Branches of Mathematics Overview

Arithmetic: Deals with basic operations (addition, subtraction, multiplication, division) and number properties.
Algebra: Involves symbols and letters to represent numbers in equations and formulas.
Geometry: Studies properties and relationships of shapes, sizes, and figures in space.
Trigonometry: Focuses on the relationships between the angles and sides of triangles.
Calculus: Concerns change and motion, including differentiation and integration.
Statistics: Involves data collection, analysis, interpretation, and presentation.
Probability: Studies the likelihood of events occurring.

Numbers: Natural numbers, whole numbers, integers, rational numbers, and irrational numbers.
Equations: Expressions that state two quantities are equal; includes linear, quadratic, and polynomial equations.
Functions: Relationships where each input has exactly one output; includes linear, quadratic, exponential, and logarithmic functions.

Points, Lines, and Angles: Basic elements of geometry; points have no dimension, lines extend indefinitely, angles measure rotation.
Shapes:
- Triangles: Classified by sides (scalene, isosceles, equilateral) and angles (acute, obtuse, right).
- Quadrilaterals: Includes squares, rectangles, parallelograms, trapezoids.
- Circles: Defined by radius, diameter, circumference, and area.

Expressions: Combinations of numbers, variables, and operators (e.g., 3x + 2).
Factoring: Breaking down an expression into simpler components (e.g., x² - 5x = x(x - 5)).
Inequalities: Statements about the relative size of values (e.g., x > 5, x ≤ 10).

Limits: Approaching values as inputs get very close to a certain point.
Derivatives: Measures how a function changes as its input changes (slope of the tangent line).
Integrals: Represents accumulation of quantities and area under curves.

Mean, Median, Mode: Measures of central tendency.
Variance and Standard Deviation: Measures of spread or dispersion within a data set.
Probability Distributions: Functions that describe the likelihood of outcomes (e.g., normal distribution).

Logic: Use of formal principles of reasoning (e.g., syllogisms).
Proof: A structured argument establishing the truth of a mathematical statement.
Theorems: Established results that can be proven based on previously known statements.

Calculators: Essential for performing complex calculations.
Graphing Tools: Used for visualizing functions and data.
Mathematical Software: Programs like MATLAB and Mathematica for advanced computations and modeling.

Arithmetic deals with basic calculations like adding, subtracting, multiplying, and dividing numbers.
Algebra uses symbols and letters to represent numbers and solve equations.
Geometry studies shapes, their sizes, and how they relate to each other in space.
Trigonometry examines the relationships between angles and sides of triangles.
Calculus focuses on change and motion by using concepts like differentiation and integration.
Statistics involves collecting, analyzing, interpreting, and presenting data.
Probability explores the likelihood of events happening.

Numbers are classified into different types, including natural numbers, whole numbers, integers, rational numbers, and irrational numbers.
Equations are expressions that state two quantities are equal. They can be linear, quadratic, or polynomial.
Functions represent relationships where each input has only one output. Examples include linear, quadratic, exponential, and logarithmic functions.

Points are locations with no dimension, lines extend infinitely, and angles measure rotation.
Triangles are categorized by their sides (scalene, isosceles, equilateral) and angles (acute, obtuse, right).
Quadrilaterals include shapes like squares, rectangles, parallelograms, and trapezoids.
Circles are defined by their radius, diameter, circumference, and area.

Expressions are combinations of numbers, variables, and operators, such as 3x + 2.
Factoring involves breaking down an expression into simpler components. For example, x² - 5x factors into x(x - 5).
Inequalities compare the relative sizes of values, like x > 5 or x ≤ 10.

Limits describe the value a function approaches as the input gets very close to a certain point.
Derivatives measure how a function changes as its input changes. They represent the slope of the tangent line.
Integrals represent accumulating quantities and calculating the area under a curve.

Mean, Median, and Mode are measures of central tendency, indicating the typical value in a data set.
Variance and Standard Deviation measure the spread or dispersion of data points.
Probability Distributions are functions that describe the likelihood of outcomes, with the normal distribution being a common example.

Logic applies formal principles to reasoning, using methods like syllogisms.
Proofs are structured arguments that establish the truth of a mathematical statement.
Theorems are established results that can be proven based on previously known statements.

### Key Mathematical Tools

Calculators are essential for performing complex calculations.
Graphing Tools help visualize functions and data.
Mathematical Software like MATLAB and Mathematica provide advanced computations and modeling capabilities.