Basic Math Concepts and Algebra Overview

Numbers
- Natural Numbers: 1, 2, 3, ...
- Whole Numbers: 0, 1, 2, 3, ...
- Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
- Rational Numbers: Numbers that can be expressed as a fraction (e.g., 1/2, 0.75).
- Irrational Numbers: Numbers that cannot be expressed as a fraction (e.g., √2, π).
Operations
- Addition (+), Subtraction (−), Multiplication (×), Division (÷)
- Order of Operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right), often abbreviated as PEMDAS.

Variables and Expressions
- Variable: A symbol (often x or y) representing an unknown value.
- Expression: A combination of numbers, variables, and operations (e.g., 2x + 3).
Equations
- An equation states that two expressions are equal (e.g., 2x + 3 = 7).
- Solving equations involves isolating the variable.
Functions
- A function relates an input to an output (e.g., f(x) = 2x + 5).
- Types of functions: linear, quadratic, polynomial, exponential.

Basic Shapes
- Triangle: Sum of angles = 180°.
- Quadrilaterals: Includes squares, rectangles, and trapezoids.
- Circle: Key terms include radius, diameter, and circumference (C = πd).
Area and Volume
- Area: Measure of space inside a shape (e.g., A = l × w for rectangles).
- Volume: Measure of space inside a 3D object (e.g., V = l × w × h for rectangular prisms).

Basic Ratios
- Sine (sin), Cosine (cos), Tangent (tan).
- Relationships in right triangles:
  - sin θ = opposite/hypotenuse
  - cos θ = adjacent/hypotenuse
  - tan θ = opposite/adjacent
Pythagorean Theorem
- a² + b² = c² (in a right triangle, where c is the hypotenuse).

Limits
- The value that a function approaches as the input approaches a certain point.
Derivatives
- Measures the rate of change of a function (slope of the tangent line).
Integrals
- The area under a curve; represents accumulation.

Descriptive Statistics
- Mean: Average of a data set.
- Median: Middle value when data is ordered.
- Mode: Most frequently occurring value.
Probability
- Likelihood of an event occurring, expressed as a number between 0 and 1.

Statements and Arguments
- A statement is a declarative sentence that can be true or false.
- An argument consists of premises leading to a conclusion.
Quantifiers
- Universal Quantifier (for all, ∀) and Existential Quantifier (there exists, ∃).

Types of Proofs
- Direct Proof: Starts with known facts and applies logical steps.
- Indirect Proof: Assumes the opposite to show a contradiction.
Common Proof Techniques
- Mathematical Induction: Proving a base case and an inductive step.

Real-world Applications
- Math is used in finance (calculating interest), engineering (designing structures), medicine (statistical studies), and computer science (algorithms).

Numbers:
- Natural numbers are positive whole numbers (1, 2, 3, ...).
- Whole numbers include zero and all natural numbers (0, 1, 2, 3, ...).
- Integers are whole numbers including negatives (... -3, -2, -1, 0, 1, 2, 3, ...).
- Rational numbers are numbers that can be expressed as a fraction, including decimals that terminate or repeat (e.g., 1/2, 0.75).
- Irrational numbers cannot be expressed as a fraction, and their decimal representations go on forever without repeating (e.g., √2, π).
Operations:
- Basic operations include addition (+), subtraction (-), multiplication (×), and division (÷).
- Order of operations is a rule that dictates the order in which operations are performed (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right - commonly remembered as PEMDAS).

Variables and Expressions:
- Variables are symbols, often represented by letters like 'x' or 'y', representing unknown values.
- Expressions are combinations of numbers, variables, and operations (e.g., 2x + 3).
Equations:
- An equation sets two expressions as equal (e.g., 2x + 3 = 7).
- Solving an equation involves isolating the variable to find its value.
Functions:
- A function expresses a relationship between an input and an output (e.g., f(x) = 2x + 5).
- Common types of functions include linear, quadratic, polynomial, and exponential.

Basic Shapes:
- A triangle has three sides and three angles, with the sum of its angles always equaling 180 degrees.
- Quadrilaterals are four-sided shapes, including squares, rectangles, and trapezoids.
- A circle is defined by its radius (distance from the center to a point on the circle) and diameter (twice the radius). The circumference of a circle is calculated using the formula C = πd.
Area and Volume:
- Area measures the space inside a two-dimensional shape. The area of a rectangle is found by multiplying its length and width (A = l × w).
- Volume measures the space inside a three-dimensional object. The volume of a rectangular prism is calculated by multiplying its length, width, and height (V = l × w × h).

Basic Ratios:
- Sine (sin), cosine (cos), tangent (tan) are trigonometric functions that represent ratios between sides of a right triangle.
- Right triangle relationships:
  - sin θ = opposite / hypotenuse
  - cos θ = adjacent / hypotenuse
  - tan θ = opposite / adjacent
Pythagorean Theorem:
- In a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b): a² + b² = c².

Limits:
- A limit describes the value that a function approaches as its input gets closer and closer to a certain point.
Derivatives:
- A derivative measures the rate of change of a function at a specific point. It represents the slope of the tangent line to the function's graph.
Integrals:
- An integral represents the area under a curve. It can be used to calculate accumulated change over a given interval.

Descriptive Statistics:
- Mean is the average of a set of data, calculated by summing all values and dividing by the number of values.
- Median is the middle value in an ordered data set.
- Mode is the value that appears most frequently in a data set.
Probability:
- Probability refers to the likelihood of an event occurring, expressed as a number between 0 and 1.

Statements and Arguments:
- A statement is a declarative sentence that can be determined as true or false.
- An argument consists of a set of premises (statements assumed to be true) that lead to a conclusion.
Quantifiers:
- Universal quantifier (∀), meaning "for all," applies a statement to every member of a set.
- Existential quantifier (∃), meaning "there exists," asserts that at least one member of a set satisfies a statement.

Types of Proofs:
- Direct proof: Starts with known facts and uses logical steps to arrive at a conclusion.
- Indirect proof: Assumes the opposite of what we want to prove and shows that this assumption leads to a contradiction.
Common Proof Techniques:
- Mathematical Induction: Proving a statement for a base case and then showing that if the statement holds for a particular case, it also holds for the next case.

Real-world Applications:
- Mathematics is used in a wide range of fields, including finance (calculating interest, managing investments), engineering (designing structures, optimizing systems), medicine (conducting statistical studies, developing treatments), and computer science (developing algorithms, designing software).