Numbers



What is a number?

A number in mathematics is the word or symbol used to designate quantities or entities that behave as quantities.

A number in mathematics is the word or symbol used to designate quantities or entities that behave as quantities.

Numbers are grouped into diverse sets or structures; each contains the one before it and is more complete than it and with greater possibilities in its operations. They are listed below.

Numbers in different languages

  1. Arbëresh
  2. Acholi
  3. Aczu Śavnecze
  4. Adyghe
  5. Afata
  6. Afrihili
  7. Afrikaans
  8. Ainu
  9. Araki
  10. Amharic
  11. Aramteskan
  12. Arabic
  13. Arhuaco
  14. Arikara
  15. Mapudungun
  16. Assiniboine
  17. Asturian
  18. Atlantean
  19. Atrian
  20. Kotava
  21. Ayeri
  22. Aymara
  23. Azazilúŝ
  24. Azerbaijani
  25. Babm
  26. Bashkir
  27. Bambara
  28. Bavarian
  29. Barsoomian
  30. Ba kom
  31. Belarusian
  32. Belter Creole
  33. Baka
  34. Bocce
  35. Bolak
  36. Breton
  37. Brooding
  38. Burushaski
  39. Bulgarian
  40. Garifuna
  41. Carrier
  42. Kaqchikel
  43. Kali’na
  44. Castithan
  45. Catalan
  46. Chavacano
  47. Ceqli
  48. Czech
  49. Chakobsa
  50. Cherokee
  51. Chuvash
  52. Koasati
  53. Klallam
  54. Lowland Oaxaca Chontal
  55. Cocama
  56. Comox
  57. Cornish
  58. Corsican
  59. Michif
  60. Crimean Tatar
  61. Ch’ol
  62. Welsh
  63. Dagbani
  64. Dai
  65. Danish
  66. Tsez
  67. German
  68. Dogrib
  69. Aukan
  70. D’ni
  71. Dothraki
  72. Dovahzul
  73. Lower Sorbian
  74. Mussau-Emira
  75. Alutiiq
  76. English
  77. Engála
  78. Engsvanyáli
  79. Eonavian
  80. Esperanto
  81. South Efate
  82. Inupiaq
  83. Estonian
  84. Basque
  85. Ewokese
  86. Faroese
  87. Persian
  88. Finnish
  89. Kven
  90. Folkspraak
  91. Digisk Folkspraak
  92. French
  93. Jèrriais
  94. North Frisian
  95. West Frisian
  96. Friulian
  97. Nigerian Fulfulde
  98. Ga
  99. Gallo
  100. Gandal
  101. Antillean Creole of Martinique
  102. Giak
  103. Gilbertese
  104. Scottish Gaelic
  105. Irish
  106. Galician
  107. Globasa
  108. Glosa
  109. Manx Gaelic
  110. Gottscheerish
  111. Grayis
  112. Swiss German
  113. Alsatian
  114. Wayuu
  115. Guarani
  116. Gujarati
  117. Gunganese
  118. Guosa
  119. G’Vunna
  120. Gwere
  121. Haida
  122. Haitian Creole
  123. Hausa
  124. Hebrew
  125. Elder Speech
  126. High Valyrian
  127. Hindi
  128. Hiuʦɑθ
  129. Hopi
  130. Hunsrik
  131. Upper Sorbian
  132. Huli
  133. Hungarian
  134. Hupa
  135. Halkomelem
  136. Huttese
  137. Armenian
  138. Hylian
  139. Igbo
  140. Idiom neutral
  141. Ido
  142. Northern Yi
  143. Interlingue
  144. Illitan
  145. Interlingua
  146. Indonesian
  147. Indojisnen
  148. Ingush
  149. Intal
  150. Interslavic
  151. Irathient
  152. Icelandic
  153. Italian
  154. Ithkuil
  155. Itláni
  156. Ingrian
  157. Jakaltek
  158. Jawaese
  159. Lojban
  160. Japanese
  161. Jaqaru
  162. Bezhta
  163. Georgian
  164. Kazakh
  165. Kabiye
  166. Cape Verdean Creole
  167. Kēlen
  168. Kerch
  169. Kiitra
  170. KiLiKi
  171. Kinuk’aaz
  172. Kyrgyz
  173. Kirmanjki
  174. Northern Kurdish
