Editing Number (section)

==Main classification{{anchor|Classification|Classification of numbers}}==
{{Redirect|Number system|systems which express numbers|Numeral system}}
{{See also|List of types of numbers}}
Numbers can be classified into [[set (mathematics)|sets]], called '''number sets''' or '''number systems''', such as the [[natural numbers]] and the [[real numbers]]. The main number systems are as follows:
{|class="wikitable" style="margin: 1em auto; max-width: 600px; overflow-x: auto"
|+ Main number systems
!Symbol
!Name
!Examples/Explanation
|-
!<math>\mathbb{N}</math>
![[Natural number]]s
| 0, 1, 2, 3, 4, 5, ... or 1, 2, 3, 4, 5, ...<br />
<math>\mathbb{N}_0</math> or <math>\mathbb{N}_1</math> are sometimes used.
|-
!<math>\mathbb{Z}</math>
![[Integer]]s
|..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ...
|-
!<math>\mathbb{Q}</math>
![[Rational number]]s
|{{sfrac|''a''|''b''}} where ''a'' and ''b'' are integers and ''b'' is not 0
|-
!<math>\mathbb{R}</math>
![[Real number]]s
|The limit of a convergent sequence of rational numbers
|-
!<math>\mathbb{C}</math>
![[Complex number]]s
|''a'' + ''bi'' where ''a'' and ''b'' are real numbers and ''i'' is a formal square root of&nbsp;−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
:<math>\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}</math>.

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

===Natural numbers===
{{Main|Natural number}}
[[File:Nat num.svg|thumb|The natural numbers, starting with 1]]
The most familiar numbers are the [[natural number]]s (sometimes called whole numbers or counting numbers): 1, 2, 3, and so on. Traditionally, the sequence of natural numbers started with&nbsp;1 (0 was not even considered a number for the Ancient Greeks.) However, in the 19th&nbsp;century, [[set theory|set theorists]] and other mathematicians started including&nbsp;0 ([[cardinality]] of the [[empty set]], i.e. 0&nbsp;elements, where&nbsp;0 is thus the smallest [[cardinal number]]) in the set of natural numbers.<ref>
{{MathWorld|title=Natural Number|id=NaturalNumber}}</ref><ref>{{Cite web |url=http://www.merriam-webster.com/dictionary/natural%20number |title=natural number |work=Merriam-Webster.com |publisher=[[Merriam-Webster]] |access-date=4 October 2014 |archive-url=https://web.archive.org/web/20191213133201/https://www.merriam-webster.com/dictionary/natural%20number |archive-date=13 December 2019 |url-status=live }}</ref> Today, different mathematicians use the term to describe both sets, including&nbsp;0 or not. The [[mathematical symbol]] for the set of all natural numbers is '''N''', also written <math>\mathbb{N}</math>, and sometimes <math>\mathbb{N}_0</math> or <math>\mathbb{N}_1</math> when it is necessary to indicate whether the set should start with 0 or 1, respectively.

In the [[base 10]] numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten [[numerical digit|digits]]: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The [[Radix|radix or base]] is the number of unique numerical digits, including zero, that a numeral system uses to represent numbers (for the decimal system, the radix is 10). In this base&nbsp;10 system, the rightmost digit of a natural number has a [[place value]] of&nbsp;1, and every other digit has a place value ten times that of the place value of the digit to its right.

In [[set theory]], which is capable of acting as an axiomatic foundation for modern mathematics,<ref>{{Cite book |last=Suppes |first=Patrick |author-link=Patrick Suppes |title=Axiomatic Set Theory |publisher=Courier Dover Publications |year=1972 |page=[https://archive.org/details/axiomaticsettheo00supp_0/page/1 1] |isbn=0-486-61630-4 |url=https://archive.org/details/axiomaticsettheo00supp_0/page/1 }}</ref> natural numbers can be represented by classes of equivalent sets. For instance, the number&nbsp;3 can be represented as the class of all sets that have exactly three elements. Alternatively, in [[Peano Arithmetic]], the number&nbsp;3 is represented as sss0, where s is the "successor" function (i.e.,&nbsp;3 is the third successor of&nbsp;0). Many different representations are possible; all that is needed to formally represent&nbsp;3 is to inscribe a certain symbol or pattern of symbols three times.

