Mathematics

Mathematics is the science that studies numbers, measurements, geometric shapes, and the logical relationships between them. It proceeds rationally: starting from premises that are accepted without proof (axioms), it then derives conclusions using valid proofs that do not depend on observation.

Mathematics has two aspects. One is inquiry pursued for its own sake, examining abstract structures. The other is the application of mathematics to other sciences: physics, engineering, economics, medicine, and computer science. The language of mathematics allows these disciplines to define rules and measure outcomes.

Major branches

The principal branches are: arithmetic, which deals with addition, subtraction, multiplication, and division; algebra, which uses variables and equations; geometry, which studies shapes and measurements; trigonometry, which examines triangles and the relationships between sides and angles; and analysis, which includes calculus, limits, derivatives, and integrals. Probability and statistics also belong to mathematics, addressing uncertainty.

Types of numbers

Numbers are arranged in levels. Natural numbers are 1, 2, 3, and so on. When zero and negative numbers are added to these, we get integers. Fractions represent a part of a whole, such as 1/2 or 3/4; decimals express the same idea in base-ten notation, such as 0.5. Numbers that can be written as fractions are called rational; those that cannot, such as √2 and π, are irrational.

Powers and operations

A power (or exponent) is when a number is multiplied by itself a fixed number of times; for example, 2 to the power of 3 equals 8. A root is the inverse operation of a power. These ideas underlie logarithms, equations involving powers, and exponential growth used to describe natural phenomena and social dynamics.

Proof and logical foundation

What distinguishes mathematics is proof. A theorem is accepted only if it is derived from previously accepted premises step by step. This method was first systematically set out by Euclid, who built his geometry on a small set of axioms. Today mathematical logic and set theory provide a general language for defining other structures.