Introduction to Mathematics

• Mathematics is the study of topics like quantity (numbers), structure, space, and change.
• Mathematicians and philosophers hold differing views regarding the precise scope and definition of mathematics.
• Mathematics aims to identify patterns and create new conjectures.
• Mathematicians use mathematical proofs to determine the truth or falsehood of conjectures.
• Mathematical reasoning offers insights and predictions about nature when mathematical structures serve as effective models of real-world phenomena.
• Mathematics evolved from counting, calculation, measurement, and systematic observation of physical objects' shapes and motions through abstraction and logic.

Areas of mathematics

• Quantity involves numbers and arithmetic operations.
• Structure encompasses algebra.
• Space includes geometry.
• Change covers calculus.

Quantity

• Mathematics originated with counting.
• Mathematics became essential for accounting, land surveying, and astronomy as civilizations progressed.
• The oldest known mathematical texts, originating from Sumer (modern Iraq), involve fractions and quadratic equations.
• The Egyptian Rhind Mathematical Papyrus and Moscow Mathematical Papyrus are the next oldest mathematical texts.
• Further advancements occurred in Greece, emphasizing deductive reasoning and mathematical rigor.
• The concept of numbers expanded to include zero, negative numbers, irrational numbers, and transcendental numbers.
• Number theory focuses on the characteristics of integers.

Structure

• Algebra involves the study of mathematical structures.
• An algebraic structure comprises a set accompanied by one or more operations.
• Abstract algebra delves into algebraic structures such as groups, rings, and fields.
• Vector spaces find application in linear algebra.

Space

• Geometry studies the shapes, sizes, relative positions of figures, and characteristics of space.
• Euclidean geometry focuses on shapes constructed using a compass and straightedge.
• Trigonometry explores the relationships between angles and sides of triangles.
• Differential geometry uses calculus to study curves and surfaces.
• Topology examines properties that remain unchanged through deformations, twisting, and stretching of objects.

Change

• Calculus is the study of change.
• Differential calculus determines the rate of change of one quantity relative to another.
• Integral calculus calculates the area or volume of an object.
• Differential equations establish a connection between a function and its derivatives.
• Calculus is used to examine functions and limits.
• Calculus is a fundamental tool in physics, engineering, and economics.

Mathematical proof

• A mathematical proof constitutes an argument demonstrating the necessary truth of a statement.
• Proofs must rely on deduction instead of inductive or empirical arguments.
• A proof should build upon axioms and previously verified statements.
• Axioms are statements accepted as true without requiring proof.
• Proofs exemplify exhaustive deductive reasoning, establishing logical certainty.
• Proofs often arise through a process of trial and error.

Uses of mathematics

• Mathematics sees widespread application in science, engineering, medicine, finance, and social sciences.
• Fields like statistics and game theory evolve in close relation to their practical uses.
• Other areas are developed mainly for theoretical interest.
• The distinction between pure and applied mathematics is not always clear.
• Applications for pure mathematics often emerge later.

Notation, terminology, and style

• Mathematics employs specific notation for mathematical symbols.
• Mathematical notation enhances conciseness in mathematics.
• Mathematical notation represents formulas clearly across different languages and cultures.
• Every mathematical concept requires a name.

The language of mathematics

• English is the primary language for mathematical discourse, with most mathematicians being fluent.
• Significant mathematical literature exists in French, German, and Russian.