Number Theory and Types of Numbers

The number systems used in mathematics form a nested hierarchy, each extending the one before it. The natural numbers are the counting numbers; adding their negatives and zero gives the integers. Ratios of integers form the rational numbers, while numbers that cannot be written as such a ratio — like the square root of two or the constant pi — are irrational. Together the rationals and irrationals make up the real numbers, which fill the number line completely. Extending further by introducing a square root of negative one produces the complex numbers.

Within these systems certain classes of numbers have special importance. Prime numbers are integers greater than one that are divisible only by one and themselves; they are the multiplicative building blocks of all integers, since every integer factors uniquely into primes. Transcendental numbers, such as pi and Euler's number, are real numbers that are not roots of any polynomial with integer coefficients, distinguishing them from algebraic irrationals like the square root of two.

Other numbers are notable for the patterns they form. The Fibonacci sequence, in which each term is the sum of the two before it, appears in many natural growth patterns and is closely connected to the golden ratio. Number theory also studies sorting and ordering of numbers and the algorithms used to arrange them, divisibility rules, modular arithmetic and the distribution of primes. Many of these ideas, though stated simply, lead to deep and still unsolved problems.

Frequently asked questions

What makes a number irrational?

An irrational number cannot be written as a ratio of two integers. Its decimal expansion neither terminates nor repeats; examples include the square root of two and pi.

What is a prime number?

A prime is an integer greater than one whose only positive divisors are one and itself. Every integer can be factored uniquely into primes.

How do transcendental numbers differ from other irrationals?

Transcendental numbers, like pi and Euler's number, are not solutions of any polynomial equation with integer coefficients, unlike algebraic irrationals such as the square root of two.