===Integers===
{{Main|Integer}}
The negative of a positive integer is defined as a number that produces&nbsp;0 when it is added to the corresponding positive integer. Negative numbers are usually written with a negative sign (a [[minus sign]]). As an example, the negative of&nbsp;7 is written&nbsp;−7, and {{nowrap|7 + (−7) {{=}} 0}}. When the [[set (mathematics)|set]] of negative numbers is combined with the set of natural numbers (including&nbsp;0), the result is defined as the set of [[integer]]s, '''Z''' also written [[Blackboard bold|<math>\mathbb{Z}</math>]]. Here the letter Z comes {{ety|de|Zahl|number}}. The set of integers forms a [[ring (mathematics)|ring]] with the operations addition and multiplication.<ref>{{Mathworld|Integer|Integer}}</ref>

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===
{{Main|Rational number}}
A rational number is a number that can be expressed as a fraction 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 {{sfrac|''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 {{sfrac|1|2}} and {{sfrac|2|4}} are equal, that is:
:<math>{1 \over 2} = {2 \over 4}.</math>

In general,
:<math>{a \over b} = {c \over d}</math> if and only if <math>{a \times d} = {c \times b}.</math>

If the [[absolute value]] of ''m'' is greater than ''n'' (supposed to be positive), then the absolute value of the fraction is greater than&nbsp;1. Fractions can be greater than, less than, or equal to&nbsp;1 and can also be positive, negative, or&nbsp;0. The set of all rational numbers includes the integers since every integer can be written as a fraction with denominator&nbsp;1. For example&nbsp;−7 can be written&nbsp;{{sfrac|−7|1}}. The symbol for the rational numbers is '''Q''' (for ''[[quotient]]''), also written [[Blackboard bold|<math>\mathbb{Q}</math>.]]

===Real numbers===
{{Main|Real number}}

The symbol for the real numbers is '''R''', also written as <math>\mathbb{R}.</math> They include all the measuring numbers. Every real number corresponds to a point on the [[number line]]. The following paragraph will focus primarily on positive real numbers. The treatment of negative real numbers is according to the general rules of arithmetic and their denotation is simply prefixing the corresponding positive numeral by a [[minus sign]], e.g. −123.456.

Most real numbers can only be ''approximated'' by [[decimal]] numerals, in which a [[decimal point]] is placed to the right of the digit with place value&nbsp;1. 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. For example, 123.456 represents {{sfrac|123456|1000}}, or, in words, one hundred, two tens, three ones, four tenths, five hundredths, and six thousandths. A real number can be expressed by a finite number of decimal digits only if it is rational and its [[fractional part]] has a denominator whose prime factors are 2 or 5 or both, because these are the prime factors of 10, the base of the decimal system. Thus, for example, one half is 0.5, one fifth is 0.2, one-tenth is 0.1, and one fiftieth is 0.02. Representing other real numbers as decimals would require an infinite sequence of digits to the right of the decimal point. If this infinite sequence of digits follows a pattern, it can be written with an ellipsis or another notation that indicates the repeating pattern. Such a decimal is called a [[repeating decimal]]. Thus {{sfrac|3}} can be written as 0.333..., with an ellipsis to indicate that the pattern continues. Forever repeating 3s are also written as 0.{{overline|3}}.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Repeating Decimal|url=https://mathworld.wolfram.com/RepeatingDecimal.html|access-date=2020-07-23|website=Wolfram MathWorld |language=en|archive-date=2020-08-05|archive-url=https://web.archive.org/web/20200805170548/https://mathworld.wolfram.com/RepeatingDecimal.html|url-status=live}}</ref>

It turns out that these repeating decimals (including the [[Trailing zero|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 number|irrational]]. A famous irrational real number is the [[pi|{{pi}}]], the ratio of the [[circumference]] of any circle to its [[diameter]]. When pi is written as
:<math>\pi = 3.14159265358979\dots,</math>
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 [[proof that pi is irrational|{{pi}} is irrational]]. Another well-known number, proven to be an irrational real number, is
:<math>\sqrt{2} = 1.41421356237\dots,</math>
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 {{nowrap|( 1 trillion {{=}} 10<sup>12</sup> {{=}} 1,000,000,000,000 )}} of digits.

