Basic Arithmetic and Number Systems

Basic Arithmetic Operations

• Addition: Combining two or more numbers to find a total. The symbol is '+'.
• Subtraction: Finding the difference between two numbers. The symbol is '-'.
• Multiplication: Repeated addition of the same number. The symbol is '×' or '*'.
• Division: Finding how many times one number goes into another. The symbol is '÷' or '/'.
• Order of operations (PEMDAS/BODMAS): A set of rules for evaluating expressions. The order is Parentheses/Brackets, Exponents/Orders, Multiplication and Division (left to right), Addition and Subtraction (left to right).

Number Systems

• Natural numbers: The counting numbers (1, 2, 3,...).
• Whole numbers: Natural numbers plus zero (0, 1, 2, 3,...).
• Integers: Whole numbers and their opposites (-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 is not zero. Examples include fractions, terminating decimals, and repeating decimals.
• Irrational numbers: Numbers that cannot be expressed as a fraction of two integers. Examples include π (pi) and the square root of 2.
• Real numbers: The set of all rational and irrational numbers.
• Imaginary numbers: Numbers involving the square root of -1, denoted as 'i'.

Basic Algebra

• Variables: Symbols used to represent unknown quantities.
• Expressions: Combinations of variables, numbers, and operations.
• Equations: Statements that two expressions are equal.
• Solving equations: Finding the value of the variable that makes the equation true.
• Inequalities: Statements comparing two expressions using symbols like <, >, ≤, ≥.
• Solving inequalities: Finding the values of the variable that make the inequality true.

Geometry

• Points, lines, and planes: Fundamental geometric objects.
• Angles: Formed by two rays sharing a common endpoint.
• Polygons: Closed figures formed by line segments.
• Triangles: Three-sided polygons.
• Quadrilaterals: Four-sided polygons.
• Circles: Set of points equidistant from a center point.
• Area: The measure of the surface enclosed by a two-dimensional shape.
• Perimeter: The distance around a two-dimensional shape.
• Volume: The measure of the space occupied by a three-dimensional object.

Measurement

• Units of measurement: Standardized units for quantifying physical quantities (e.g., meters, kilograms, seconds).
• Length, width, height: Dimensions of objects.
• Area calculation formulas for various shapes (e.g., rectangle, triangle, circle).
• Volume calculation formulas for various shapes (e.g., cube, rectangular prism, cylinder).
• Converting units of measurement (e.g., inches to centimeters).

Data Analysis

• Data collection: Gathering information.
• Data representation: Presenting data visually (e.g., charts, graphs).
• Measures of central tendency: Mean, median, mode.
• Measures of variability: Range, standard deviation.
• Data interpretation: Analyzing trends and patterns in the data.

Functions

• Relation: A set of ordered pairs.
• Functions: A special type of relation where each input (x-value) has only one output (y-value). A function assigns a unique output to each input.
• Domain: The set of all possible input values.
• Range: The set of all possible output values.
• Function notation: Expressing a function in the form 'f(x)' to indicate the output when the input is 'x'.

Probability

• Probability: The measure of the likelihood of an event occurring.
• Sample space: The set of all possible outcomes.
• Event: A subset of the sample space.
• Simple probability: Calculating the probability of a single event.
• Compound probability: Calculating the probability of multiple events.
• Conditional probability: Probability of an event given that another event has occurred.

Exponents and Roots

• Exponents: Short cuts for repeated multiplication.
• Roots: Opposite to exponents. Finds the base number when the power is known.

Sets

• Sets: Collections of objects.
• Union, intersection, complements of sets. Fundamental operations among sets.

Complex Numbers

• Complex numbers are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit (i² = -1).

Matrices

• Matrices: Rectangular arrays of numbers.
• Matrix operations: Addition, subtraction, multiplication, etc.

Trigonometry

• Trigonometric functions (sine, cosine, tangent) relate angles in a right-angled triangle to the ratios of its sides.
• Trigonometric identities: Equations involving trigonometric functions that are true for all angles.
• Applications of trigonometry: Solving triangles, finding distances and angles.