Number

Set inclusions between the natural numbers (ℕ), the integers (ℤ), the rational numbers (ℚ), the real numbers (ℝ), and the complex numbers

(ℂ)

A number is a

). In common usage, a numeral is not clearly distinguished from the number that it represents.

In mathematics, the notion of number has been extended over the centuries to include zero (0),^[3] negative numbers,^[4] rational numbers such as one half ${\displaystyle \left({\tfrac {1}{2}}\right)}$, real numbers such as the square root of 2 ${\displaystyle \left({\sqrt {2}}\right)}$ and

, the study of the properties of numbers.

Besides their practical uses, numbers have cultural significance throughout the world.

During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, and may be seen as extending the concept. Among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. In modern mathematics, number systems are considered important special examples of more general algebraic structures such as rings and fields, and the application of the term "number" is a matter of convention, without fundamental significance.^[10]

History

First use of numbers

Bones and other artifacts have been discovered with marks cut into them that many believe are tally marks.^[11] These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals.

A tallying system has no concept of place value (as in modern decimal notation), which limits its representation of large numbers. Nonetheless, tallying systems are considered the first kind of abstract numeral system.

The first known system with place value was the Mesopotamian base 60 system (c. 3400 BC) and the earliest known base 10 system dates to 3100 BC in Egypt.^[12]

Numerals

Numbers should be distinguished from numerals, the symbols used to represent numbers. The Egyptians invented the first ciphered numeral system, and the Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets.

Zero

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The first known documented use of

Islamic world

.

Brahmagupta's Brāhmasphuṭasiddhānta is the first book that mentions zero as a number, hence Brahmagupta is usually considered the first to formulate the concept of zero. He gave rules of using zero with negative and positive numbers, such as "zero plus a positive number is a positive number, and a negative number plus zero is the negative number". The Brāhmasphuṭasiddhānta is the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number as was done by the Babylonians or as a symbol for a lack of quantity as was done by Ptolemy and the Romans.

The use of 0 as a number should be distinguished from its use as a placeholder numeral in

).

There are other uses of zero before Brahmagupta, though the documentation is not as complete as it is in the Brāhmasphuṭasiddhānta.

Records show that the

1

was a number.)

The late

better source needed

]

By 130 AD, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for 0 (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals. Because it was used alone, not as just a placeholder, this Hellenistic zero was the first documented use of a true zero in the Old World. In later Byzantine manuscripts of his Syntaxis Mathematica (Almagest), the Hellenistic zero had morphed into the Greek letter Omicron (otherwise meaning 70).

Another true zero was used in tables alongside

or a colleague about 725, a true zero symbol.

Negative numbers

Further information:

History of negative numbers

The abstract concept of negative numbers was recognized as early as 100–50 BC in China. The Nine Chapters on the Mathematical Art contains methods for finding the areas of figures; red rods were used to denote positive coefficients, black for negative.^[17] The first reference in a Western work was in the 3rd century AD in Greece. Diophantus referred to the equation equivalent to 4x + 20 = 0 (the solution is negative) in Arithmetica, saying that the equation gave an absurd result.

During the 600s, negative numbers were in use in India to represent debts. Diophantus' previous reference was discussed more explicitly by Indian mathematician Brahmagupta, in Brāhmasphuṭasiddhānta in 628, who used negative numbers to produce the general form quadratic formula that remains in use today. However, in the 12th century in India, Bhaskara gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots".

European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although

, but referred to them as "absurd numbers".

As recently as the 18th century, it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless.

Rational numbers

It is likely that the concept of fractional numbers dates to

, which also covers number theory as part of a general study of mathematics.

The concept of

] Similarly, Babylonian math texts used sexagesimal (base 60) fractions with great frequency.

Irrational numbers

Further information:

History of irrational numbers

The earliest known use of irrational numbers was in the

better source needed

]

The 16th century brought final European acceptance of

and Méray.

The search for roots of

.

in the theory of Kettenbruchdeterminanten.

Transcendental numbers and reals

Further information:

History of π

The existence of

countably infinite

, so there is an uncountably infinite number of transcendental numbers.

Infinity and infinitesimals

Further information:

History of infinity

The earliest known conception of mathematical

Jain

mathematicians c. 400 BC. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually. The symbol ${\displaystyle {\text{∞}}}$ is often used to represent an infinite quantity.

.

In the 1960s,

Leibniz

.

A modern geometrical version of infinity is given by projective geometry, which introduces "ideal points at infinity", one for each spatial direction. Each family of parallel lines in a given direction is postulated to converge to the corresponding ideal point. This is closely related to the idea of vanishing points in perspective drawing.

