Introduction to Mathematics

• Mathematics is the study of numbers, shapes, quantities, and patterns.
• It is a fundamental science used in various fields, including natural science, engineering, medicine, finance, and social sciences.
• Mathematics provides essential tools for understanding and modeling the world.

Core Areas of Mathematics

• Arithmetic involves basic operations on numbers: addition, subtraction, multiplication, and division.
• Algebra deals with symbols and the rules for manipulating these symbols; it is a generalization of arithmetic.
• Geometry studies shapes, sizes, positions, and properties of space.
• Calculus is concerned with continuous change, including rates and accumulation; it is divided into differential and integral calculus.
• Trigonometry studies relationships between angles and sides of triangles.
• Statistics involves collecting, analyzing, interpreting, presenting, and organizing data.
• Probability deals with the likelihood of events occurring.

Arithmetic

• Whole numbers are non-negative integers (0, 1, 2, ...).
• Integers include all whole numbers and their negatives (... -2, -1, 0, 1, 2, ...).
• Rational numbers can be expressed as a fraction p/q, where p and q are integers and q is not zero.
• Irrational numbers cannot be expressed as a fraction, such as pi (π) or the square root of 2.
• Real numbers include all rational and irrational numbers.
• Complex numbers have a real and imaginary part, in the form a + bi, where i is the imaginary unit (i² = -1).
• Addition is combining two numbers to find their sum.
• Subtraction is finding the difference between two numbers.
• Multiplication is repeated addition of a number.
• Division is splitting a number into equal parts.
• Order of operations is typically remembered by the acronym PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).

Algebra

• Variables are symbols (usually letters) representing unknown or changing quantities.
• Expressions are combinations of variables, numbers, and operations.
• Equations state the equality of two expressions.
• Linear equations involve variables raised to the first power; they can be written in the form y = mx + b.
• Quadratic equations involve variables raised to the second power; they can be written in the form ax² + bx + c = 0.
• Polynomials are expressions containing variables raised to non-negative integer powers.
• Factoring is the process of breaking down an expression into its factors.
• Solving equations involves finding the values of the variables that make the equation true.
• Systems of equations are sets of two or more equations with the same variables.
• Matrices are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns.

Geometry

• Points are locations in space.
• Lines are straight paths extending infinitely in both directions.
• Planes are flat surfaces extending infinitely in all directions.
• Angles are formed by two rays sharing a common endpoint (vertex).
• Triangles are three-sided polygons; they can be classified as equilateral, isosceles, or scalene based on their side lengths, and as acute, right, or obtuse based on their angles.
• Quadrilaterals are four-sided polygons; examples include squares, rectangles, parallelograms, and trapezoids.
• Circles are sets of points equidistant from a center point.
• Perimeter is the distance around a two-dimensional shape.
• Area is the amount of surface covered by a two-dimensional shape.
• Volume is the amount of space occupied by a three-dimensional object.
• Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (a² + b² = c²).
• Coordinate geometry uses a coordinate system to represent geometric shapes and solve geometric problems.

Calculus

• Functions describe relationships between variables, where each input has a unique output.
• Limits describe the value that a function approaches as the input approaches some value.
• Derivatives measure the instantaneous rate of change of a function.
• Integrals measure the accumulation of a function over an interval.
• Differential calculus deals with finding derivatives.
• Integral calculus deals with finding integrals.
• The fundamental theorem of calculus connects differentiation and integration.

Trigonometry

• Trigonometric functions relate angles of a right triangle to ratios of its sides.
• Sine (sin) is the ratio of the opposite side to the hypotenuse.
• Cosine (cos) is the ratio of the adjacent side to the hypotenuse.
• Tangent (tan) is the ratio of the opposite side to the adjacent side.
• These functions are periodic and used to model wave phenomena.
• Inverse trigonometric functions find the angle given a trigonometric ratio.

Statistics

• Descriptive statistics summarize and present data using measures like mean, median, mode, and standard deviation.
• Mean is the average of a set of numbers.
• Median is the middle value in a sorted set of numbers.
• Mode is the value that appears most frequently in a set of numbers.
• Standard deviation measures the spread or dispersion of data around the mean.
• Inferential statistics draw conclusions and make predictions based on a sample of data.
• Hypothesis testing is a method for testing claims about a population based on a sample.
• Regression analysis examines the relationship between variables.

Probability

• Probability is the measure of the likelihood that an event will occur.
• It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
• Events are outcomes of random phenomena.
• Independent events are events whose outcomes do not affect each other.
• Dependent events are events whose outcomes affect each other.
• Conditional probability measures the probability of an event given that another event has occurred.

Mathematical Reasoning and Proof

• Mathematical statements are declarative sentences that are either true or false.
• Axioms are statements that are assumed to be true without proof.
• Theorems are statements that have been proven to be true based on axioms and previously proven theorems.
• Proofs are logical arguments that establish the truth of a theorem.
• Direct proof starts with the assumptions and uses logical steps to arrive at the conclusion.
• Indirect proof (proof by contradiction) assumes the negation of the statement and shows that it leads to a contradiction.
• Mathematical induction proves statements that hold for all natural numbers.

Mathematics is the study of numbers, shapes, patterns, and change. It provides tools for understanding the world and is used in various fields. Core areas include arithmetic, algebra, geometry, and calculus.