Introduction to Mathematics

• Mathematics is the study of topics such as quantity, structure, space, and change.
• It has no generally accepted definition.

History

• Mathematical study began with arithmetic, geometry, and trigonometry.
• Mathematics has since expanded to cover a wider variety of fields.
• Some areas of study include set theory, number theory, algebra, analysis, and mathematical foundations.

Arithmetic

• Arithmetic involves the study of numbers, especially the properties of the traditional operations between them: addition, subtraction, multiplication, and division.
• Any set of objects upon which all of these operations can be performed is known as a field.

Algebra

• Algebra is the study of mathematical symbols and the rules for manipulating these symbols.
• It is a unifying thread of almost all of mathematics.
• Elementary algebra deals with solving equations using variables which represent numbers.
• Abstract algebra studies algebraic structures such as groups, rings, and fields.

Geometry

• Geometry is one of the oldest branches of mathematics.
• It is concerned with properties of space that are related with distance, shape, size, and relative position of figures.
• Plane geometry deals with flat shapes such as lines, circles, and triangles.
• Solid geometry studies three dimensional objects such as cubes, spheres, and pyramids.

Trigonometry

• Trigonometry is a branch of mathematics that studies relationships between angles and side lengths of triangles.
• Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves.

Calculus

• Calculus encompasses the mathematical study of continuous change.
• It has two major branches, differential calculus and integral calculus.
• Differential calculus concerns instantaneous rates of change, and the slopes of curves.
• Integral calculus concerns accumulation of quantities, and the areas under or between curves.

Analysis

• Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.
• It includes the study of real and complex numbers, algebra, calculus, geometry, etc.

Discrete Mathematics

• Discrete mathematics is the study of mathematical structures that are discrete rather than continuous.
• It includes topics such as set theory, combinatorics, logic, graph theory, and information theory.

Mathematical Foundations

• Mathematical foundations aims to put the whole of mathematics on a solid logical footing.
• It includes set theory, model theory, proof theory, and type theory.