Complex numbers

The earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventor

. It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers.

This was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time. When René Descartes coined the term "imaginary" for these quantities in 1637, he intended it as derogatory. (See imaginary number for a discussion of the "reality" of complex numbers.) A further source of confusion was that the equation

${\displaystyle \left({\sqrt {-1}}\right)^{2}={\sqrt {-1}}{\sqrt {-1}}=-1}$

seemed capriciously inconsistent with the algebraic identity

${\displaystyle {\sqrt {a}}{\sqrt {b}}={\sqrt {ab}},}$

which is valid for positive real numbers a and b, and was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity, and the related identity

${\displaystyle {\frac {1}{\sqrt {a}}}={\sqrt {\frac {1}{a}}}}$

in the case when both a and b are negative even bedeviled

This difficulty eventually led him to the convention of using the special symbol i in place of ${\displaystyle {\sqrt {-1}}}$ to guard against this mistake.

The 18th century saw the work of Abraham de Moivre and Leonhard Euler. De Moivre's formula (1730) states:

${\displaystyle (\cos \theta +i\sin \theta )^{n}=\cos n\theta +i\sin n\theta }$

while Euler's formula of complex analysis (1748) gave us:

${\displaystyle \cos \theta +i\sin \theta =e^{i\theta }.}$

The existence of complex numbers was not completely accepted until Caspar Wessel described the geometrical interpretation in 1799. Carl Friedrich Gauss rediscovered and popularized it several years later, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's De algebra tractatus.

In the same year, Gauss provided the first generally accepted proof of the

in 1893.

In 1850

extended complex plane

.

Prime numbers

Prime numbers have been studied throughout recorded history.^{[citation needed]} They are positive integers that are divisible only by 1 and themselves. Euclid devoted one book of the Elements to the theory of primes; in it he proved the infinitude of the primes and the fundamental theorem of arithmetic, and presented the Euclidean algorithm for finding the greatest common divisor of two numbers.

In 240 BC, Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers. But most further development of the theory of primes in Europe dates to the Renaissance and later eras.^{[citation needed]}

In 1796,

in 1896. Goldbach and Riemann's conjectures remain unproven and unrefuted.

Main classification

Numbers can be classified into

. The main number systems are as follows:

Main number systems

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

Each of these number systems is a subset of the next one. So, for example, a rational number is also a real number, and every real number is also a complex number. This can be expressed symbolically as

${\displaystyle \mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R} \subset \mathbb {C} }$.

A more complete list of number sets appears in the following diagram.

Number systems Complex ${\displaystyle :\;\mathbb {C} }$ Real ${\displaystyle :\;\mathbb {R} }$ Rational ${\displaystyle :\;\mathbb {Q} }$ Integer ${\displaystyle :\;\mathbb {Z} }$ Natural ${\displaystyle :\;\mathbb {N} }$ Zero : 0 One : 1 Prime numbers Composite numbers Negative integers Fraction Finite decimal Dyadic (finite binary) Repeating decimal Irrational Algebraic irrational Transcendental Imaginary

Natural numbers

The most familiar numbers are the

for the set of all natural numbers is N, also written ${\displaystyle \mathbb {N} }$, and sometimes ${\displaystyle \mathbb {N} _{0}}$ or ${\displaystyle \mathbb {N} _{1}}$ when it is necessary to indicate whether the set should start with 0 or 1, respectively.

In the

of 1, and every other digit has a place value ten times that of the place value of the digit to its right.

In

, the number 3 is represented as sss0, where s is the "successor" function (i.e., 3 is the third successor of 0). Many different representations are possible; all that is needed to formally represent 3 is to inscribe a certain symbol or pattern of symbols three times.

Integers

The

, Z also written ${\displaystyle \mathbb {Z} }$. Here the letter Z comes from German Zahl 'number'. The set of integers forms a ring with the operations addition and multiplication.^[35]

The natural numbers form a subset of the integers. As there is no common standard for the inclusion or not of zero in the natural numbers, the natural numbers without zero are commonly referred to as positive integers, and the natural numbers with zero are referred to as non-negative integers.

