Introduction to Mathematics and Core concepts

• Mathematics is the abstract science of number, quantity, and space.
• Mathematics may be studied in purely abstract terms or as applied to other disciplines.

Core Areas of Mathematics

• Arithmetic involves the study of numbers and basic operations like addition, subtraction, multiplication, and division.
• Algebra involves the study of mathematical symbols and the rules for manipulating these symbols.
• Geometry involves the study of shapes, sizes, and positions of figures.
• Calculus involves the study of continuous change, rates of change, and accumulation.
• Trigonometry involves the study of relationships between angles and sides of triangles
• Statistics involves the collection, analysis, interpretation, presentation, and organization of data.
• Probability involves the measure of the likelihood that an event will occur.

Arithmetic

• The fundamental operations are addition, subtraction, multiplication, and division.
• Number types include natural numbers, integers, rational numbers, and real numbers.
• Order of operations follows the PEMDAS/BODMAS rule (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).
• Fractions represent parts of a whole and involve numerators and denominators.
• Decimals are a way to represent numbers that are not whole, using a decimal point.
• Percentages express a number as a fraction of 100.
• Ratios compare two quantities.
• Proportions state that two ratios are equal.

Algebra

• Variables are symbols representing unknown values.
• Expressions are combinations of variables, numbers, and operations.
• Equations state that two expressions are equal.
• Solving equations finds the value(s) of the variable(s) that make the equation true.
• Linear equations involve variables raised to the power of 1.
• Quadratic equations involve variables raised to the power of 2.
• Systems of equations involve two or more equations with the same variables.
• Inequalities compare expressions using symbols like <, >, ≤, and ≥.
• Functions are relations that map each input to a unique output.
• Polynomials are expressions with variables raised to non-negative integer powers.

Geometry

• Points, lines, and planes are fundamental undefined terms in geometry.
• Angles are formed by two rays sharing a common endpoint (vertex).
• Types of angles include acute, obtuse, right, and straight angles.
• Triangles are three-sided polygons; classified by sides (equilateral, isosceles, scalene) and angles (acute, obtuse, right).
• Quadrilaterals are four-sided polygons; types include squares, rectangles, parallelograms, and trapezoids.
• Circles are sets of points equidistant from a center.
• Area measures the extent of a two-dimensional surface.
• Volume measures the space occupied by a three-dimensional object.
• Pythagorean theorem relates the sides of a right triangle: a² + b² = c².

Calculus

• Limits describe the behavior of a function as its input approaches a certain value.
• Derivatives measure the rate of change of a function.
• Integrals calculate the accumulation of a function's values.
• Differentiation finds the derivative of a function.
• Integration finds the integral of a function.
• Applications of calculus include optimization, related rates, and area/volume calculations.
• Fundamental Theorem of Calculus connects differentiation and integration.

Trigonometry

• Trigonometric functions (sine, cosine, tangent) relate angles of a right triangle to the ratios of its sides.
• Unit circle provides a visual representation of trigonometric functions.
• Trigonometric identities are equations that are true for all values of the variables.
• Inverse trigonometric functions find the angle given a trigonometric ratio.
• Law of Sines and Law of Cosines relate sides and angles in non-right triangles.
• Applications include navigation, surveying, and physics.

Statistics

• Data collection involves gathering information.
• Descriptive statistics summarize and describe data using measures like mean, median, and mode.
• Inferential statistics makes inferences and predictions about a population based on a sample.
• Probability distributions describe the likelihood of different outcomes.
• Hypothesis testing assesses the evidence for or against a claim about a population.
• Regression analysis models the relationship between variables.

Probability

• Sample space is the set of all possible outcomes of an experiment.
• Events are subsets of the sample space.
• Probability of an event is the measure of the likelihood that the event will occur.
• Independent events are events whose outcomes do not affect each other.
• Conditional probability is the probability of an event given that another event has occurred.
• Expected value is the average outcome of an experiment.
• Combinations and permutations count the number of ways to choose or arrange items.