Introduction to Mathematics

• Math is the science and study of quantity, structure, space, and change.
• It uses patterns to formulate new conjectures and establishes truth by rigorous deduction from suitably chosen axioms and definitions.
• There are varied opinions among mathematicians and philosophers as to the exact scope and definition of mathematics.

Fields of Mathematics

• Arithmetic studies numbers and operations on them.
• Algebra studies algebraic structures, relations, and quantities.
• Geometry studies shapes, sizes, and positions of figures.
• Trigonometry studies relationships between angles and sides of triangles.
• Calculus studies continuous change, limits, functions, derivatives, integrals, and infinite series.
• Statistics studies the collection, analysis, interpretation, presentation, and organization of data.
• Probability studies the likelihood that an event will occur.
• Topology studies properties of spaces that are preserved under continuous deformations, such as stretching and bending.
• Number theory focuses on the properties and relationships of numbers, especially integers

Mathematical Concepts

• Axioms are statements taken to be true, serving as a starting point for deduction.
• Theorems are statements that have been proven to be true based on axioms and previously proven theorems.
• Proofs are logical arguments that demonstrate the truth of a theorem.
• Conjectures are statements proposed as true, but not yet proven.
• Algorithms are step-by-step procedures for solving a problem.
• Functions are relationships that map inputs to outputs.
• Variables are symbols representing quantities that can change.
• Constants are symbols representing fixed quantities.
• Equations are statements of equality between two expressions.
• Inequalities are statements that compare two expressions using symbols like <, >, ≤, or ≥.

Numbers

• Natural numbers are positive whole numbers (1, 2, 3,...).
• Integers include natural numbers, zero, and their negatives (..., -2, -1, 0, 1, 2, ...).
• Rational numbers can be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
• Irrational numbers cannot be expressed as a fraction of two integers (e.g., √2, π).
• Real numbers include all rational and irrational numbers.
• Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1).

Basic Operations

• Addition is the process of combining two or more numbers.
• Subtraction is the process of finding the difference between two numbers.
• Multiplication is the process of repeated addition.
• Division is the process of splitting a number into equal parts.
• Exponentiation is the process of raising a number to a power.
• Root extraction is the process of finding a number that, when raised to a power, equals a given number.

Algebra

• Variables are symbols that represent unknown or changeable values.
• Expressions are combinations of variables, constants, and operations.
• Equations are statements that two expressions are equal.
• Polynomials are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• Factoring is the process of breaking down a polynomial into simpler expressions.
• Solving equations involves finding the values of the variables that make the equation true.

Geometry

• Points are locations in space, represented by coordinates.
• Lines are straight paths extending infinitely in both directions.
• Planes are flat surfaces extending infinitely in all directions.
• Angles are formed by two rays sharing a common endpoint (vertex).
• Triangles are polygons with three sides and three angles.
• Quadrilaterals are polygons with four sides and four angles.
• Circles are sets of points equidistant from a center point.
• Polygons are closed figures formed by line segments.
• Volume is the amount of space occupied by a three-dimensional object.
• Area is the amount of surface within a two-dimensional shape.
• Perimeter is the total length of the sides of a two-dimensional shape.

Calculus

• Limits describe the value that a function approaches as the input approaches some value.
• Derivatives measure the rate of change of a function.
• Integrals are used to find the area under a curve.
• Functions are relations between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
• Series are the sum of a sequence of terms.

Mathematical Notation

• Symbols are used to represent mathematical operations, relations, and quantities.
• Common symbols include +, -, ×, ÷, =, <, >, ≤, ≥, √, π, Σ, ∫.
• Understanding notation is crucial for interpreting and communicating mathematical ideas.

Mathematical Reasoning

• Deductive reasoning is the process of drawing conclusions based on established facts and rules.
• Inductive reasoning is the process of making generalizations based on observations.
• Proofs are logical arguments that demonstrate the truth of a statement.
• Counterexamples are examples that disprove a statement.

Applications of Math

• Physics uses math to model the physical world.
• Engineering uses math to design and build structures and systems.
• Computer science uses math to develop algorithms and software.
• Economics uses math to model economic systems.
• Finance uses math to manage money and investments.
• Statistics is used across many disciplines to analyze data and draw conclusions.