Introduction to Arithmetic and Algebra

• Mathematics is the abstract science of number, quantity, and space.
• Math can be studied in its own right (pure mathematics), or as it is applied to other disciplines such as physics and engineering (applied mathematics).

Arithmetic

• Arithmetic involves the study of numbers, especially the properties of the traditional operations between them—addition, subtraction, multiplication, and division.
• Addition is the most basic operation of arithmetic.
• It combines two numbers into a single number, the sum.
• Subtraction is the inverse of addition.
• It finds the difference between two numbers.
• Multiplication is equivalent to adding a number to itself a certain number of times.
• Division is the inverse of multiplication.
• It finds how many times one number is contained in another.

Algebra

• Algebra is a generalization of arithmetic that uses symbols to represent numbers and other mathematical objects.
• It uses variables to represent quantities without fixed values.
• Equations are mathematical statements that express the equality of two expressions.
• Solving an equation involves finding the value(s) of the variable(s) that make the equation true.
• Linear equations are equations in which the highest power of the variable is 1.
• Quadratic equations are equations in which the highest power of the variable is 2.
• Factoring is the process of breaking down a number or expression into its factors.
• The quadratic formula provides the solutions to any quadratic equation in the form ax^2 + bx + c = 0.

Geometry

• Geometry is concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.
• Euclidean geometry is based on axioms and theorems of Euclid.
• Key concepts include points, lines, angles, and shapes.
• A point is a location in space.
• A line is a straight, one-dimensional figure extending infinitely in both directions.
• An angle is formed by two lines that share a common endpoint (vertex).
• Triangles, quadrilaterals, circles, and polygons are fundamental shapes in geometry.
• The Pythagorean theorem relates the sides of a right triangle: a^2 + b^2 = c^2, where c is the hypotenuse.
• Area calculations include determining the space occupied by two-dimensional shapes.
• Volume calculations determine the space occupied by three-dimensional objects.
• Coordinate geometry uses a coordinate system to represent geometric shapes and figures.

Trigonometry

• Trigonometry studies the relationships between the angles and sides of triangles.
• It focuses on trigonometric functions such as sine, cosine, and tangent.
• Sine (sin) is the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine (cos) is the ratio of the adjacent side to the hypotenuse in a right triangle.
• Tangent (tan) is the ratio of the opposite side to the adjacent side in a right triangle.
• The unit circle provides a visual representation of trigonometric functions.
• Trigonometric identities are equations involving trigonometric functions that are always true.
• Trigonometry is used in navigation, surveying, and physics.

Calculus

• Calculus is the study of continuous change.
• It is divided into differential calculus and integral calculus.
• Differential calculus deals with the rate of change of a function.
• The derivative of a function represents its instantaneous rate of change at a point.
• Integral calculus deals with the accumulation of quantities.
• The integral of a function represents the area under its curve.
• Limits are a fundamental concept in calculus.
• They describe the behavior of a function as its input approaches a particular value.
• The fundamental theorem of calculus relates differentiation and integration.

Statistics and Probability

• Statistics is the science of collecting, analyzing, interpreting, and presenting data.
• Descriptive statistics summarize and describe the main features of a dataset.
• Inferential statistics use sample data to make inferences about a population.
• Probability is a measure of the likelihood that an event will occur.
• Probability values range from 0 to 1.
• A probability of 0 means the event is impossible.
• A probability of 1 means the event is certain.
• The normal distribution is a common probability distribution in statistics.
• Hypothesis testing involves using statistical methods to test a claim or hypothesis about a population.

Discrete Mathematics

• Discrete mathematics deals with mathematical structures that are fundamentally discrete rather than continuous.
• It includes topics such as logic, set theory, combinatorics, graph theory, and number theory.
• Logic studies the principles of valid reasoning and inference.
• Set theory deals with collections of objects called sets.
• Combinatorics is concerned with counting and arranging objects.
• Graph theory studies the properties of graphs, which are mathematical structures used to model relationships between objects.
• Number theory deals with the properties of integers.

Real Analysis

• The branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions.
• It provides a rigorous foundation for calculus.
• Focuses on concepts like convergence, continuity, differentiation, and integration within the real number system.
• Important theorems include the Bolzano-Weierstrass theorem, the Intermediate Value Theorem, and the Mean Value Theorem.
• Sequences and Series: Examination of the convergence and divergence of infinite sequences and series.
• Functions of a Real Variable: Study of the properties of real-valued functions, including continuity, differentiability, and integrability.

Complex Analysis

• The branch of mathematical analysis that investigates functions of complex numbers.
• Complex numbers extend the real number system by including the imaginary unit "i," where i² = -1.
• Complex functions can be differentiated and integrated, leading to powerful results such as Cauchy's integral theorem.
• Holomorphic Functions: Functions that are complex differentiable in an open set.
• Complex Integration: Integration of complex functions along paths in the complex plane.
• Residue Theorem: A fundamental result that relates the integral of a complex function around a closed curve to the residues of its singularities.

Topology

• A branch of mathematics that 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:
  • Basic concepts include open sets, closed sets, neighborhoods, limit points, and continuity.
  • Study of topological spaces and their properties, such as compactness, connectedness, and separation axioms.
• Algebraic Topology:
  • Uses algebraic tools to study topological spaces.
  • Fundamental groups, homology, and cohomology are used to classify topological spaces.

Numerical Analysis

• An area of mathematics and computer science devoted to developing, analyzing, and implementing algorithms for solving mathematical problems numerically.
• Approximation Theory:
  • Deals with approximating functions or data using simpler functions, such as polynomials or splines.
  • Interpolation and curve fitting techniques.
• Numerical Linear Algebra:
  • Numerical methods for solving linear systems, eigenvalue problems, and singular value decomposition.
  • Direct methods (e.g., Gaussian elimination) and iterative methods (e.g., Jacobi, Gauss-Seidel).