Introduction to Maths

• Maths 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.
• Maths is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, finance, and the social sciences.
• Applied maths concerns itself with the use of maths tools to solve problems in natural science, engineering, medicine, finance, business, and the social sciences.
• maths studies theoretical concepts without regard to any application outside maths.
• The term "maths" came into English from Old English during the Middle English period, influenced by the Old French mathematique
• The word's ultimate origin is the Greek μάθημα (máthema), meaning "learning, study, science".
• The Greek word μάθημα is derived from μανθάνω (manthánō), itself related to the Proto-Indo-European root *mendʰ-, meaning "to learn, to know".

Areas of Maths Study

• Arithmetic: Studies numbers and operations on them.
• Algebra: Studies the generalizations of arithmetic operations and mathematical objects.
• Geometry: Studies spatial relationships and shapes.
• Trigonometry: Studies relationships between angles and sides of triangles.
• Calculus: Studies continuous change and limits, including derivatives and integrals.
• Statistics: Studies the collection, analysis, interpretation, presentation, and organization of data.
• Probability: Studies the likelihood of events occurring.
• Topology: Studies properties of space that are preserved under continuous deformations, such as stretching and bending.
• Set Theory: Studies collections of objects.
• Number Theory: Studies properties and relationships of numbers, especially integers.
• Discrete Maths: Studies mathematical structures that are fundamentally discrete rather than continuous.

Key Concepts in Maths

• Axioms: Basic assumptions or self-evident truths upon which mathematical theories are built.
• Theorems: Statements that have been proven to be true based on axioms and previously established theorems.
• Proofs: Logical arguments that demonstrate the truth of a theorem.
• Conjectures: Statements that are proposed as true but have not yet been proven.
• Algorithms: Step-by-step procedures for solving mathematical problems.
• Models: Mathematical representations of real-world phenomena.

Numbers

• Natural Numbers: The counting numbers (1, 2, 3, ...).
• Integers: Whole numbers, including negative numbers and zero (... -2, -1, 0, 1, 2, ...).
• Rational Numbers: Numbers that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
• Irrational Numbers: Numbers that cannot be expressed as a fraction of two integers (e.g., √2, π).
• Real Numbers: All rational and irrational numbers.
• Complex Numbers: Numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1).

Operations

• Addition: Combining two or more numbers to find their sum.
• Subtraction: Finding the difference between two numbers.
• Multiplication: Repeated addition of a number to itself.
• Division: Splitting a number into equal parts.
• Exponentiation: Raising a number to a power.

Equations and Expressions

• Equation: A statement that two expressions are equal, indicated by an equals sign (=).
• Expression: A combination of numbers, variables, and operations.
• Variable: A symbol representing an unknown quantity.
• Constant: A fixed value that does not change.

Algebra

• Solving Equations: Finding the value(s) of the variable(s) that make an equation true.
• Linear Equations: Equations where the highest power of the variable is 1.
• Quadratic Equations: Equations where the highest power of the variable is 2.
• Systems of Equations: A set of two or more equations that are solved together.
• Polynomials: Expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

Geometry

• Points: A location in space, with no dimension.
• Lines: A straight path that extends infinitely in both directions.
• Planes: A flat surface that extends infinitely in all directions.
• Angles: The measure of the space between two intersecting lines or surfaces.
• Shapes: Two-dimensional figures with specific properties (e.g., triangles, squares, circles).
• Solids: Three-dimensional objects with specific properties (e.g., cubes, spheres, pyramids).
• Area: The amount of surface covered by a two-dimensional shape.
• Volume: The amount of space occupied by a three-dimensional object.

Calculus

• Limits: The value that a function approaches as the input approaches some value.
• Derivatives: The rate of change of a function with respect to its input.
• Integrals: The area under a curve, representing the accumulation of a quantity.
• Functions: A relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
• Series: The sum of a sequence of terms.

Statistics

• Data: Information collected for analysis.
• Mean: The average value of a set of numbers.
• Median: The middle value in a set of numbers when arranged in order.
• Mode: The value that appears most frequently in a set of numbers.
• Standard Deviation: A measure of the spread or dispersion of data around the mean.
• Distributions: The way in which data is spread out or grouped.
• Hypothesis Testing: A method for making statistical inferences about a population based on sample data.

Applied maths

• maths is heavily used in physics for modeling and theoretical frameworks, in engineering for designing structures and systems, and in computer science for algorithm design and analysis.
• Financial Maths: Uses maths models to analyze financial markets and manage risk.
• Actuarial Science: Applies maths and statistical methods to assess risk in the insurance and finance industries.
• Operations Research: Uses maths models to optimize decision-making in organizations.
• maths is also important in medical imaging, drug development, and epidemiology.
• Climate Modeling: maths models are used to predict climate change and its impact on the environment.
• Image Processing: maths techniques are used to enhance and analyze images in various fields, including medicine and security.
• Cryptography: Uses mathematical algorithms to secure communications and data.