Basic Concepts in Mathematics

Arithmetic
- Operations: Addition, Subtraction, Multiplication, Division
- Order of operations: PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)
Algebra
- Variables: Symbols representing numbers
- Expressions: Combinations of numbers and variables (e.g., 3x + 2)
- Equations: Statements of equality (e.g., 2x + 3 = 7)
- Functions: Relationships between variables, often expressed as f(x)
Geometry
- Shapes: Circles, triangles, squares, polygons
- Properties: Perimeter, area, volume
- Theorems: Pythagorean theorem, properties of angles (e.g., complementary, supplementary)
Trigonometry
- Ratios: Sine, cosine, tangent
- Right triangles: Relationships between angles and sides
- Unit circle: Circle of radius 1 used to define trigonometric functions
Statistics
- Data types: Qualitative vs. quantitative
- Measures: Mean, median, mode, range
- Probability: Likelihood of events occurring, calculated as favorable outcomes over total outcomes
Calculus
- Limits: Approach of a function as it nears a point
- Derivatives: Measure of how a function changes as its input changes
- Integrals: Area under a curve, accumulation of quantities
Number Theory
- Properties of integers: Even, odd, prime, composite
- Divisibility rules: Conditions under which one number can be divided by another without a remainder
- Greatest common divisor (GCD) and least common multiple (LCM)
Mathematical Logic
- Statements: Propositions that can be true or false
- Logical operators: AND, OR, NOT
- Proofs: Methods of demonstrating the truth of mathematical statements (e.g., direct proof, contradiction)
Set Theory
- Sets: Collections of distinct objects
- Operations: Union, intersection, difference
- Venn diagrams: Visual representation of set relationships

Key Mathematical Tools

Calculators: For computations and graphing
Graphs: Visual representations of functions and data
Software: Tools like MATLAB, Python, and Excel for complex calculations and modeling

Important Mathematical Constants

π (pi): Ratio of a circle's circumference to its diameter, approximately 3.14
e (Euler's number): Base of natural logarithms, approximately 2.718

Problem-Solving Strategies

Understand the problem: Read carefully, identify knowns and unknowns
Develop a plan: Choose a method or formula
Execute the plan: Calculate step by step
Review/reflect: Check for accuracy and reasonableness of the solution

Arithmetic

Fundamental operations include addition, subtraction, multiplication, and division.
Order of operations is governed by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Algebra

Variables are symbols (like x or y) that represent numbers in mathematical expressions.
An expression is a combination of numbers and variables, such as 3x + 2.
An equation is a statement asserting the equality of two expressions, e.g., 2x + 3 = 7.
Functions establish relationships across variables and can be represented as f(x).

Geometry

Basic shapes include circles, triangles, squares, and various polygons.
Essential properties to consider are perimeter (the length around a shape), area (the space contained within a shape), and volume (the space contained within a three-dimensional object).
Important theorems include the Pythagorean theorem and properties concerning angles such as complementary (sum to 90°) and supplementary (sum to 180°).

Trigonometry

Key trigonometric ratios are sine, cosine, and tangent, which relate the angles and sides of right triangles.
Right triangles feature crucial relationships between their angles and side lengths.
The unit circle, a circle with a radius of 1, is critical for defining trigonometric functions.

Statistics

Data is categorized into qualitative (non-numeric) and quantitative (numeric).
Key measures of central tendency include mean (average), median (middle value), mode (most frequent value), and range (difference between highest and lowest values).
Probability assesses the chance of events, calculated as the ratio of favorable outcomes to total outcomes.

Calculus

Limits describe how a function approaches a specific point as its input nears a value.
Derivatives measure how a function's output changes in response to changes in the input.
Integrals calculate the area beneath a curve, representing the accumulation of quantities.

Number Theory

Key properties of integers include classifications as even (divisible by 2), odd (not divisible by 2), prime (only divisible by 1 and itself), and composite (has divisors other than 1 and itself).
Divisibility rules dictate when one integer can be evenly divided by another.
GCD refers to the greatest common divisor, while LCM denotes the least common multiple of a set of numbers.

Mathematical Logic

Statements, or propositions, can either be true or false.
Common logical operators include AND (both conditions must be true), OR (at least one condition must be true), and NOT (negation of a condition).
Proofs are structured methods to validate mathematical statements, with techniques such as direct proof and proof by contradiction.

Set Theory

A set is a collection of distinct objects, defined by its elements.
Key operations involve union (combining sets), intersection (common elements), and difference (elements in one set but not in another).
Venn diagrams visually illustrate the relationships between sets.

Key Mathematical Tools

Calculators assist in computations and graphing functions and data.
Graphs serve as visual aids for understanding functions and relationships in data.
Software tools like MATLAB, Python, and Excel facilitate more complex calculations and mathematical modeling.

Important Mathematical Constants

π (pi) represents the ratio of a circle's circumference to its diameter, approximately valued at 3.14.
e (Euler's number) is crucial in calculus as the base of natural logarithms, approximately valued at 2.718.

Problem-Solving Strategies

Initially, understand the problem by carefully reading it and identifying knowns and unknowns.
Formulate a plan, selecting appropriate methods or formulas for solving the problem.
Execute the plan methodically, calculating step by step.
Conclude by reviewing the solution for accuracy and ensuring it is reasonable.