Number

A number is a mathematical object used to count, measure, and label. The simplest numbers we learn are the natural numbers: 1, 2, 3, and so on. Numbers can be spoken as words, like “eleven,” or written as symbols called numerals, such as “11.” Because we cannot remember endless symbols, we use systems like the Hindu–Arabic numeral system, which uses ten basic digits (0–9) to represent any number.

Over time, mathematicians have expanded what we consider a number. Today, numbers include not just whole counts but also zero, negative numbers, fractions like one half, and even special numbers like the square root of 2 and pi. These numbers help us solve many kinds of problems, from sharing cookies equally to understanding the shape of the Earth.

Working with numbers involves basic operations such as addition, subtraction, multiplication, and division. The study of numbers and their patterns is called arithmetic, while deeper exploration of their properties falls under number theory. Numbers are everywhere—in everyday life, science, and even in advanced mathematics.

History

For broader coverage of this topic, see History of mathematics.

First use of numbers

Numbers have a long history, with early humans using marks on bones or other objects to count things. Some of the oldest known objects with these marks are the Lebombo bone and the Ishango bone, which may have been used to count days or animals. These early systems were simple and did not use place value like our modern decimal system.

The first clear examples of numbers come from ancient Mesopotamia, where a base-60 system was used around 3400 BC. Later, around 3100 BC, the Egyptians developed a base-10 system. These early systems helped people solve practical problems and record information.

Numerals

Numerals are the symbols we use to write numbers. The Egyptians created one of the first systems of numerals, and later the Greeks used letters of their alphabet to represent numbers. Roman numerals, using letters like I, V, and X, were common in Europe until the Hindu–Arabic numeral system spread in the late 14th century. This system, which includes the symbol for zero, was developed in ancient India around 500 AD and is the most widely used system today.

Zero

The concept of zero as a number was first clearly described by the Indian mathematician Brahmagupta in 628 AD. He explained how zero could be used in calculations, including adding and subtracting it from other numbers. Before this, zero was mainly used as a placeholder in number systems. The idea of zero spread from India to other parts of the world over time.

Negative numbers

The idea of negative numbers appeared early in China around 100–50 BC. These numbers were used to represent debts or losses. Later, Indian mathematicians like Brahmagupta also used negative numbers in their work. However, many European mathematicians were slow to accept negative numbers, and it wasn't until the 17th century that they became widely used in mathematics.

Rational numbers

Fractional numbers, or rational numbers, have been used since ancient times. The Ancient Egyptians used fractions in their mathematical texts, and Greek and Indian mathematicians also studied them. These numbers are important for solving many kinds of problems, from measuring land to calculating areas.

Real numbers and irrational numbers

Irrational numbers, which cannot be written as simple fractions, were first recognized by ancient mathematicians. The Greeks discovered that the square root of 2 is irrational, meaning it cannot be expressed as a fraction. In the 16th and 17th centuries, mathematicians began to study these numbers more systematically. By the 19th century, they were defined more rigorously.

Transcendental numbers and reals

Transcendental numbers are numbers that are not solutions to any algebraic equations with integer coefficients. The existence of these numbers was proven in the 19th century. Important examples include the number π and Euler's number e.

Infinity and infinitesimals

Infinity is a concept used to describe something that is endless. Ancient Indian mathematicians discussed infinity, and later European thinkers like Aristotle and Galileo Galilei also explored the idea. In the 19th century, Georg Cantor developed a new way to understand infinity using set theory.

Complex numbers

Complex numbers include numbers with a square root of negative one, written as i. These numbers were first used in solving equations that had no real solutions. By the 18th century, mathematicians like Leonhard Euler developed formulas that helped explain how these numbers work. Today, complex numbers are important in many areas of mathematics and science.

Prime numbers

Prime numbers are numbers greater than 1 that can only be divided evenly by 1 and themselves. Ancient Greek mathematicians like Euclid studied prime numbers, and later mathematicians developed methods to find and understand them better. Today, prime numbers are important in areas like computer science and cryptography.

Cultural and symbolic significance

Numbers often have cultural and symbolic meanings. In many cultures, certain numbers are considered lucky or unlucky. For example, the number 13 is often seen as unlucky in Western cultures, while the number eight is considered lucky in Chinese culture. These beliefs show how numbers are part of our everyday thinking and traditions.

Main classification

See also: List of types of numbers

Numbers can be classified into number sets or number systems, such as the natural numbers and the real numbers. The main number systems are as follows:

Each of these number systems extends the preceding one. For example, a rational number is also a real number, and every real number is also a complex number.

Natural numbers

Main article: Natural number

The most familiar numbers are the natural numbers (sometimes called whole numbers or counting numbers): 1, 2, 3, and so on. Traditionally, the sequence of natural numbers started with 1. However, in the 19th century, mathematicians started including 0 in the set of natural numbers. Today, different mathematicians use the term to describe both sets, including 0 or not. The mathematical symbol for the set of all natural numbers is N.

