Basic Arithmetic and Number Systems

Basic Arithmetic Operations

• Addition: Combining two or more numbers to find a total. Commutative property: order doesn't matter (a + b = b + a). Associative property: grouping doesn't matter ((a + b) + c = a + (b + c)).
• Subtraction: Finding the difference between two numbers. Inverse operation of addition.
• Multiplication: Repeated addition of the same number. Commutative and associative properties also apply. Distributive property: a(b + c) = ab + ac.
• Division: Finding how many times one number is contained within another. Inverse operation of multiplication. Division by zero is undefined.

Number Systems

• Natural numbers (counting numbers): 1, 2, 3,...
• Whole numbers: 0, 1, 2, 3,... Includes natural numbers and zero.
• Integers:..., -3, -2, -1, 0, 1, 2, 3,... Includes whole numbers and their negative counterparts.
• Rational numbers: Numbers that can be expressed as a fraction p/q, where p and q are integers and q is not zero. Includes terminating and repeating decimals.
• Irrational numbers: Numbers that cannot be expressed as a fraction of two integers. Examples include π and the square root of 2.
• Real numbers: The set of all rational and irrational numbers.

Algebra

• Variables: Symbols (like x, y, z) used to represent unknown values.
• Equations: Statements that show the equality of two expressions. Examples: 2x + 3 = 7, y² = 9.
• Inequalities: Statements that show the relationship between two expressions that are not necessarily equal. Examples: x > 5, 2y ≤ 10.
• Linear equations: Equations that form a straight line on a graph. General form: Ax + By = C.
• Polynomials: Expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication.
• Factoring: Expressing a polynomial as a product of simpler polynomials.
• Exponents: Repeated multiplication of a number by itself.
• Roots: The opposite of an exponent.

Geometry

• Points: Basic building blocks of geometry.
• Lines: One-dimensional figures with no thickness.
• Planes: Two-dimensional figures.
• Angles: Formed by two rays sharing a common endpoint.
• Polygons: Figures formed by line segments.
• Triangles: Three-sided polygons. Types include equilateral, isosceles, scalene, right, acute, obtuse.
• Quadrilaterals: Four-sided polygons.
• Circles: A set of points equidistant from a center point.
• Perimeter: The distance around a two-dimensional shape.
• Area: The space enclosed by a two-dimensional shape.
• Volume: The space occupied by a three-dimensional shape.
• Pythagorean theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²).

Data Analysis

• Measures of central tendency: Mean (average), median (middle value), mode (most frequent value).
• Measures of dispersion: Range, variance, standard deviation (how spread out the data is).
• Data representation: Tables, charts, graphs (histograms, bar charts, scatter plots).

Set Theory

• Sets: Collections of objects.
• Subsets: A set whose members are also members of another set.
• Union: Combining sets to form a larger set containing all elements.
• Intersection: Finding the elements that are common to two or more sets.

Functions

• Functions: A relation where each input has only one output. Represented with notation like f(x) =...
• Domain and range: Domain is the set of possible inputs (x-values); range is the set of possible outputs (y-values).
• Linear functions: Functions that graph as straight lines.
• Quadratic functions: Functions that graph as parabolas.
• Exponential functions: Functions where the input appears as an exponent.

Trigonometry

• Trigonometric functions: Relate angles and sides of right-angled triangles (sine, cosine, tangent, etc.).
• Unit circle: A circle with a radius of 1 unit centered at the origin in a coordinate plane. Trigonometric functions are defined using this circle.
• Special angles: Common angles like 30°, 45°, 60°, and 90° have specific values for trigonometric functions.
• Applications: Trigonometry is used in navigation, surveying, and engineering.

Calculus

• Limits: Approaching a value.
• Derivatives: Rate of change.
• Integrals: Accumulated change.
• Applications: Calculus is used in physics, engineering, and economics.