Fundamentals of Mathematics

• Mathematics involves studying numbers, shapes, quantities, and patterns.
• It provides a framework for understanding the world.
• Mathematics applies to science, engineering, economics, and computer science.

Core Areas

• Arithmetic involves basic operations on numbers.
• Algebra uses symbols and rules to manipulate them.
• Geometry studies shapes, sizes, and positions.
• Calculus deals with continuous change, rates, and accumulation.
• Trigonometry studies relationships between angles and sides of triangles.
• Statistics involves data collection, analysis, interpretation, presentation, and organization.
• Probability measures how likely events are to occur.

Arithmetic

• Numbers form the basis of mathematics.
• Integers are whole numbers, including negative numbers and zero.
• Rational numbers can be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
• Real numbers include both rational and irrational numbers.
• Complex Numbers have a real and an imaginary part, in the form a + bi, where i is the imaginary unit (√-1).
• Operations include addition (+), subtraction (-), multiplication (×), and division (÷).
• Order of Operations follows PEMDAS/BODMAS.
• Properties include:
  • Commutative Property: a + b = b + a and a × b = b × a.
  • Associative Property: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
  • Distributive Property: a × (b + c) = a × b + a × c.

Algebra

• Variables are symbols representing unknown quantities.
• Expressions combine variables, numbers, and operations.
• Equations state that two expressions are equal.
• Linear Equations have a variable with the highest power of 1.
• Quadratic Equations have a variable with the highest power of 2 (ax² + bx + c = 0).
• Polynomials contain variables and coefficients with non-negative integer exponents.
• Factoring breaks down an expression into simpler expressions.
• Solving Equations involves finding variable values that make the equation true.
• Systems of Equations comprise two or more equations with the same variables.

Geometry

• Points, Lines, and Planes are basic geometric elements.
• Angles are formed by two rays sharing a vertex.
• Triangles are three-sided polygons.
  • Types of triangles: Equilateral, Isosceles, Scalene, Right-angled.
  • Pythagorean Theorem: a² + b² = c² in a right-angled triangle, where c is the hypotenuse.
• Quadrilaterals are four-sided polygons.
  • Types of quadrilaterals: Square, Rectangle, Parallelogram, Trapezoid.
• Circles include points equidistant from a center.
  • Radius: Distance from center to a point on the circle.
  • Diameter: Distance across the circle through the center (2 × radius).
  • Circumference: Distance around the circle (2πr).
  • Area: πr².
• 3D Shapes have length, width, and height.
  • Types of 3D shapes: Cube, Sphere, Cylinder, Cone.
• Area measures the surface of a 2D shape.
• Volume measures the space occupied by a 3D shape.

Calculus

• Limits are values that a function approaches.
• Derivatives measure the instantaneous rate of change of a function.
• Integrals calculate the area under a curve.
  • Definite Integral calculates the area between two points.
  • Indefinite Integral finds the antiderivative of a function.
• Fundamental Theorem of Calculus connects differentiation and integration.
• Calculus is applied in optimization problems, related rates, and area/volume calculations.

Trigonometry

• Trigonometric Functions relate angles of a triangle to the ratios of its sides.
  • Sine (sin): Opposite / Hypotenuse.
  • Cosine (cos): Adjacent / Hypotenuse.
  • Tangent (tan): Opposite / Adjacent.
• Unit Circle is used to define trigonometric functions.
• Trigonometric Identities are equations involving trigonometric functions.
• Trigonometry is applied in solving triangles and modeling periodic phenomena.

Statistics

• Data Collection involves gathering information through surveys, experiments, and observations.
• Data Analysis examines data to draw conclusions.
  • Descriptive Statistics summarize data (mean, median, mode, standard deviation).
  • Inferential Statistics make predictions or inferences based on data.
• Probability Distributions describe likelihood of different outcomes.
  • Normal Distribution is a symmetric bell-shaped distribution.
• Hypothesis Testing tests claims using sample data.
• Regression Analysis determines variable relationships.

Probability

• Events are outcomes of an experiment.
• Sample Space includes all possible outcomes.
• Probability of an Event measures the likelihood of an event occurring (between 0 and 1).
• Independent Events' outcomes do not affect each other.
• Conditional Probability is the probability of an event given another event has occurred.
• Bayes' Theorem updates the probability of a hypothesis based on new evidence.

Mathematical Reasoning

• Logic includes principles of valid reasoning.
• Proofs are arguments demonstrating truth.
  • Direct Proof shows conclusion from premises.
  • Indirect Proof (Proof by Contradiction) assumes the negation of the conclusion and shows that it leads to a contradiction.
• Mathematical Induction proves a statement for all natural numbers.

Key Concepts

• Functions relate inputs to outputs, with each input having only one output.
• Sets are collections of distinct objects.
  • Set Theory studies sets.
• Relations are sets of ordered pairs.
• Mappings transfer elements from one set to another.

Applications of Mathematics

• Physics uses math to model phenomena and derive equations.
• Engineering uses math in design, analysis, and problem-solving.
• Computer Science applies math to algorithms, data structures, and cryptography.
• Economics uses math to model systems and make predictions.
• Finance uses math in pricing assets, managing risk, and making investment decisions.