Rational numbers

A rational number is a number that can be expressed as a

with an integer numerator and a positive integer denominator. Negative denominators are allowed, but are commonly avoided, as every rational number is equal to a fraction with positive 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; for example 1/2 and 2/4 are equal, that is:

${\displaystyle {1 \over 2}={2 \over 4}.}$

In general,

${\displaystyle {a \over b}={c \over d}}$ if and only if ${\displaystyle {a\times d}={c\times b}.}$

If the absolute value of m is greater than n (supposed to be positive), then the absolute value of the fraction is greater than 1. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or 0. The set of all rational numbers includes the integers since every integer can be written as a fraction with denominator 1. For example −7 can be written −7/1. The symbol for the rational numbers is Q (for quotient), also written ${\displaystyle \mathbb {Q} }$.

Real numbers

The symbol for the real numbers is R, also written as ${\displaystyle \mathbb {R} .}$ They include all the measuring numbers. Every real number corresponds to a point on the

, e.g. −123.456.

Most real numbers can only be approximated by

It turns out that these repeating decimals (including the repetition of zeroes) denote exactly the rational numbers, i.e., all rational numbers are also real numbers, but it is not the case that every real number is rational. A real number that is not rational is called irrational. A famous irrational real number is the π, the ratio of the circumference of any circle to its diameter. When pi is written as

${\displaystyle \pi =3.14159265358979\dots ,}$

as it sometimes is, the ellipsis does not mean that the decimals repeat (they do not), but rather that there is no end to them. It has been proved that

. Another well-known number, proven to be an irrational real number, is

${\displaystyle {\sqrt {2}}=1.41421356237\dots ,}$

the square root of 2, that is, the unique positive real number whose square is 2. Both these numbers have been approximated (by computer) to trillions ( 1 trillion = 10¹² = 1,000,000,000,000 ) of digits.

Not only these prominent examples but

are measured as 1.23 m and 4.56 m, then multiplication gives an area for the rectangle between 5.614591 m² and 5.603011 m². Since not even the second digit after the decimal place is preserved, the following digits are not significant. Therefore, the result is usually rounded to 5.61.

Just as the same fraction can be written in more than one way, the same real number may have more than one decimal representation. For example, 0.999..., 1.0, 1.00, 1.000, ..., all represent the natural number 1. A given real number has only the following decimal representations: an approximation to some finite number of decimal places, an approximation in which a pattern is established that continues for an unlimited number of decimal places or an exact value with only finitely many decimal places. In this last case, the last non-zero digit may be replaced by the digit one smaller followed by an unlimited number of 9's, or the last non-zero digit may be followed by an unlimited number of zeros. Thus the exact real number 3.74 can also be written 3.7399999999... and 3.74000000000.... Similarly, a decimal numeral with an unlimited number of 0's can be rewritten by dropping the 0's to the right of the rightmost nonzero digit, and a decimal numeral with an unlimited number of 9's can be rewritten by increasing by one the rightmost digit less than 9, and changing all the 9's to the right of that digit to 0's. Finally, an unlimited sequence of 0's to the right of a decimal place can be dropped. For example, 6.849999999999... = 6.85 and 6.850000000000... = 6.85. Finally, if all of the digits in a numeral are 0, the number is 0, and if all of the digits in a numeral are an unending string of 9's, you can drop the nines to the right of the decimal place, and add one to the string of 9s to the left of the decimal place. For example, 99.999... = 100.

The real numbers also have an important but highly technical property called the

property.

It can be shown that any

) to the algebraic equation ${\displaystyle x^{2}+1=0}$.

Complex numbers

Moving to a greater level of abstraction, the real numbers can be extended to the complex numbers. This set of numbers arose historically from trying to find closed formulas for the roots of cubic and quadratic polynomials. This led to expressions involving the square roots of negative numbers, and eventually to the definition of a new number: a square root of −1, denoted by i, a symbol assigned by Leonhard Euler, and called the imaginary unit. The complex numbers consist of all numbers of the form

${\displaystyle \,a+bi}$

where a and b are real numbers. Because of this, complex numbers correspond to points on the

. The symbol for the complex numbers is C or ${\displaystyle \mathbb {C} }$.

The

.

Subclasses of the integers

Even and odd numbers

Main article:

Even and odd numbers

An even number is an integer that is "evenly divisible" by two, that is

. Similarly, the first non-negative even numbers are {0, 2, 4, 6, ...}.

Prime numbers

A prime number, often shortened to just prime, is an integer greater than 1 that is not the product of two smaller positive integers. The first few prime numbers are 2, 3, 5, 7, and 11. There is no such simple formula as for odd and even numbers to generate the prime numbers. The primes have been widely studied for more than 2000 years and have led to many questions, only some of which have been answered. The study of these questions belongs to number theory. Goldbach's conjecture is an example of a still unanswered question: "Is every even number the sum of two primes?"

