Key Concepts in Mathematics

Branches of Mathematics
- Arithmetic: Basics of numbers, addition, subtraction, multiplication, division.
- Algebra: Use of symbols and letters to represent numbers and relationships; includes equations and inequalities.
- Geometry: Study of shapes, sizes, and properties of space; involves points, lines, surfaces, and solids.
- Trigonometry: Focuses on the relationships between angles and sides of triangles; key functions: sine, cosine, tangent.
- Calculus: Studies change and motion; includes differentiation and integration.
- Statistics: Analyzes data; involves mean, median, mode, variance, and probability.
- Number Theory: Study of integers and their properties; includes primes, divisibility, and modular arithmetic.
- Discrete Mathematics: Involves study of mathematical structures that are fundamentally discrete rather than continuous; includes graph theory and combinatorics.
Mathematical Operations
- Addition (+): Combining two or more numbers.
- Subtraction (−): Finding the difference between numbers.
- Multiplication (×): Repeated addition of a number.
- Division (÷): Splitting into equal parts or groups.
Properties of Operations
- Commutative Property: ( a + b = b + a ) (Addition and multiplication).
- Associative Property: ( (a + b) + c = a + (b + c) ) (Addition and multiplication).
- Distributive Property: ( a(b + c) = ab + ac ).
Key Formulas
- Area of a Circle: ( A = \pi r^2 ).
- Circumference of a Circle: ( C = 2\pi r ).
- Pythagorean Theorem: ( a^2 + b^2 = c^2 ) (in right triangles).
- Quadratic Formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).
Functions and Graphs
- Function: A relation that assigns exactly one output for each input; notation ( f(x) ).
- Linear Function: Form ( y = mx + b ); represents a straight line.
- Quadratic Function: Form ( y = ax^2 + bx + c ); represents a parabola.
- Exponential Function: Form ( y = ab^x ); growth or decay processes.
Set Theory
- Set: A collection of distinct objects.
- Union: Combine two sets; ( A \cup B ).
- Intersection: Common elements; ( A \cap B ).
- Complement: Elements not in the set; ( A' ).
Probability Basics
- Independent Events: The outcome of one event does not affect the other.
- Dependent Events: The outcome of one event affects the other.
- Basic Probability Formula: ( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} ).
Measurement Units
- Length: Meters (m), centimeters (cm), inches (in).
- Area: Square meters (m²), square feet (ft²).
- Volume: Cubic meters (m³), liters (L).
- Weight: Kilograms (kg), grams (g), pounds (lb).

These notes cover foundational concepts and are designed to aid in the understanding of various mathematical principles and operations.

Branches of Mathematics

Arithmetic: Deals with basic operations like addition, subtraction, multiplication, and division with numbers.
Algebra: Represents numbers and relationships using symbols and letters; involves solving equations and inequalities.
Geometry: Focuses on the study of shapes, sizes, and properties of space including points, lines, surfaces, and solids.
Trigonometry: Studies relationships between angles and sides of triangles using functions like sine, cosine, and tangent.
Calculus: Examines change and motion through differentiation and integration.
Statistics: Analyzes data using measures like mean, median, mode, variance, and probability.
Number Theory: Investigates properties of integers including primes, divisibility, and modular arithmetic.
Discrete Mathematics: Deals with discrete mathematical structures like graph theory and combinatorics.

Mathematical Operations

Addition (+): Combining two or more numbers.
Subtraction (−): Finding the difference between two numbers.
Multiplication (×): Repeated addition of a number.
Division (÷): Splitting into equal parts or groups.

Properties of Operations

Commutative Property: The order of operation doesn't matter for addition and multiplication ( ( a + b = b + a ) and ( a × b = b × a )).
Associative Property: Grouping of numbers doesn't affect the result for addition and multiplication ( ( (a + b) + c = a + (b + c) ) and ( (a × b) × c = a × (b × c) )).
Distributive Property: Multiplication distributes over addition ( ( a(b + c) = ab + ac )).

Key Formulas

Area of a Circle: ( A = \pi r^2 ) (where ( r ) is the radius).
Circumference of a Circle: ( C = 2\pi r ).
Pythagorean Theorem: ( a^2 + b^2 = c^2 ) (where ( a ) and ( b ) are the legs of a right triangle, and ( c ) is the hypotenuse).
Quadratic Formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) (used to solve quadratic equations of the form ( ax^2 + bx + c = 0 )).

Functions and Graphs

Function: A relationship assigning exactly one output for each input, often expressed as ( f(x) ).
Linear Function: Represented by ( y = mx + b ), where ( m ) is the slope and ( b ) is the y-intercept, graphing as a straight line.
Quadratic Function: Expressed as ( y = ax^2 + bx + c ), resulting in a parabolic curve.
Exponential Function: Takes the form ( y = ab^x ); used to model growth or decay processes.

Set Theory

Set: A collection of distinct objects.
Union: Combining two sets; denoted by ( A \cup B ).
Intersection: Elements common to both sets; represented by ( A \cap B ).
Complement: Elements not present in the set; symbolized as ( A' ).

Probability Basics

Independent Events: The outcome of one event does not influence the outcome of another.
Dependent Events: The outcome of one event affects the outcome of another.
Basic Probability Formula: ( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} ).

Measurement Units

Length: Measured in units like meters (m), centimeters (cm), and inches (in).
Area: Expressed in square meters (m²), square feet (ft²), or other similar units.
Volume: Measured in cubic meters (m³), liters (L), and other relevant units.
Weight: Measured in kilograms (kg), grams (g), or pounds (lb).