Introduction to Mathematics: Arithmetic and Algebra

• Mathematics is the study of quantity (numbers), structure, space, and change.

Arithmetic

• Arithmetic involves the basic operations of addition, subtraction, multiplication, and division.
• Addition is the process of combining two or more numbers to find their total, known as the sum.
• Subtraction is the process of finding the difference between two numbers.
• Multiplication is the process of repeated addition, resulting in a product.
• Division is the process of splitting a number into equal parts, resulting in a quotient.
• The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which mathematical operations should be performed.

Algebra

• Algebra uses symbols and letters to represent numbers and quantities in formulas and equations.
• Variables are symbols (usually letters) that represent unknown values.
• Expressions are combinations of numbers, variables, and operations.
• Equations are statements that show the equality between two expressions.
• Solving an equation involves finding the value(s) of the variable(s) that make the equation true.
• Linear equations are equations where the highest power of the variable is 1.
• Quadratic equations are equations where the highest power of the variable is 2.
• Systems of equations involve two or more equations with the same variables.

Geometry

• Geometry deals with the study of shapes, sizes, positions, and properties of space.
• Points are basic elements with no dimension, usually represented by a dot.
• Lines are one-dimensional figures with infinite length but no width.
• Planes are two-dimensional flat surfaces that extend infinitely in all directions.
• Angles are formed by two rays sharing a common endpoint (vertex), measured in degrees or radians.
• Triangles are three-sided polygons, classified by sides (equilateral, isosceles, scalene) and angles (acute, obtuse, right).
• Quadrilaterals are four-sided polygons (e.g., squares, rectangles, parallelograms, trapezoids).
• Circles are sets of points equidistant from a center point.
• The Pythagorean theorem relates the sides of a right triangle: a² + b² = c², where c is the hypotenuse.
• Area is the measure of the surface enclosed by a two-dimensional shape, expressed in square units.
• Perimeter is the total length of the sides of a two-dimensional shape.
• Volume is the measure of the space occupied by a three-dimensional object, expressed in cubic units.

Trigonometry

• Trigonometry studies the relationships between angles and sides of triangles.
• Sine (sin), cosine (cos), and tangent (tan) are fundamental trigonometric functions.
• These functions relate angles to the ratios of sides in a right triangle.
• sin(θ) = opposite / hypotenuse
• cos(θ) = adjacent / hypotenuse
• tan(θ) = opposite / adjacent
• The unit circle is a circle with a radius of 1, used to visualize trigonometric functions for all angles.
• Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables.

Calculus

• Calculus is the study of continuous change, divided into differential and integral calculus.
• Differential calculus deals with rates of change and slopes of curves.
• The derivative of a function measures the instantaneous rate of change of the function.
• Integral calculus deals with the accumulation of quantities and areas under curves.
• The integral of a function represents the area under the curve of the function.
• Limits are fundamental concepts in calculus that describe the behavior of a function as it approaches a certain value.

Statistics

• Statistics is the science of collecting, analyzing, interpreting, and presenting data.
• Descriptive statistics involves summarizing and presenting data using measures like mean, median, mode, and standard deviation.
• Mean is the average of a set of numbers (sum of values divided by the number of values).
• 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 a set of data around the mean.
• Inferential statistics involves making inferences and generalizations about a population based on a sample.
• Probability is the measure of the likelihood that an event will occur, expressed as a number between 0 and 1.

Number Theory

• Number theory is a branch of mathematics devoted primarily to the study of the integers.
• Prime numbers are integers greater than 1 that have only two distinct positive divisors: 1 and themselves.
• Composite numbers are integers greater than 1 that have more than two divisors.
• The fundamental theorem of arithmetic states that every integer greater than 1 can be uniquely expressed as a product of prime numbers.
• Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value (the modulus).

Discrete Mathematics

• Discrete mathematics studies mathematical structures that are fundamentally discrete rather than continuous.
• Logic is the study of reasoning and argumentation, including propositional logic and predicate logic.
• Set theory deals with the properties and relationships of sets, which are collections of objects.
• Combinatorics involves counting and arranging objects, including permutations and combinations.
• Graph theory studies graphs, which are mathematical structures used to model pairwise relations between objects.

Mathematical Analysis

• Mathematical analysis is a branch of mathematics that deals with the rigorous study of calculus, real and complex numbers, and related topics.
• Real analysis focuses on the properties of real numbers, sequences, series, and functions.
• Complex analysis extends the concepts of calculus to functions of complex numbers.
• Functional analysis studies vector spaces and operators acting on them.

Topology

• Topology studies the properties of spaces that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending, but not tearing or gluing.
• Point-set topology (or general topology) studies the properties of open sets, closed sets, continuity, and convergence in abstract spaces.
• Algebraic topology uses algebraic tools to study topological spaces, such as homology and homotopy groups.

Numerical Analysis

• Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis.
• It involves designing and analyzing algorithms for solving mathematical problems that cannot be solved analytically.
• Common topics include root-finding, numerical integration, and numerical solutions of differential equations.