Introduction to Arithmetic

• Math is the science and study of quantity, structure, space, and change.
• Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.
• Math is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, finance, and social sciences.
• Applied mathematics concerns itself with the use of mathematical tools to solve problems in natural science, engineering, medicine, finance, social sciences etc.
• Pure mathematics concerns itself with the study of mathematical concepts themselves.

Arithmetic

• Arithmetic is the oldest and most elementary branch of mathematics.
• It deals with basic operations on numbers.
• Basic operations include addition, subtraction, multiplication, and division.
• More advanced operations include exponentiation, extraction of roots, and taking logarithms.
• Arithmetic operations are performed according to a standard order of operations.
• The order of operations is Parentheses, Exponents, Multiplication and Division, Addition and Subtraction (PEMDAS).

Algebra

• Algebra is a branch of mathematics that generalizes arithmetic.
• It introduces variables (symbols) representing numbers or quantities.
• Algebra explores relationships between these variables.
• Key concepts include algebraic expressions, equations, and inequalities.
• Algebra is foundational for further study in mathematics, science, and engineering.
• Elementary algebra is essential for anyone wishing to pursue a career in a STEM field.

Geometry

• Geometry is one of the oldest branches of mathematics.
• It is concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.
• Euclidean geometry is the study of geometry based on Euclid's axioms.
• Non-Euclidean geometry includes hyperbolic and elliptic geometry.
• Geometry has many practical applications in fields such as architecture, engineering, and computer graphics.

Trigonometry

• Trigonometry studies relationships between angles and sides of triangles.
• Trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) are central to trigonometry.
• Trigonometry is used to solve problems involving triangles.
• It has applications in surveying, navigation, and physics.
• Trigonometry is closely related to geometry and is often used in conjunction with algebraic techniques.

Calculus

• Calculus is a branch of mathematics that deals with continuous change.
• It has two major branches: differential calculus and integral calculus.
• Differential calculus concerns derivatives and rates of change.
• Integral calculus concerns integrals and accumulation of quantities.
• Calculus is used to solve problems in physics, engineering, economics, and other fields.
• It is an essential tool for modeling and understanding complex systems.

Statistics

• Statistics is the science of collecting, analyzing, interpreting, and presenting data.
• It involves methods for summarizing and drawing inferences from data.
• Key concepts include mean, median, mode, standard deviation, and variance.
• Statistical methods are used in many fields, including science, business, and government.
• Understanding statistics is essential for making informed decisions in a data-driven world.

Probability

• Probability is the measure of the likelihood that an event will occur.
• It is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
• Probability theory provides a framework for analyzing random events.
• Key concepts include sample space, events, and probability distributions.
• Probability concepts are applied in statistics, finance, and risk management.

Discrete Mathematics

• Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.
• It deals with objects that can assume only distinct, separated values.
• Topics include logic, set theory, combinatorics, graph theory, and number theory.
• Discrete mathematics is essential for computer science and information technology.
• It provides the mathematical foundations for algorithms, data structures, and cryptography.

Logic

• Logic is the study of reasoning and argumentation.
• It provides rules and principles for distinguishing valid from invalid inferences.
• Propositional logic and predicate logic are two main branches of logic.
• Logical reasoning is essential for mathematics, computer science, and philosophy.
• Understanding logic helps in constructing sound arguments and avoiding fallacies.

Set Theory

• Set theory is a branch of mathematics that studies sets.
• A set is a collection of distinct objects, considered as an object in its own right.
• Set theory provides a foundation for other branches of mathematics.
• Key concepts include union, intersection, complement, and subset.
• Set theory is used in computer science, logic, and statistics.

Number Theory

• Number theory is a branch of mathematics that studies the properties of integers.
• It deals with topics such as prime numbers, divisibility, and congruences.
• Number theory has applications in cryptography and computer science.
• Key concepts include prime factorization, greatest common divisor, and modular arithmetic.
• Number theory is one of the oldest branches of mathematics and has fascinated mathematicians for centuries.

Topology

• Topology is a branch of mathematics that studies properties preserved through deformations, twistings, and stretching of objects.
• It deals with properties that do not change under continuous transformations.
• Topology has applications in physics, computer science, and data analysis.
• Key concepts include open sets, continuity, and connectedness.
• Topology provides a framework for studying abstract spaces and their properties.