Introduction to Mathematics

• Mathematics is the study of topics such as quantity (numbers), structure, space, and change.
• There are different views among mathematicians and philosophers as to the exact scope and definition of mathematics.
• Mathematicians seek out patterns and formulate new conjectures.
• Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their correctness.
• Mathematical research is required to solve mathematical problems.

Fields of Mathematics

• Arithmetic: Studies numbers and operations on them.
• Algebra: Studies algebraic structures, relations, and quantities.
• Geometry: Studies shapes, sizes, and properties of space.
• Trigonometry: Studies relationships between angles and sides of triangles.
• Calculus: Studies continuous change and rates of change.
• Statistics: Studies data collection, analysis, interpretation, presentation, and organization.
• Logic: Studies valid reasoning and inference.
• Topology: Studies properties of space that are preserved under continuous deformations, such as stretching or bending

Basic Concepts

• Number: A mathematical object used to count, measure, and label.
• Variable: A symbol representing a quantity that can change or vary.
• Equation: A statement that asserts the equality of two expressions.
• Function: 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.
• Theorem: A statement that has been proven on the basis of previously established statements.
• Proof: A sequence of logical statements used to demonstrate the truth of a theorem.
• Set: A well-defined collection of distinct objects, considered as an object in its own right.

Arithmetic

• Deals with fundamental operations: addition, subtraction, multiplication, and division.
• Integers: Whole numbers and their negatives.
• Rational Numbers: Numbers that can be expressed as a fraction of two integers.
• Real Numbers: Numbers that include rational and irrational numbers.
• Complex Numbers: Numbers that have a real and imaginary part.

Algebra

• Involves variables and algebraic expressions.
• Linear Equations: Equations where the highest power of the variable is 1.
• Quadratic Equations: Equations where the highest power of the variable is 2.
• Polynomials: Expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• Matrices: Rectangular arrays of numbers arranged in rows and columns.

Geometry

• Euclidean Geometry: Deals with shapes and their properties in a plane or in space based on a set of axioms.
• Coordinate Geometry: Uses a coordinate system to represent geometric shapes and solve geometric problems algebraically.
• Trigonometry: The branch of mathematics dealing with the relationships between the sides and angles of triangles.

Calculus

• Differential Calculus: Deals with the rate of change of functions and the slope of curves.
• Integral Calculus: Deals with the accumulation of quantities and the area under curves.
• Limits: The value that a function approaches as the input approaches some value.
• Derivatives: Measure the rate of change of a function.
• Integrals: Measure the area under a curve.

Statistics

• Descriptive Statistics: Summarizes and describes the main features of a dataset.
• Inferential Statistics: Uses sample data to make inferences and predictions about a population.
• Probability: Measures the likelihood of an event occurring.
• Distributions: Describe the spread and shape of data.

Mathematical Logic

• Propositional Logic: Deals with propositions and their logical connectives.
• Predicate Logic: Extends propositional logic to include predicates and quantifiers.
• Set Theory: Studies sets and their properties.

Problem Solving Strategies

• Understand the Problem: Read the problem carefully and identify what is being asked.
• Devise a Plan: Choose a strategy, such as drawing a diagram, looking for a pattern, or using a formula.
• Carry Out the Plan: Implement the chosen strategy and perform the necessary calculations.
• Look Back: Check the solution and make sure it makes sense in the context of the problem.

Mathematics explores quantity, structure, space, and change. Mathematicians identify patterns, create conjectures, and use mathematical proofs to verify them. Mathematical research is essential for solving problems.