Introduction to Mathematics

• Mathematics is the abstract science of number, quantity, and space.
• Mathematics may be used as a tool for solving problems.
• Applied mathematics is the branch of mathematics concerned with the application of mathematical knowledge to other fields.
• Pure mathematics is mathematics studied for its own sake.

Branches of Mathematics

• Arithmetic studies numbers and operations on them.
• Algebra studies mathematical symbols and the rules for manipulating these symbols.
• Geometry is concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.
• Trigonometry studies relationships between angles and sides of triangles.
• Calculus studies continuous change.
• Statistics is the study of the collection, analysis, interpretation, presentation, and organization of data.

Key Concepts

• Number: A fundamental concept representing quantity.
• Variable: A symbol representing an unknown quantity.
• Equation: A statement that asserts the equality of two expressions.
• Function: A relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
• Theorem: A statement that has been proven to be true based on previously established statements.
• Proof: A logical argument that establishes the truth of a statement.

Number Systems

• Natural Numbers: The set of positive integers (1, 2, 3, ...).
• Integers: The set of whole numbers and their negatives (... -2, -1, 0, 1, 2, ...).
• Rational Numbers: Numbers that can be expressed as a fraction p/q, where p and q are integers and q is not zero.
• Irrational Numbers: Numbers that cannot be expressed as a fraction of two integers.
• Real Numbers: The set of all rational and irrational numbers.
• Complex Numbers: Numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit (i² = -1).

Arithmetic Operations

• Addition: Combining two numbers to find their sum.
• Subtraction: Finding the difference between two numbers.
• Multiplication: Repeated addition of a number.
• Division: Splitting a number into equal parts.
• Exponentiation: Raising a number to a power.
• Root Extraction: Finding the root of a number.

Algebraic Concepts

• Expressions: Combinations of variables, constants, and operations.
• Polynomials: Expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• Factoring: Decomposing an expression into a product of simpler expressions.
• Solving Equations: Finding the values of variables that make an equation true.
• Inequalities: Mathematical statements that compare two expressions using symbols like <, >, ≤, or ≥.

Geometric Concepts

• Points: A location in space.
• Lines: A straight, one-dimensional figure extending infinitely in both directions.
• Planes: A flat, two-dimensional surface extending infinitely.
• Angles: The measure of the rotation between two lines or rays.
• Shapes: Two-dimensional figures with defined boundaries.
• Solids: Three-dimensional objects with volume.
• Congruence: The property of two figures having the same shape and size.
• Similarity: The property of two figures having the same shape but different sizes.

Trigonometric Concepts

• Trigonometric Ratios: Ratios relating the sides of a right triangle to its angles (sine, cosine, tangent, etc.).
• Unit Circle: A circle with a radius of 1 used to define trigonometric functions for all real numbers.
• Trigonometric Identities: Equations that are true for all values of the variables involved.
• Solving Triangles: Finding the unknown angles and sides of a triangle using trigonometric ratios.

Calculus Concepts

• Limits: The value that a function approaches as the input approaches some value.
• Derivatives: The rate of change of a function with respect to its input.
• Integrals: The accumulation of a function over an interval.
• Fundamental Theorem of Calculus: Connects differentiation and integration.
• Applications: Optimization, related rates, area/volume calculations.

Statistical Concepts

• Data: Information collected for analysis.
• Descriptive Statistics: Methods for summarizing and presenting data.
• Inferential Statistics: Methods for drawing conclusions about a population based on a sample.
• Probability: The measure of the likelihood that an event will occur.
• Distributions: Mathematical functions that describe the probability of different outcomes in a population.
• Hypothesis Testing: A method for testing a claim about a population based on sample data.