Fundamental Concepts of Mathematics

Fundamental Concepts

• Arithmetic is the foundation of mathematics, encompassing basic operations like addition, subtraction, multiplication, and division.
• Algebra involves using symbols (variables) to represent unknown quantities and solve equations.
• Geometry deals with shapes, their properties, and relationships in space.
• Calculus focuses on continuous change and includes differential and integral calculus.

Number Systems

• Natural numbers (counting numbers): 1, 2, 3, ...
• Whole numbers: 0, 1, 2, 3, ...
• Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
• Rational numbers: numbers that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
• Irrational numbers: numbers that cannot be expressed as a fraction of two integers. Examples include π and √2.
• Real numbers: the set of all rational and irrational numbers.
• Imaginary numbers: numbers involving the square root of -1 (represented by 'i').
• Complex numbers: numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.

Basic Operations

• Addition combines two or more numbers to find a sum.
• Subtraction finds the difference between two numbers.
• Multiplication combines equal groups of numbers to find a product.
• Division splits a number into equal parts to find a quotient.
• Order of operations (PEMDAS/BODMAS): dictates the sequence in which calculations are performed (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).

Algebraic Equations

• An equation states that two mathematical expressions are equal.
• Variables represent unknown quantities.
• Solving equations involves isolating the variable to find its value.
• Linear equations have the form ax + b = 0, where a and b are constants and x is a variable.
• Quadratic equations have the form ax² + bx + c = 0, where a, b, and c are constants and x is a variable.

Geometric Shapes

• Lines: straight paths that extend infinitely in both directions.
• Angles: formed by two rays sharing a common endpoint.
• Triangles: polygons with three sides and three angles.
• Circles: sets of points equidistant from a central point.
• Polygons: closed figures with straight sides.
• Solids: three-dimensional shapes.

Measurement

• Units of measurement are used to quantify physical quantities.
• Length, area, volume, and time are examples of various measurements.
• Common units include meters, centimeters, kilometers (length); square meters (area); cubic meters (volume); and seconds, minutes, hours (time).

Sets

• Sets are collections of objects.
• Set operations include union, intersection, and complement.
• Venn diagrams are used to visualize relationships between sets.

Logic

• Premises and conclusions
• Inductive and deductive reasoning
• True statements
• False statements
• Conditional statements; and their logical equivalents (e.g., contrapositive)

Functions

• A function maps inputs to outputs.
• Functions are often expressed using a formula.
• Graphing functions helps visualize their behavior.
• Linear functions, Quadratic functions, Exponential functions and logarithmic functions are common types.

Statistics

• Organizing, analyzing, and interpreting data
• Measures of central tendency (mean, median, mode)
• Measures of spread (range, standard deviation)

Probability

• Probability quantifies the likelihood of events.
• Sample space encompasses all possible outcomes in an experiment.
• Probability values range from 0 (impossible) to 1 (certain).

Discrete Mathematics

• Deals with objects that can be counted (discrete).
• Includes topics like graph theory, combinatorics, and logic.