  175. Korean
  176. Karelian
  177. Kutenai
  178. Awa Pit
  179. Lango
  180. Langue nouvelle
  181. Latin
  182. Latino sine flexione
  183. Latvian
  184. Láadan
  185. Lezgian
  186. Lingua Franca Nova
  187. Lingwa de planeta
  188. Lingala
  189. Lithuanian
  190. Livonian
  191. Livyáni
  192. Kiliwa
  193. Lakota
  194. Llanito
  195. Ladin
  196. Lombard (Milanese)
  197. Loglan
  198. Luxembourgish
  199. Laz
  200. Marshallese
  201. Mandalorian
  202. Mazahua
  203. Kristang
  204. Menominee
  205. Mauritian Creole
  206. Miami-Illinois
  207. Micmac
  208. Minangkabau
  209. Macedonian
  210. Kituba
  211. Malagasy
  212. Maltese
  213. Mwotlap
  214. Moloko
  215. Mandinka
  216. Innu
  217. Mohawk
  218. Mondial
  219. Mondir
  220. Mondlango
  221. Māori
  222. Totontepec Mixe
  223. Tezoatlán Mixtec
  224. Navajo
  225. Na’vi
  226. Nêlêmwa
  227. Nengone
  228. Nìmpyèshiu
  229. Norwegian (Bokmål)
  230. Nove Latina
  231. Ndom
  232. Nyungwe
  233. Occitan
  234. Ojibwa
  235. Okanagan
  236. Oneida
  237. Odia
  238. Oromo
  239. Sierra Otomi
  240. Neimoidian
  241. Pandunia
  242. Timbisha
  243. Picard
  244. Pennsylvania German
  245. Plautdietsch
  246. Proto-Indo-European
  247. Polari
  248. Polish
  249. Portuguese (Brazil)
  250. Portuguese (Portugal)
  251. Malecite-Passamaquoddy
  252. Paicî
  253. Purépecha
  254. Punu
  255. Quetzaltepec Mixe
  256. Southern Quechua
  257. Quenya
  258. Rapa Nui
  259. Ravkan
  260. Rohingya
  261. Dzambazi Romani
  262. Caló
  263. Kalderash Romani
  264. Ro
  265. Romansh
  266. Romani
  267. Romanid
  268. Romulan
  269. Russian
  270. Sango
  271. Yakut
  272. Scots
  273. Shiväisith
  274. Shuswap
  275. Shyriiwook
  276. Siinyamda
  277. Pite Sami
  278. Sindarin
  279. Ume Sami
  280. Lushootseed
  281. Slovak
  282. Slovio
  283. Slovene
  284. Southern Sami
  285. Northern Sami
  286. Lule Sami
  287. Inari Sami
  288. Skolt Sami
  289. Shona
  290. Soninke
  291. Solresol
  292. Somali
  293. Sona
  294. Spanish
  295. Spokil
  296. Albanian
  297. Squamish
  298. Sardinian
  299. Sranan Tongo
  300. Serbian
  301. Saterland Frisian
  302. Saanich
  303. Sunúz
  304. Susu
  305. Swahili
  306. Swedish
  307. Tahitian
  308. Central Tarahumara
  309. Tetun Dili
  310. Telugu
  311. Nume
  312. Tukudede
  313. Klingon
  314. Tlingit
  315. Toki Pona
  316. Tolowa
  317. Siletz dee-ni
  318. Tongan (telephone-style)
  319. Tpaalha
  320. Tok Pisin
  321. Copala Triqui
  322. Trigedasleng
  323. Tswana
  324. Tsonga
  325. Tsolyáni
  326. Tüchte
  327. Tunica
  328. Turkish
  329. Tutonish
  330. Tamazight
  331. Uyghur
  332. Ukrainian
  333. Universalglot
  334. Uropi
  335. Va Ehenív
  336. Venetian
  337. Veda
  338. Veps
  339. Verdurian
  340. Makhuwa
  341. Volapük
  342. Votic
  343. Vulcan
  344. Wardwesân
  345. Mwani
  346. Wóxtjanato
  347. Wymysorys
  348. Xhosa
  349. Soga
  350. Mohegan-Pequot
  351. Yán Koryáni
  352. Yao
  353. Yiddish
  354. Yup’ik
  355. Santa Ana Yareni Zapotec
  356. Isthmus Zapotec
  357. Aloápam Zapotec
  358. Rincón Zapotec
  359. Choapan Zapotec
  360. Lachixío Zapotec
  361. Zulu

