Introduction to Mathematics

Core areas of Mathematics

• Arithmetic involves basic operations like addition, subtraction, multiplication, and division.
• Algebra uses symbols and letters to represent numbers and quantities in formulas and equations.
• Geometry studies shapes, sizes, positions, and properties of space.
• Trigonometry focuses on the relationships between angles and sides of triangles.
• Calculus deals with continuous change, rates of change, and accumulation.

Number Systems

• Natural Numbers are positive whole numbers (1, 2, 3, ...).
• Whole Numbers include natural numbers and zero (0, 1, 2, 3, ...).
• Integers consist of whole numbers and their negatives (... -3, -2, -1, 0, 1, 2, 3, ...).
• 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, examples include √2 or π.
• Real Numbers encompass all rational and irrational numbers.
• Complex Numbers are in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1).

Basic Operations

• Addition combines two or more numbers to find their sum (e.g., 2 + 3 = 5).
• Subtraction finds the difference between two numbers (e.g., 5 - 2 = 3).
• Multiplication is the repeated addition of a number (e.g., 2 x 3 = 6).
• Division splits a number into equal parts (e.g., 6 ÷ 2 = 3).
• Order of Operations follows PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).

Algebra Basics

• Variables are symbols (usually letters) representing unknown quantities.
• Expressions combine numbers, variables, and operations (e.g., 3x + 2y - 5).
• Equations are statements showing equality between two expressions (e.g., 3x + 2 = 8).
• Solving Equations involves finding the value(s) of the variable(s) that make the equation true.
• Linear Equations are equations where the highest power of the variable is 1 (e.g., ax + b = c).
• Quadratic Equations are equations where the highest power of the variable is 2 (e.g., ax² + bx + c = 0).

Geometry Fundamentals

• Points are exact locations in space, represented by dots.
• Lines are straight paths extending infinitely in both directions.
• Line Segments are parts of a line with two endpoints.
• Rays are parts of a line with one endpoint, extending infinitely in one direction.
• Angles are formed by two rays sharing a common endpoint (vertex), measured in degrees or radians.
• Triangles are three-sided polygons.
• Quadrilaterals are four-sided polygons.
• Circles are sets of all points equidistant from a center point.

Measurement

• Length is the distance between two points, measured in units like meters, feet, inches, etc.
• Area represents the amount of surface covered by a two-dimensional shape, measured in square units.
• Volume indicates the amount of space occupied by a three-dimensional object, measured in cubic units.
• Mass is the amount of matter in an object, measured in kilograms, grams, etc.
• Time is the duration of events, measured in seconds, minutes, hours, etc.

Data Handling and Statistics

• Data Collection involves gathering information through surveys, experiments, or observations.
• Data Representation presents data using tables, charts, graphs (bar graphs, pie charts, line graphs).
• Mean is the average of a set of numbers calculated as the sum of numbers divided by the count of numbers.
• Median is the middle value in a sorted set of numbers.
• Mode is the value that appears most frequently in a set of numbers.
• Range is the difference between the largest and smallest values in a set of numbers.

Trigonometry Fundamentals

• Trigonometric Ratios are ratios of sides in a right-angled triangle, including sine, cosine, and tangent.
• Sine (sin) is calculated as Opposite / Hypotenuse.
• Cosine (cos) is calculated as Adjacent / Hypotenuse.
• Tangent (tan) is calculated as Opposite / Adjacent.
• Pythagorean Theorem states that in a right-angled triangle, a² + b² = c² (where c is the hypotenuse).
• Angles of Elevation and Depression are angles formed by a horizontal line and the line of sight.

Calculus Introduction

• Limits are the value that a function approaches as the input approaches some value.
• Derivatives measure how a function changes as its input changes (rate of change).
• Integrals measure the area under a curve (accumulation).
• Differentiation is the process of finding the derivative of a function.
• Integration is the process of finding the integral of a function.

Mathematical Reasoning

• Deductive Reasoning involves drawing conclusions based on logical rules and given information.
• Inductive Reasoning involves making generalizations based on observations and patterns.
• Proof by Contradiction assumes the opposite of what is to be proven and shows that it leads to a contradiction.
• Mathematical Induction proves a statement for all natural numbers by showing it holds for the base case and then proving the inductive step.

Sets and Logic

• Set Theory studies sets, which are collections of objects.
• Set Operations include union, intersection, complement, and difference.
• Logic studies reasoning and argumentation.
• Propositional Logic uses truth tables and logical operators (AND, OR, NOT, etc.) to analyze statements.

Coordinate Geometry

• Cartesian Plane is a plane with two perpendicular axes (x-axis and y-axis) used to locate points.
• Coordinates are ordered pairs (x, y) that specify the position of a point on the Cartesian plane.
• Distance Formula is √((x₂ - x₁)² + (y₂ - y₁)²).
• Slope measures the steepness of a line (rise over run).
• Equation of a Line is y = mx + c (where m is the slope and c is the y-intercept).

Advanced Topics (Overview)

• Linear Algebra studies vectors, matrices, and linear transformations.
• Abstract Algebra studies algebraic structures like groups, rings, and fields.
• Real Analysis is a rigorous study of real numbers, sequences, series, and functions.
• Complex Analysis studies complex numbers and functions.
• Topology studies the properties of spaces that are preserved under continuous deformations.

Application of Maths

• Science: Physics, chemistry, biology all rely heavily on mathematical models and techniques.
• Engineering: Crucial for design, analysis, and optimization in all engineering disciplines.
• Computer Science: Algorithms, data structures, and cryptography are based on mathematical principles.
• Economics: Mathematical models are used for forecasting, optimization, and understanding economic phenomena.
• Finance: Used in pricing derivatives, managing risk, and portfolio optimization.
• Statistics: Essential for data analysis, inference, and decision-making in various fields.