Editing 3 (section)

== Mathematics ==
=== Divisibility rule ===
A [[natural number]] is [[divisible]] by 3 if the [[digital root|sum of its digits]] in [[base 10]] is also divisible by 3. This known as the [[divisibility rule]] of 3. Because of this, the reverse of any number that is divisible by three (or indeed, any [[permutation]] of its digits) is also divisible by three. This divisibility rule works in any [[positional notation|positional numeral system]] whose [[radix|base]] divided by three leaves a remainder of one (bases 4, 7, 10, etc.).{{Citation needed|date=February 2025}}

=== Properties ===
3 is the second smallest [[prime number]] and the first [[Parity (mathematics)|odd]] prime number. 3 is a [[twin prime]] with [[5]], and a [[cousin prime]] with [[7]].

A [[triangle]] is made of three [[Edge (geometry)|sides]]. It is the smallest non-self-intersecting [[polygon]] and the only polygon not to have proper [[diagonals]]. When doing quick estimates, 3 is a rough approximation of [[pi|{{pi}}]], 3.1415..., and a very rough approximation of [[E (mathematical constant)|''e'']], 2.71828...

3 is the first [[Mersenne prime]]. 3 is also the first of five known [[Fermat prime]]s. It is the second [[Fibonacci number|Fibonacci prime]] (and the second [[Lucas prime]]), the second [[Sophie Germain prime]], and the second [[factorial prime]].

3 is the second and only prime [[triangular number]],<ref>{{Cite web |title=A000217 - OEIS |url=https://oeis.org/A000217 |access-date=2024-11-28 |website=oeis.org}}</ref> and [[Gauss]] proved that every integer is the sum of at most 3 [[triangular numbers]].

Three is the only prime which is one less than a [[square number|perfect square]]. Any other number which is <math>n^2</math> − 1 for some integer <math>n</math> is not prime, since it is (<math>n</math> − 1)(<math>n</math> + 1). This is true for 3 as well (with <math>n</math> = 2), but in this case the smaller factor is 1. If <math>n</math> is greater than 2, both <math>n</math> − 1 and <math>n</math> + 1 are greater than 1 so their product is not prime.

=== Numeral systems ===
There is some evidence to suggest that early man may have used counting systems which consisted of "One, Two, Three" and thereafter "Many" to describe counting limits. Early peoples had a word to describe the quantities of one, two, and three but any quantity beyond was simply denoted as "Many". This is most likely based on the prevalence of this phenomenon among people in such disparate regions as the deep Amazon and Borneo jungles, where western civilization's explorers have historical records of their first encounters with these indigenous people.<ref>{{cite book | last1 = Gribbin | first1 = Mary  | last2 = Gribbin | first2 = John R. | last3 = Edney | first3 = Ralph | last4 = Halliday | first4 = Nicholas | title = Big numbers | publisher = Wizard | location = Cambridge | year = 2003 | isbn = 1840464313 }}</ref>

=== List of basic calculations ===
{|class="wikitable" style="text-align: center; background: white"
|-
!width="105px"|[[Multiplication]]
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
!12
!13
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!100
!1000
!10000
|-
|'''3 × ''x'''''
|'''3'''
|[[6 (number)|6]]
|[[9 (number)|9]]
|[[12 (number)|12]]
|[[15 (number)|15]]
|[[18 (number)|18]]
|[[21 (number)|21]]
|[[24 (number)|24]]
|[[27 (number)|27]]
|[[30 (number)|30]]
|[[33 (number)|33]]
|[[36 (number)|36]]
|[[39 (number)|39]]
|[[42 (number)|42]]
|[[45 (number)|45]]
|[[48 (number)|48]]
|[[51 (number)|51]]
|[[54 (number)|54]]
|[[57 (number)|57]]
|[[60 (number)|60]]
|[[63 (number)|63]]
|[[66 (number)|66]]
|[[69 (number)|69]]
|[[72 (number)|72]]
|[[75 (number)|75]]
|[[150 (number)|150]]
|[[300 (number)|300]]
|[[3000 (number)|3000]]
|[[30000 (number)|30000]]
|}

{|class="wikitable" style="text-align: center; background: white"
|-
!width="105px"|[[Division (mathematics)|Division]]
!1
!2
!3
!4
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!7
!8
!9
!10
!width="5px"|
!11
!12
!13
!14
!15
!16
!17
!18
!19
!20
|-
|'''3 ÷ ''x'''''
|'''3'''
|1.5
|1
|0.75
|0.6
|0.5
|0.{{overline|428571}}
|0.375
|0.{{overline|3}}
|0.3
!
|0.{{overline|27}}
|0.25
|0.{{overline|230769}}
|0.2{{overline|142857}}
|0.2
|0.1875
|0.1{{overline|7647058823529411}}
|0.1{{overline|6}}
|0.1{{overline|57894736842105263}}
|0.15
|-
|'''''x'' ÷ 3'''
|0.{{overline|3}}
|0.{{overline|6}}
|1
|1.{{overline|3}}
|1.{{overline|6}}
|2
|2.{{overline|3}}
|2.{{overline|6}}
|'''3'''
|3.{{overline|3}}
!
|3.{{overline|6}}
|4
|4.{{overline|3}}
|4.{{overline|6}}
|5
|5.{{overline|3}}
|5.{{overline|6}}
|6
|6.{{overline|3}}
|6.{{overline|6}}
|}

{|class="wikitable" style="text-align: center; background: white"
|-
!width="105px"|[[Exponentiation]]
!1
!2
!3
!4
!5
!6
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!9
!10
!width="5px"|
!11
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!13
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!19
!20
|-
|'''3{{sup|''x''}}'''
|'''3'''
|9
|27
|[[81 (number)|81]]
|[[243 (number)|243]]
|729
|2187
|6561
|19683
|59049
!
|177147
|531441
|1594323
|4782969
|14348907
|43046721
|129140163
|387420489
|1162261467
|3486784401
|-
|'''''x''{{sup|3}}'''
|1
|[[8 (number)|8]]
|27
|[[64 (number)|64]]
|[[125 (number)|125]]
|[[216 (number)|216]]
|[[343 (number)|343]]
|[[512 (number)|512]]
|729
|[[1000 (number)|1000]]
!
|1331
|1728
|2197
|2744
|3375
|4096
|4913
|5832
|6859
|8000
|}