This is the mathematical notion of fundamental importance, introduced more or less consciously since antiquity, in order to be able to operate on quantities of elements constituting sets or on quantities expressing measures of material entities. Many numerical sets can be introduced axiomatically together with the corresponding operations, such as algebraic and topological particulars. Vice versa, one can proceed constructively, introducing successively larger numerical sets.

Types of numbers: brief introduction

.

The natural numbers 1, 2, 3,... are introduced as cardinal or as ordinal, i.e. as entities in a position to represent the order of finite sets and the positions of sequences (Peano axioms); zero is introduced as the order of the empty set. 

Zero and the natural numbers constitute the set of non-negative numbers. Negative numbers are introduced as the inverses of the positive numbers with respect to addition, and in order to be able to perform unrestricted subtraction. 

Rational numbers are introduced in order to perform unrestricted division. The extension to algebraic numbers is done to guarantee the existence of zeros of polynomials with integer coefficients. 

Real numbers are introduced in order to be able to perform with minimal restrictions operations passing to the limit. 

Finally, the real field is extended to that of the complex numbers to guarantee the existence of n roots for each polynomial of degree n.

- Polynomials with integer coefficients are introduced so that operations can be performed with minimal restrictions on passing to the limit.

- Fermat's number: Any number of the form 22n+1, for each n=1,2,3, ... It has been shown that its author's first conjecture that these numbers were all prime is not true.

- Perfect number: positive integer equal to the sum of its positive divisors, excluding itself. It is not known whether perfect odd numbers exist.

- Polygonal number: natural number of the sequence n0 = 1, n1 ... nr ... where nr = nr-1 + (m-2)r +1, where m is a natural number greater than two. For m = 3,4,5..., we obtain the triangular, quadrangular, pentagonal numbers.... The number nr is the number of points marked in a geometrical scheme formed with triangles, squares, pentagons..., respectively.

- Transfinite number: cardinal number which is not integer.

- Transcendent number: number that is not the root of any algebraic equation with rational coefficients.

- Triangular number: natural number of the sequence n0 = 1, n1 ... nr ... in which nr = nr-1 + r +1, .  The number nr is the number of marked points in a geometric scheme formed with triangles.

- Friendly numbers: pair of positive integers such that the sum of the positive divisors of each number less than itself is equal to the other number.

- Pythagorean numbers: triads of positive integers such that the square of one of them is equal to the sum of the squares of the other two. If the lengths of the two sides of a triangle are integers and Pythagorean, the triangle is right-angled.

The natural numbers

These are the ones used to count the elements of sets:

N = {0, 1, 2, 2, 3,..., 9, 10, 11, 11, 12,...}

There are infinities. They can be added and multiplied and with both operations the result is, in all cases, a natural number. However, they can not always be subtracted or divided (neither 3 - 7 nor 7 : 4 are natural numbers).

The integers

These are the natural numbers and the corresponding negatives:

Z = {..., -11, -10, -9,..., -3, -2, -1, 0, 1, 2, 3,..., 9, 10, 11,...}

In addition to being added and multiplied in all cases, they can be subtracted, so this structure improves on that of the naturals. However, in general, two integers cannot be divided. This is why we move on to the following number structure.

Rational numbers

These are the ones that can be expressed as a quotient of two integers. The set Q of rational numbers is composed of the integers and the fractional numbers. They can be added, subtracted, multiplied and divided (except by zero) and the result of all these operations between two rational numbers is always another rational number.

The real numbers

Unlike the natural numbers and the integers, the rational numbers are not arranged in such a way that they can be ordered one at a time. That is, there is no "next" rational number, because between any two rational numbers there are infinitely many others, so that if they are represented on a line, it is densely occupied by them: if we take a piece of line, a segment, however small, contains infinitely many rational numbers. However, in between these numbers densely located on the line there are also infinite other points that are not occupied by rationals. These are the irrational numbers.

The set formed by all rational and irrational numbers is the set of real numbers, so that all the numbers mentioned so far (natural, integer, rational, irrational) are real. These numbers occupy the number line point by point, so it is called real line.

The same operations are defined among the real numbers as among the rationals (addition, subtraction, multiplication and division, except for zero).

The imaginary numbers

The product of a real number by itself is always 0 or positive, so the equation x2 = -1 has no solution in the real number system. If one wants to give a value to x, such that x = Á, this cannot be a real value, no longer in a mathematical sense but also not in a technical sense. A new set of numbers (different from that of the real numbers), that of the imaginary numbers, is used for this purpose. The symbol i represents the unit of the imaginary numbers and is equivalent to Á. These numbers allow one to find, for example, the solution of the equation , which can be written as

x = 3 × i or x = 3i


The numbers bi,b ≠ 0, are called pure imaginary numbers.

An imaginary number is obtained by adding together a real number and a pure imaginary number.

The complex numbers

In its general form, a complex number is represented as a+ bi, where a and b are real numbers. The set of complex numbers consists of all the real numbers and all the imaginary numbers.

Complex numbers are usually represented in the so-called Argand diagram. The real and imaginary parts of a complex number are placed as points on two perpendicular lines or axes. In this way, a complex number is represented as a single point on a plane, known as the complex plane.

Complex numbers are of great use in the theory of alternating electric current as well as in other branches of physics, in engineering and in the natural sciences.

Complex numbers are also useful in the theory of alternating electric current as well as in other branches of physics, in engineering and in the natural sciences.

List of numbers from 1 to 1000