Introduction to Mathematics

• Mathematics involves the study of topics such as quantity, structure, space, and change.
• It has no generally accepted definition.
• Mathematicians seek and use patterns to formulate new conjectures; they resolve the truth or falsity of conjectures by mathematical proofs.
• Mathematics 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, business, computer science, and social sciences.
• Mathematical fields are often classified into pure mathematics and applied mathematics
• However, pure mathematics topics often find applications later.
• The history of mathematics stretches back thousands of years, with basic arithmetic developing in ancient civilizations.
• Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements.
• Mathematics continued to develop, with significant contributions from China, India, and the Islamic world.
• During the Renaissance, new mathematical discoveries interacted with new scientific discoveries, leading to an acceleration in research.
• Today, mathematics is used in nearly every field.

Arithmetic

• Arithmetic is the oldest and most elementary branch of mathematics.
• It deals with basic operations on numbers such as addition, subtraction, multiplication, and division.
• Arithmetic is the foundation for more advanced mathematical concepts.
• Key topics include number systems, fractions, decimals, percentages, and basic problem-solving.

Algebra

• Algebra is a branch of mathematics generalizing arithmetic.
• It uses symbols or letters to represent numbers or quantities.
• Algebra is used to solve equations and explore relationships between variables.
• Key topics include linear equations, quadratic equations, polynomials, and systems of equations.

Geometry

• Geometry is one of the oldest branches of mathematics.
• It is concerned with properties and relationships of points, lines, surfaces, solids, and higher dimensional analogs.
• Geometry deals with shapes, sizes, and relative positions of figures.
• Key topics include Euclidean geometry, trigonometry, analytic geometry, and differential geometry.

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 instantaneous rates of change and slopes of curves.
• Integral calculus concerns accumulation of quantities and the areas under and between curves.
• Key topics include limits, derivatives, integrals, and differential equations.

Trigonometry

• Trigonometry studies relationships between angles and sides of triangles.
• Trigonometric functions (sine, cosine, tangent, etc.) are used to describe these relationships.
• Trigonometry is used in navigation, surveying, and other fields.

Statistics

• Statistics involves the collection, analysis, interpretation, presentation, and organization of data.
• It deals with all aspects of data, including the planning of data collection in terms of the design of surveys and experiments.
• Key concepts include probability, distributions, hypothesis testing, and regression analysis.

Probability

• Probability theory is the branch of mathematics that deals with chance.
• It quantifies uncertainty.
• Probability ranges from 0 to 1, where 0 indicates impossibility and 1 indicates certainty.
• It's used extensively in statistics, finance, and science.