One answered question, as to whether every integer greater than one is a product of primes in only one way, except for a rearrangement of the primes, was confirmed; this proven claim is called the fundamental theorem of arithmetic. A proof appears in Euclid's Elements.

Other classes of integers

Many subsets of the natural numbers have been the subject of specific studies and have been named, often after the first mathematician that has studied them. Example of such sets of integers are

.

Subclasses of the complex numbers

Algebraic, irrational and transcendental numbers

Algebraic numbers are those that are a solution to a polynomial equation with integer coefficients. Real numbers that are not rational numbers are called irrational numbers. Complex numbers which are not algebraic are called transcendental numbers. The algebraic numbers that are solutions of a monic polynomial equation with integer coefficients are called algebraic integers.

Constructible numbers

Motivated by the classical problems of constructions with straightedge and compass, the constructible numbers are those complex numbers whose real and imaginary parts can be constructed using straightedge and compass, starting from a given segment of unit length, in a finite number of steps.

Computable numbers

A computable number, also known as recursive number, is a

.

The computable numbers may be viewed as the real numbers that may be exactly represented in a computer: a computable number is exactly represented by its first digits and a program for computing further digits. However, the computable numbers are rarely used in practice. One reason is that there is no algorithm for testing the equality of two computable numbers. More precisely, there cannot exist any algorithm which takes any computable number as an input, and decides in every case if this number is equal to zero or not.

The set of computable numbers has the same cardinality as the natural numbers. Therefore, almost all real numbers are non-computable. However, it is very difficult to produce explicitly a real number that is not computable.

Extensions of the concept

p-adic numbers

The p-adic numbers may have infinitely long expansions to the left of the decimal point, in the same way that real numbers may have infinitely long expansions to the right. The number system that results depends on what base is used for the digits: any base is possible, but a prime number base provides the best mathematical properties. The set of the p-adic numbers contains the rational numbers, but is not contained in the complex numbers.

The elements of an

). Therefore, they are often regarded as numbers by number theorists. The p-adic numbers play an important role in this analogy.

Hypercomplex numbers

Some number systems that are not included in the complex numbers may be constructed from the real numbers in a way that generalize the construction of the complex numbers. They are sometimes called

, neither associative nor commutative.

Transfinite numbers

For dealing with infinite sets, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. The former gives the ordering of the set, while the latter gives its size. For finite sets, both ordinal and cardinal numbers are identified with the natural numbers. In the infinite case, many ordinal numbers correspond to the same cardinal number.

Nonstandard numbers

statements about R to be reinterpreted as true first-order statements about *R.

Superreal and surreal numbers extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form fields.

See also

Notes

References

• Tobias Dantzig, Number, the language of science; a critical survey written for the cultured non-mathematician, New York, The Macmillan Company, 1930.^{[ISBN missing]}
• Erich Friedman, What's special about this number? Archived 2018-02-23 at the Wayback Machine
• Steven Galovich, Introduction to Mathematical Structures, Harcourt Brace Javanovich, 1989,
  ISBN 0-15-543468-3
  .
• ISBN 0-387-90092-6
  .
• ISBN 978-0195061352
• Alfred North Whitehead and Bertrand Russell, Principia Mathematica to *56, Cambridge University Press, 1910.^{[ISBN missing]}
• Leo Cory, A Brief History of Numbers, Oxford University Press, 2015,
  ISBN 978-0-19-870259-7
  .

External links

Wikimedia Commons has media related to Numbers.

Wikiquote has quotations related to Number.

Look up number in Wiktionary, the free dictionary.

Wikiversity has learning resources about Primary mathematics: Numbers

• Nechaev, V.I. (2001) [1994]. "Number". Encyclopedia of Mathematics. EMS Press.
• Tallant, Jonathan. "Do Numbers Exist". Numberphile. Brady Haran. Archived from the original on 8 March 2016. Retrieved 6 April 2013.
• In Our Time: Negative Numbers. BBC Radio 4. 9 March 2006. Archived from the original on 31 May 2022.
• Robin Wilson (7 November 2007). "4000 Years of Numbers". Gresham College. Archived from the original on 8 April 2022.
• Krulwich, Robert (22 July 2011). "What's the World's Favorite Number?". NPR. Archived from the original on 18 May 2021. Retrieved 17 September 2011.; "Cuddling With 9, Smooching With 8, Winking At 7". NPR. 21 August 2011. Archived from the original on 6 November 2018. Retrieved 17 September 2011.
• Online Encyclopedia of Integer Sequences