Basic Concepts in Mathematics

Numbers
- Natural Numbers: Positive integers (1, 2, 3, ...)
- Whole Numbers: Natural numbers plus zero (0, 1, 2, ...)
- Integers: Whole numbers and their negatives (..., -3, -2, -1, 0, 1, 2, 3, ...)
- Rational Numbers: Numbers that can be expressed as a fraction (e.g., 1/2, 3/4)
- Irrational Numbers: Numbers that cannot be expressed as a simple fraction (e.g., √2, π)
Basic Operations
- Addition (+)
- Subtraction (−)
- Multiplication (×)
- Division (÷)

Variables and Expressions
- Variables: Symbols representing numbers (e.g., x, y)
- Expressions: Combinations of numbers and variables (e.g., 2x + 3)
Equations
- An equation states that two expressions are equal (e.g., 2x + 3 = 7)
- Solutions: Values of the variable that make the equation true
Functions
- Definition: A relation where each input has exactly one output (e.g., f(x) = x^2)
- Types: Linear, quadratic, polynomial, exponential, logarithmic

Shapes and Properties
- Common Shapes: Circle, triangle, square, rectangle
- Properties: Area, perimeter, volume
Theorems
- Pythagorean Theorem: In a right triangle, a² + b² = c² (where c is the hypotenuse)
- Properties of Angles: Complementary (sum = 90°), supplementary (sum = 180°)

Functions
- Sine (sin), cosine (cos), tangent (tan)
- Relationships in right triangles:
  - sin(θ) = opposite/hypotenuse
  - cos(θ) = adjacent/hypotenuse
  - tan(θ) = opposite/adjacent
Unit Circle
- Circle with radius 1, used to define trigonometric functions for all angles.

Limits
- Concept of approaching a value (e.g., lim (x→a) f(x))
Derivatives
- Measure of how a function changes as its input changes (f'(x))
- Applications: Tangent lines, rates of change
Integrals
- Represents the accumulation of quantities (e.g., area under a curve)
- Fundamental Theorem of Calculus: Links differentiation and integration.

Descriptive Statistics
- Mean: Average value
- Median: Middle value in a sorted list
- Mode: Most frequently occurring value
- Range: Difference between the highest and lowest values
Probability
- Definition: Measure of likelihood of an event occurring
- Basic Formula: P(A) = Number of favorable outcomes / Total number of outcomes

Logic
- Statements: Propositions that can be true or false
- Logical Operators: AND, OR, NOT
Proofs
- Direct Proof: Demonstrating truth through logical steps
- Indirect Proof: Assuming the opposite to show a contradiction

Understanding the Problem
- Read carefully, identify what is known and what needs to be found.
Devising a Plan
- Choose appropriate strategies (e.g., drawing diagrams, creating equations).
Carrying out the Plan
- Implement chosen strategies step by step.
Reviewing/Reflecting
- Check if the solution makes sense and is correct.

Natural Numbers: Positive integers starting from 1 (1, 2, 3,...).
Whole Numbers: Includes natural numbers plus zero (0, 1, 2,...).
Integers: Comprises whole numbers and their negative counterparts (..., -3, -2, -1, 0, 1, 2, 3,...).
Rational Numbers: Can be expressed as fractions, such as 1/2 or 3/4.
Irrational Numbers: Cannot be represented as a fraction, examples include √2 and π.
Basic operations include Addition (+), Subtraction (−), Multiplication (×), and Division (÷).

Variables represent numbers, commonly denoted by symbols like x or y.
Expressions are combinations involving numbers and variables, such as 2x + 3.
An Equation signifies equality between two expressions (e.g., 2x + 3 = 7).
Solutions are values that satisfy an equation.
A function links each input to exactly one output, exemplified by f(x) = x².
Types of functions include Linear, Quadratic, Polynomial, Exponential, and Logarithmic.

Common shapes studied include Circle, Triangle, Square, and Rectangle.
Properties of shapes involve Area, Perimeter, and Volume.
Pythagorean Theorem: In a right triangle, a² + b² = c², with c as the hypotenuse.
Angle properties: Complementary angles sum to 90°, while supplementary angles sum to 180°.

Trigonometric functions include Sine (sin), Cosine (cos), and Tangent (tan).
Relationships in right triangles are defined as:
- sin(θ) = opposite/hypotenuse
- cos(θ) = adjacent/hypotenuse
- tan(θ) = opposite/adjacent.
Unit Circle: A circle with a radius of 1, crucial for defining trigonometric functions for all angles.

Limits refer to values that a function approaches (e.g., lim (x→a) f(x)).
Derivatives indicate how a function's output changes as its input varies, denoted as f'(x).
Applications of derivatives include finding tangent lines and evaluating rates of change.
Integrals signify the accumulation of quantities, often calculating area under a curve.
Fundamental Theorem of Calculus connects differentiation with integration.

Descriptive Statistics includes:
- Mean: The average of a dataset.
- Median: The middle value in a sorted list of numbers.
- Mode: The most frequently occurring value in a dataset.
- Range: The difference between the highest and lowest values in a dataset.
Probability measures the likelihood of an event occurring.
Basic probability formula: P(A) = Number of favorable outcomes / Total number of outcomes.

Logic involves statements that can either be true or false.
Logical Operators include AND, OR, and NOT for constructing more complex statements.
Proofs can be Direct, showing truth through sequential logic, or Indirect, demonstrating truth by contradiction.

Understanding the Problem: Carefully read and identify known information versus what needs to be found.
Devising a Plan: Select suitable strategies, such as diagrams or equations, to tackle the problem.
Carrying out the Plan: Implement the chosen strategies step by step for clarity and accuracy.
Reviewing/Reflecting: Check if the solution is logical, reasonable, and correct, allowing for adjustments if necessary.