In the base 10 numeral system, symbols for natural numbers are written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Integers

Main article: Integer

The negative of a positive integer is defined as a number that produces 0 when it is added to the corresponding positive integer. Negative numbers are usually written with a negative sign. As an example, the negative of 7 is written −7, and 7 + (−7) = 0. When the set of negative numbers is combined with the set of natural numbers (including 0), the result is defined as the set of integers, Z.

Rational numbers

Main article: Rational number

A rational number is a number that can be expressed as a fraction with an integer numerator and a positive integer denominator. Fractions are written as two integers, the numerator and the denominator, with a dividing bar between them. The fraction ⁠_m_/_n_⁠ represents m parts of a whole divided into n equal parts. Two different fractions may correspond to the same rational number.

Real numbers

Main article: Real number

The symbol for the real numbers is R. They include all the measuring numbers. Every real number corresponds to a point on the number line. Each digit to the right of the decimal point has a place value one-tenth of the place value of the digit to its left.

Repeating decimal

If the fractional part of a real number has an infinite sequence of digits that follows a cyclical pattern, it can be written with an ellipsis or another notation that indicates the repeating pattern.

Irrational numbers

For real numbers that are not rational numbers, representing them as decimals would require an infinite sequence of varying digits to the right of the decimal point. These real numbers are called irrational. A famous irrational real number is π, the ratio of the circumference of any circle to its diameter.

Complex numbers

Main article: Complex number

Moving to a greater level of abstraction, the real numbers can be extended to the complex numbers. The complete solution set of a polynomial of degree two or higher can include the square roots of negative numbers. To conveniently represent this, the square root of −1 is denoted by i, called the imaginary unit. Hence, complex numbers consist of all values of the form:

a + b i

where a and b are real numbers. Because of this, complex numbers correspond to points on the complex plane, a vector space of two real dimensions.

Main number systems

Symbol	Name	Examples/Explanation
N {\displaystyle \mathbb {N} }	Natural numbers	0, 1, 2, 3, 4, 5, ... or 1, 2, 3, 4, 5, ... N 0 {\displaystyle \mathbb {N} _{0}} or N 1 {\displaystyle \mathbb {N} _{1}} are sometimes used.
Z {\displaystyle \mathbb {Z} }	Integers	..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ...
Q {\displaystyle \mathbb {Q} }	Rational numbers	⁠a/b⁠ where a and b are integers and b is not 0
R {\displaystyle \mathbb {R} }	Real numbers	The limit of a convergent sequence of rational numbers
C {\displaystyle \mathbb {C} }	Complex numbers	a + bi where a and b are real numbers and i is a formal square root of −1

Subclasses of the integers

Even and odd numbers

Main article: Even and odd numbers

An even number is an integer that can be divided by two without leaving a remainder. An odd number is an integer that cannot be evenly divided by two. For example, 2, 4, and 6 are even numbers, while 1, 3, and 5 are odd numbers.

Prime numbers

Main article: Prime number

A prime number is an integer greater than 1 that cannot be made by multiplying two smaller positive integers. The first few prime numbers are 2, 3, 5, 7, and 11. Prime numbers are important in many areas of modern technology, such as in keeping information safe online and in creating special codes for computers.

Other classes of integers

Many special groups of numbers have been studied and named by mathematicians. Examples include Bernoulli numbers, Fibonacci numbers, Lucas numbers, and perfect numbers.

Subclasses of the complex numbers

Algebraic numbers are solutions to equations with whole number coefficients. Numbers that can't be expressed as simple fractions are called irrational numbers. Those that aren't algebraic are known as transcendental numbers.

Periods are special complex numbers that come from integrating algebraic functions over algebraic domains. They include important constants like π. Exponential periods extend this idea and include numbers like e.

Constructible numbers can be created using just a straightedge and compass, starting from a unit length. Computable numbers are those that a computer could theoretically calculate to any desired precision, though most real numbers can't be computed this way.

Extensions of the concept

p-adic numbers

Main article: p-adic number

The p-adic numbers can have very long expansions to the left of the decimal point, much like how real numbers can have long expansions to the right. The system of p-adic numbers depends on the base used for its digits, with prime number bases offering special mathematical benefits. These numbers include rational numbers and are studied in number theory.

Hypercomplex numbers

Main article: hypercomplex number

Beyond the usual real and complex numbers, mathematicians have created higher-dimensional systems. These include quaternions, introduced by William Rowan Hamilton, and octonions. These systems expand on complex numbers and have unique properties, such as non-commutative multiplication in quaternions. They are useful in areas like computer graphics and physics.

Transfinite numbers

Main article: transfinite number

To handle very large sets, mathematicians developed ordinal and cardinal numbers. Ordinal numbers describe the order of elements in a set, while cardinal numbers describe the size. For finite sets, these two types of numbers match the natural numbers, but for infinite sets, they differ.

Nonstandard numbers

Hyperreal numbers are used in a special area of math called non-standard analysis. They extend the real numbers to include very small and very large values, while keeping important mathematical properties.