Branches of Mathematics Overview

Arithmetic: Study of numbers and basic operations (addition, subtraction, multiplication, division).
Algebra: Involves variables and expressions; solving equations, inequalities, and polynomials.
Geometry: Study of shapes, sizes, and properties of space; includes points, lines, angles, surfaces, and solids.
Trigonometry: Focuses on the relationships between angles and sides of triangles; uses functions like sine, cosine, and tangent.
Calculus: Studies change and motion; includes differentiation (rates of change) and integration (area under curves).
Statistics: Collection, analysis, interpretation, and presentation of data; involves probability theory.
Discrete Mathematics: Focus on countable, distinct structures; includes graph theory, combinatorics.

Numbers:
- Natural Numbers: 1, 2, 3, ...
- Integers: ..., -2, -1, 0, 1, 2, ...
- Rational Numbers: Fractions, terminating or repeating decimals.
- Irrational Numbers: Non-repeating, non-terminating decimals (e.g., √2, π).
Equations:
- Linear Equations: Form y = mx + b.
- Quadratic Equations: Form ax² + bx + c = 0.
- Polynomial Equations: Involves terms with variables raised to whole number powers.
Functions:
- Definition: A relation that assigns exactly one output for each input.
- Types: Linear, quadratic, exponential, logarithmic, trigonometric.
Geometry Basics:
- Types of Angles: Acute (< 90°), Right (90°), Obtuse (> 90° and < 180°).
- Triangle Types: Equilateral, Isosceles, Scalene based on sides; Acute, Right, Obtuse based on angles.
- Circle: Important terms include radius, diameter, and circumference.

Order of Operations: PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
Factoring: Breaking down expressions into products of simpler factors.
Exponent Rules:
- Product Rule: a^m × a^n = a^(m+n)
- Quotient Rule: a^m / a^n = a^(m-n)
- Power Rule: (a^m)^n = a^(m*n)

Pythagorean Theorem: In a right triangle, a² + b² = c² (where c is the hypotenuse).
Fundamental Theorem of Calculus: Links differentiation and integration.

Real-world Applications: Used in finance (interest calculations), engineering (designs and structures), physics (motion and forces), and computer science (algorithms and data structures).

Arithmetic: Focuses on fundamental operations with numbers: addition, subtraction, multiplication, and division.
Algebra: Involves manipulating variables and expressions to solve equations and inequalities, including polynomials.
Geometry: Examines properties and relationships of shapes, lines, angles, surfaces, and solids in space.
Trigonometry: Investigates angle and side relationships in triangles using functions such as sine, cosine, and tangent.
Calculus: Analyzes concepts of change and motion through differentiation (finding rates of change) and integration (calculating areas).
Statistics: Encompasses data collection, analysis, interpretation, and presentation, integrating probability theory.
Discrete Mathematics: Studies distinct, countable structures, emphasizing areas like graph theory and combinatorics.

Natural Numbers: Counting numbers starting from 1 (1, 2, 3,...).
Integers: Whole numbers that include negative and positive values, plus zero (..., -2, -1, 0, 1, 2,...).
Rational Numbers: Any number that can be expressed as a fraction, including terminating and repeating decimals.
Irrational Numbers: Numbers that cannot be expressed as simple fractions, shown by non-repeating, non-terminating decimals (e.g., √2, π).
Linear Equations: Structured as y = mx + b, where m is the slope and b is the y-intercept.
Quadratic Equations: Formed as ax² + bx + c = 0, involving squared variables.
Polynomial Equations: Expressions that include variables raised to whole number powers across multiple terms.
Functions: Relationships that assign exactly one output for each input; categorized into types such as linear, quadratic, exponential, logarithmic, and trigonometric.
Types of Angles:
- Acute: Less than 90°
- Right: Exactly 90°
- Obtuse: More than 90° but less than 180°
Triangle Classifications:
- By sides: Equilateral (all sides equal), Isosceles (two sides equal), Scalene (no sides equal)
- By angles: Acute (all angles < 90°), Right (one angle = 90°), Obtuse (one angle > 90°)
Circle Terms: Important definitions include the radius (distance from center to edge), diameter (twice the radius), and circumference (perimeter of the circle).

Order of Operations: PEMDAS rules dictate priority: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.
Factoring: Process of expressing an algebraic expression as a product of its simpler factors.
Exponent Rules:
- Product Rule: For multiplying like bases, add the exponents (a^m × a^n = a^(m+n))
- Quotient Rule: For dividing like bases, subtract the exponents (a^m / a^n = a^(m-n))
- Power Rule: For raising a power to another power, multiply the exponents ((a^m)^n = a^(m*n)).

Mean: Calculated as the sum of data values divided by the number of values; represents the average.
Median: Identifies the middle value when a data set is arranged in ascending order.
Mode: The value that appears most frequently within a data set.
Range: Represents the gap between the highest and lowest values in a data set.

Pythagorean Theorem: States that in a right triangle, the sum of the squares of the legs equals the square of the hypotenuse (a² + b² = c²).
Fundamental Theorem of Calculus: Establishes the relationship between differentiation and integration, two core processes in calculus.

Real-world Uses: Mathematics is integral in various fields such as finance (calculating interest rates), engineering (design and structural analysis), physics (understanding motion and forces), and computer science (development of algorithms and management of data structures).