Editing Number (section)

===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.