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.” We use systems like the Hindu–Arabic numeral system to represent numbers with just ten basic digits (0–9).

Mathematicians have found many kinds of numbers. These include zero, negative numbers, fractions like one half, and special numbers like the square root of 2 and pi. Numbers help us solve many problems, from sharing things equally to understanding shapes.

We work with numbers using addition, subtraction, multiplication, and division. The study of numbers and their patterns is called arithmetic, while deeper exploration of their properties is number theory. Numbers are used everywhere—in daily life, science, and mathematics.

History

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

First use of numbers

Numbers have been used for a long time. Early humans used 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. 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 problems and record information.

Numerals

Numerals are the symbols we use to write numbers. The Egyptians created one of the first systems of numerals. 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 includes the symbol for zero. It 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. 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.

Real numbers and irrational numbers

Irrational numbers cannot be written as simple fractions. They 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 later centuries, mathematicians studied these numbers more systematically.

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 grouped into number sets or number systems, like the natural numbers and the real numbers. Here are the main number systems:

Each number system includes the ones before it. For example, a rational number is also a real number, and every real number is a complex number.

Natural numbers

Main article: Natural number

The numbers we use most are natural numbers: 1, 2, 3, and so on. Sometimes these are called whole numbers or counting numbers. In the past, natural numbers started with 1. But now, some people include 0 as a natural number, too. The mathematical symbol for all natural numbers is N.

In the base 10 system, we write natural numbers using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Integers

Main article: Integer

A negative number is what you get when you take a positive number and change its sign. For example, the negative of 7 is written −7, and 7 + (−7) = 0. When we add negative numbers to natural numbers (including 0), we get the set of integers, Z.

Rational numbers

Main article: Rational number

A rational number can be written as a fraction. A fraction has a top number (numerator) and a bottom number (denominator), separated by a line. The fraction ⁠_m_/_n_⁠ means m parts of a whole that is split into n equal parts. Different fractions can sometimes be the same rational number.

Real numbers

Main article: Real number

The symbol for real numbers is R. They include all numbers we use for measuring things. Every real number matches a point on the number line. Each digit after the decimal point is one-tenth of the value of the digit before it.

Repeating decimal

If the part after the decimal point in a real number repeats forever in a pattern, we can write it in a special way to show the repeating part.

Irrational numbers

Some real numbers cannot be written as fractions. Their decimal parts go on forever without repeating. These are called irrational numbers. A well-known irrational number is π, the ratio of the circumference of any circle to its diameter.

Complex numbers

Main article: Complex number

We can extend real numbers to complex numbers. This helps us solve certain equations. The square root of −1 is written as i, called the imaginary unit. So, complex numbers look like:

a + b i

where a and b are real numbers. Complex numbers match points on the complex plane, which is a type of vector space with two directions.

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 a number that can be divided by two with no remainder. An odd number is a number that cannot be divided by two evenly. 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 a number greater than 1 that cannot be made by multiplying two smaller whole numbers. The first few prime numbers are 2, 3, 5, 7, and 11. Prime numbers help keep information safe on the internet and are used to create special codes for computers.

Other classes of integers

Many special groups of numbers have been studied 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 written 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 long expansions to the left of the decimal point, just like real numbers can have long expansions to the right. These numbers depend on the base used for their digits, and prime number bases have special benefits. They include rational numbers and are studied in number theory.

Hypercomplex numbers

Main article: hypercomplex number

Beyond 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 small 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.