Overview of Mathematics

Definition: The study of numbers, quantities, shapes, and patterns.
Branches:
- Arithmetic: Basics of number operations (addition, subtraction, multiplication, division).
- Algebra: Study of symbols and the rules for manipulating those symbols.
- Geometry: Study of shapes, sizes, and properties of space.
- Trigonometry: Study of relationships in triangles, particularly regarding angles and lengths.
- Calculus: Study of change, involving derivatives and integrals.

Numbers:
- Types:
  - Natural Numbers: 1, 2, 3, ...
  - Whole Numbers: 0, 1, 2, 3, ...
  - Integers: ... -2, -1, 0, 1, 2, ...
  - Rational Numbers: Fractions a/b where a and b are integers, b ≠ 0.
  - Irrational Numbers: Non-repeating, non-terminating decimals (e.g., √2, π).
Arithmetic Operations:
- Addition (+): Combining quantities.
- Subtraction (−): Finding the difference between quantities.
- Multiplication (×): Repeated addition of a number.
- Division (÷): Splitting a number into equal parts.
Algebra:
- Expressions: Combinations of variables and constants (e.g., 3x + 2).
- Equations: Statements that two expressions are equal (e.g., 2x + 3 = 7).
- Functions: Relations that assign exactly one output for each input.
Geometry:
- Shapes: Points, lines, angles, polygons (triangles, rectangles), circles.
- Theorems: Pythagorean theorem, properties of parallel lines and angles.
- Area and Volume: Formulas for calculating the space within shapes.
Trigonometry:
- Functions: Sine (sin), Cosine (cos), Tangent (tan) related to right-angled triangles.
- Relationships: SOH-CAH-TOA (Sine = opposite/hypotenuse, Cosine = adjacent/hypotenuse, Tangent = opposite/adjacent).
Calculus:
- Limits: Fundamental concept for understanding calculus.
- Derivatives: Measure of how a function changes as its input changes.
- Integrals: Represent accumulation of quantities, area under a curve.

Mathematics is the study of numbers, quantities, shapes, and patterns.
It has several branches including arithmetic, algebra, geometry, trigonometry, and calculus.

Deals with basic number operations such as addition, subtraction, multiplication, and division.

Involves using symbols to represent unknown quantities and developing rules for manipulating these symbols.
Expressions combine variables and constants.
Equations state that two expressions are equal.
Functions relate inputs and outputs.

Focuses on shapes, sizes, and properties of space.
Explores points, lines, angles, polygons, and circles.
Includes theorems like the Pythagorean theorem and properties of parallel lines and angles.
Calculates area and volume for shapes.

Studies relationships in triangles, especially angles and lengths.
Uses trigonometric functions like sine (sin), cosine (cos), and tangent (tan) for calculations involving right triangles.
Remember SOH-CAH-TOA, which stands for Sine = opposite/hypotenuse, Cosine = adjacent/hypotenuse, Tangent = opposite/adjacent.

In science: Used to model physical phenomena and carry out statistical analysis.
In engineering: Essential for design calculations and optimization problems.
In economics: Used in financial models, cost analysis, and forecasting.
In daily life: Helps with budgeting, cooking measurements, and time management.