Fundamental Operations and Number Systems

Fundamental Operations

• Addition: Combining two or more numbers to find their sum. The result is called the sum.
• Subtraction: Finding the difference between two numbers. The result is called the difference.
• Multiplication: Repeated addition of the same number. The result is called the product.
• Division: Breaking a number into equal parts. The result is called the quotient.

Number Systems

• Natural Numbers (ℕ): Positive integers (1, 2, 3,...).
• Whole Numbers (ℤ⁺): Non-negative integers (0, 1, 2, 3,...).
• Integers (ℤ): Whole numbers (positive, negative, and zero).
• Rational Numbers (ℚ): 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.
• Real Numbers (ℝ): The set of all rational and irrational numbers.
• Complex Numbers (ℂ): Numbers that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit (i² = -1).

Basic Algebra

• Variables: Symbols used to represent unknown values (e.g., x, y, z).
• Equations: Statements that show that two mathematical expressions are equal (e.g., x + 2 = 5).
• Inequalities: Statements that show that two mathematical expressions are not equal (e.g., x < 5).
• Polynomials: Expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication (e.g., 2x² + 3x - 1).
• Factoring: Breaking down an expression into simpler expressions that multiply together to create the original expression.
• Solving Equations: Finding the value of the variable that makes the equation true.
• Order of Operations (PEMDAS/BODMAS): The specific sequence for evaluating expressions containing multiple operations (Parentheses/Brackets, Exponents/Orders, Multiplication and Division from left to right, Addition and Subtraction from left to right).

Geometry

• Points: Basic building blocks of geometry, often represented by a dot.
• Lines: Straight paths extending infinitely in both directions.
• Angles: Formed by two lines or rays that share a common endpoint.
• Triangles: Polygons with three sides and three angles. Types include equilateral, isosceles, and scalene.
• Quadrilaterals: Polygons with four sides and four angles. Types include parallelograms, rectangles, squares, rhombuses, and trapezoids.
• Circles: A set of points equidistant from a central point.
• Perimeter: The total distance around a two-dimensional shape.
• Area: The amount of space enclosed within a two-dimensional shape.
• Volume: The amount of space occupied by a three-dimensional object.

Statistics and Probability

• Data: Sets of observations or measurements.
• Mean: The average of a set of data.
• Median: The middle value in a sorted set of data.
• Mode: The most frequent value in a set of data.
• Standard Deviation: A measure of the spread of data around the mean.
• Probability: The likelihood of an event occurring.
• Sample Space: Set of all possible outcomes of an event.

Trigonometry

• Trigonometric Functions: Functions relating the angles and sides of a right triangle (sine, cosine, tangent, etc.)
• Right Triangles: Triangles with a 90-degree angle.
• Angle Measurement: Using degrees or radians to quantify angles.
• Applications: Solving for unknown sides or angles in right triangles; modeling periodic phenomena.