cofactor expansion along first row.

27.4 Inverse Matrix

A⁻¹ = (1/|A|) × adj(A)
A·A⁻¹ = I (identity matrix)
Exists only if |A| ≠ 0

27.5 Eigenvalues and Eigenvectors

Av = λv
(A − λI)v = 0
det(A − λI) = 0 → characteristic equation

27.6 Vector Spaces

Basis: Linearly independent set that spans the space
Rank: Number of linearly independent rows/columns
Null Space: All solutions to Ax = 0

Chapter 28: Complex Numbers

28.1 Definition

z = a + bi, where a = real part, b = imaginary part
i = √(−1), i² = −1, i³ = −i, i⁴ = 1

28.2 Arithmetic

Addition: (a+bi) + (c+di) = (a+c) + (b+d)i

Multiplication: (a+bi)(c+di) = (ac−bd) + (ad+bc)i

Conjugate: z̄ = a − bi

Division: (a+bi)/(c+di) = (a+bi)(c−di)/(c²+d²)

28.3 Polar Form

z = r(cos θ + i sin θ) = re^(iθ)
r = |z| = √(a²+b²) (modulus)
θ = arg(z) = arctan(b/a) (argument)

28.4 De Moivre's Theorem

(r(cos θ + i sin θ))ⁿ = rⁿ(cos nθ + i sin nθ)

28.5 Euler's Formula

e^(iθ) = cos θ + i sin θ
e^(iπ) + 1 = 0 (Euler's Identity)

Chapter 29: Number Theory

29.1 Divisibility and Primes

Number theory studies the properties of integers, especially prime numbers.

Fundamental Theorem of Arithmetic: Every integer > 1 can be uniquely expressed as a product of primes.

29.2 Euclidean Algorithm

GCD(a, b) = GCD(b, a mod b) until b = 0

Example:

GCD(48, 18):
48 = 2 × 18 + 12 → GCD(18, 12)
18 = 1 × 12 + 6 → GCD(12, 6)
12 = 2 × 6 + 0 → GCD = 6

29.3 Modular Arithmetic

a ≡ b (mod m) means m divides (a − b)
Properties: if a ≡ b and c ≡ d (mod m), then
a+c ≡ b+d (mod m) and ac ≡ bd (mod m)

29.4 Fermat's Little Theorem

If p is prime and gcd(a, p) = 1, then:
aᵖ⁻¹ ≡ 1 (mod p)

29.5 Chinese Remainder Theorem

If m₁, m₂, ..., mₖ are pairwise coprime, then a system of simultaneous congruences has a unique solution modulo m₁m₂...mₖ.

Chapter 30: Set Theory & Logic

30.1 Sets

A set is a collection of distinct objects called elements.

Notation: A = {1, 2, 3, 4, 5}
x ∈ A means x is in A
x ∉ A means x is not in A
|A| = cardinality (number of elements)

30.2 Set Operations

Operation	Symbol	Meaning
Union	A ∪ B	Elements in A or B (or both)
Intersection	A ∩ B	Elements in both A and B
Difference	A − B	Elements in A but not B
Complement	A'	Elements not in A
Subset	A ⊆ B	Every element of A is in B
Power Set	𝒫(A)	Set of all subsets of A

30.3 Venn Diagrams

Overlapping circles representing sets and their relationships visually.

30.4 De Morgan's Laws

(A ∪ B)' = A' ∩ B'
(A ∩ B)' = A' ∪ B'

30.5 Logic and Propositional Calculus

Operator	Symbol	Truth
AND (Conjunction)	∧	True only when both are true
OR (Disjunction)	∨	True when at least one is true
NOT (Negation)	¬	Flips truth value
Implication	p → q	False only when p=T, q=F
Biconditional	p ↔ q	True when p and q same value

30.6 Truth Tables

p	q	p∧q	p∨q	¬p	p→q	p↔q
T	T	T	T	F	T	T
T	F	F	T	F	F	F
F	T	F	T	T	T	F
F	F	F	F	T	T	T

30.7 Proof Techniques

Direct Proof: Assume hypothesis, deduce conclusion step by step
Proof by Contradiction: Assume negation of conclusion, derive contradiction
Proof by Induction: Base case + inductive step
Contrapositive: Prove ¬q → ¬p instead of p → q

📐 Complete Mathematics

Table of Contents

Chapter 1: Number Systems

1.1 Types of Numbers

1.2 Properties of Real Numbers

1.3 Number Line

1.4 Absolute Value

Chapter 2: Arithmetic

2.1 Four Basic Operations

2.2 Order of Operations (PEMDAS/BODMAS)

2.3 Factors and Multiples

2.4 Prime Numbers

2.5 LCM and GCD

2.6 Divisibility Rules

Chapter 3: Fractions & Decimals

3.1 Fractions

Types of Fractions

Operations on Fractions

3.2 Decimals

Converting Fractions to Decimals

Rounding Decimals

Chapter 4: Percentages & Ratios

4.1 Percentages

4.2 Ratios

4.3 Proportions

Chapter 5: Basic Algebra

5.1 Variables and Expressions

5.2 Simplifying Expressions

5.3 Expanding (Distributive Law)

5.4 Factoring

Chapter 6: Linear Equations

6.1 One-Variable Linear Equations

6.2 Linear Equations in Two Variables

6.3 Slope

6.4 Systems of Linear Equations

Chapter 7: Quadratic Equations

7.1 Standard Form

7.2 Methods of Solving

Factoring

Quadratic Formula

Completing the Square

7.3 Discriminant

7.4 Vertex Form

Chapter 8: Polynomials

8.1 Definition

8.2 Degree and Classification

8.3 Operations on Polynomials

8.4 Remainder and Factor Theorems

Chapter 9: Inequalities

9.1 Inequality Symbols

9.2 Solving Linear Inequalities

9.3 Compound Inequalities

9.4 Absolute Value Inequalities

Chapter 10: Functions

10.1 Definition

10.2 Function Notation

10.3 Types of Functions

10.4 Domain and Range

10.5 Composite and Inverse Functions

Chapter 11: Sequences & Series

11.1 Sequences

11.2 Arithmetic Sequences

11.3 Geometric Sequences

11.4 Sigma Notation

Chapter 12: Exponents & Logarithms

12.1 Laws of Exponents

12.2 Logarithms

12.3 Laws of Logarithms

Chapter 13: Plane Geometry

13.1 Basic Concepts

13.2 Types of Angles

13.3 Angle Relationships

13.4 Polygons - Area and Perimeter

Chapter 14: Triangles

14.1 Types of Triangles

14.2 Triangle Properties

14.3 Pythagorean Theorem

14.4 Heron's Formula

14.5 Triangle Congruence (SSS, SAS, ASA, AAS, RHS)

14.6 Similar Triangles