Not only these prominent examples but [[almost all]] real numbers are irrational and therefore have no repeating patterns and hence no corresponding decimal numeral. They can only be approximated by decimal numerals, denoting [[rounding|rounded]] or [[truncation|truncated]] real numbers. Any rounded or truncated number is necessarily a rational number, of which there are only [[countably many]]. All measurements are, by their nature, approximations, and always have a [[margin of error]]. Thus 123.456 is considered an approximation of any real number greater or equal to {{sfrac|1234555|10000}} and strictly less than {{sfrac|1234565|10000}} (rounding to 3 decimals), or of any real number greater or equal to {{sfrac|123456|1000}} and strictly less than {{sfrac|123457|1000}} (truncation after the 3. decimal). Digits that suggest a greater accuracy than the measurement itself does, should be removed. The remaining digits are then called [[significant digits]]. For example, measurements with a ruler can seldom be made without a margin of error of at least 0.001 [[Metre|m]]. If the sides of a rectangle are measured as 1.23&nbsp;m and 4.56&nbsp;m, then multiplication gives an area for the rectangle between {{nowrap|5.614591 m<sup>2</sup>}} and {{nowrap|5.603011 m<sup>2</sup>}}. 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&nbsp;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 9s, 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 0s can be rewritten by dropping the 0s to the right of the rightmost nonzero digit, and a decimal numeral with an unlimited number of 9s can be rewritten by increasing by one the rightmost digit less than 9, and changing all the 9s to the right of that digit to 0s. Finally, an unlimited sequence of 0s 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 9s, 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 [[least upper bound]] property.

It can be shown that any [[ordered field]], which is also [[completeness of the real numbers|complete]], is isomorphic to the real numbers. The real numbers are not, however, an [[algebraically closed field]], because they do not include a solution (often called a [[square root of minus one]]) to the algebraic equation <math> x^2+1=0</math>.

===Complex numbers===
{{Main|Complex number}}
Moving to a greater level of abstraction, the real numbers can be extended to the [[complex number]]s. This set of numbers arose historically from trying to find closed formulas for the roots of [[cubic function|cubic]] and [[quadratic function|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&nbsp;−1, denoted by ''[[imaginary unit|i]]'', a symbol assigned by [[Leonhard Euler]], and called the [[imaginary unit]]. The complex numbers consist of all numbers of the form
:<math>\,a + b i</math>
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 [[dimension]]s. In the expression {{nowrap|''a'' + ''bi''}}, the real number ''a'' is called the [[real part]] and ''b'' is called the [[imaginary part]]. If the real part of a complex number is&nbsp;0, then the number is called an [[imaginary number]] or is referred to as ''purely imaginary''; if the imaginary part is&nbsp;0, then the number is a real number. Thus the real numbers are a [[subset]] of the complex numbers. If the real and imaginary parts of a complex number are both integers, then the number is called a [[Gaussian integer]]. The symbol for the complex numbers is '''C''' or <math>\mathbb{C}</math>.

The [[fundamental theorem of algebra]] asserts that the complex numbers form an [[algebraically closed field]], meaning that every [[polynomial]] with complex coefficients has a [[zero of a function|root]] in the complex numbers. Like the reals, the complex numbers form a [[field (mathematics)|field]], which is [[complete space|complete]], but unlike the real numbers, it is not [[total order|ordered]]. That is, there is no consistent meaning assignable to saying that ''i'' is greater than&nbsp;1, nor is there any meaning in saying that ''i'' is less than&nbsp;1. In technical terms, the complex numbers lack a [[total order]] that is [[ordered field|compatible with field operations]].