Introduction to Core Mathematics

• Mathematics is the abstract study of topics such as quantity, structure, space, and change.

Core Areas of Mathematics

• Arithmetic studies numbers and basic operations like addition, subtraction, multiplication, and division.
• Algebra studies mathematical symbols and the rules for manipulating these symbols.
• Geometry studies shapes, sizes, positions of figures, and properties of space.
• Calculus studies continuous change and includes topics like limits, derivatives, integrals, and functions.
• Trigonometry studies relationships between angles and sides of triangles.

Number Systems

• Natural Numbers (N): Positive integers starting from 1 (1, 2, 3, ...).
• Whole Numbers (W): Natural numbers including 0 (0, 1, 2, 3, ...).
• Integers (Z): Whole numbers and their negatives (... -3, -2, -1, 0, 1, 2, 3, ...).
• Rational Numbers (Q): Numbers that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0 (e.g., 1/2, -3/4, 5).
• Irrational Numbers: Numbers that cannot be expressed as a fraction of two integers, having non-repeating, non-terminating decimal representations (e.g., √2, π).
• Real Numbers (R): Includes both rational and irrational numbers, representing all points on a number line.
• Complex Numbers (C): Numbers in the form a + bi, where a and b are real numbers, and i is the imaginary unit (√-1).

Basic Operations

• Addition: Combining two or more numbers to find their sum.
• Subtraction: Finding the difference between two numbers.
• Multiplication: Repeated addition of a number.
• Division: Splitting a number into equal parts.
• Exponentiation: Raising a number to a power.
• Root Extraction: Finding the root of a number.

Algebra Fundamentals

• Variables: Symbols representing unknown values.
• Expressions: Combinations of variables, numbers, and operations.
• Equations: Statements asserting the equality of two expressions.
• Polynomials: Expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

Geometry Concepts

• Points: Exact locations in space.
• Lines: Straight paths extending infinitely in both directions.
• Planes: Flat surfaces extending infinitely in all directions.
• Angles: Formed by two rays sharing a common endpoint (vertex).
• Triangles: Three-sided polygons.
• Quadrilaterals: Four-sided polygons.
• Circles: Set of all points equidistant from a center point.

Calculus Principles

• Limits: The value that a function approaches as the input approaches some value.
• Derivatives: Measure the instantaneous rate of change of a function.
• Integrals: Calculate the area under a curve.
• Functions: Relationships between input and output values.

Trigonometry Basics

• Sine (sin): Ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.
• Cosine (cos): Ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle.
• Tangent (tan): Ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle.
• Pythagorean Theorem: In a right-angled triangle, a² + b² = c², where a and b are the lengths of the legs and c is the length of the hypotenuse.