Tagrumma (tusnakt)

 

Deg tusnakt, tagrumma d talɣa tajerrumt i d-yettwafken s kra n iferdisen d yiwen n usaḍuf. Iferdisen-a zemren ad ilin d imḍanen, d tiɣawin neɣ d izamulen nniḍen. Asaḍuf-a yettwasemma d "timernit n tegrumma" (neɣ "timernit"), yettakk-d yiwen uferdis n tegrumma seg sin iferdisen nniḍen. Timernit n tegrumma tesɛa krad (3) n tẓenziwin tigejdanin:

1. Tisedrewt : i yal aferdis a,b, c n tegrumma ${\displaystyle (a*b)*c=a*(b*c)}$
2. Aferdis n tulut : yella yiwen uferdis, isem-is ${\displaystyle e}$ , i yal aferdis ${\displaystyle a}$ , nesɛa: ${\displaystyle a*e=e*a=a}$
3. Amitti : i yal aferdis ${\displaystyle a}$ , yella yiwen uferdis ${\displaystyle {\acute {a}}}$ (qqaren-as "amitti n a") akken ${\displaystyle a*{\acute {a}}={\acute {a}}*a=e}$
4. Tagrumma tezmer ad tili tafessast neɣ d taqsuḍt. Tagrumma taqsuḍt tesɛa taẓenẓit nniḍen : Tasunfelt : i tal sin iferdisen ${\displaystyle a}$ akked ${\displaystyle b}$ nesɛa : ${\displaystyle a*b=b*a}$

Tagrumma d yiwet seg talɣiwin tigejdanin n tusnakt tamzarugt. Deg tusnakt tatrart, tagrumma tettwaqbel d amedya n talɣa tajerrumt. Isenfaren n tegrummiwin ttbanen-d deg waṭas n yiḥricen n tusnakt, am tesnaka n tegrummiwin, tasleḍt n tegrummiwin, tasleḍt tajerrumt, tasnukyist, taseddast tajerrumt, tasleḍt talgelgalt, d wiyaḍ.

imḍanen igmawiyen ( ${\displaystyle \mathbb {N} }$* ) : ${\displaystyle \mathbb {N} *={1,2,3,4,...}}$

• Tagrumma yagi tetwasexdem i usiḍen
• ur yesɛi ara amiḍan 0

imḍanen ummiden ${\displaystyle \mathbb {N} }$

${\displaystyle \mathbb {N} ={0,1,2,3,4,5,...}}$

• Yemgarad ɣef ${\displaystyle \mathbb {N} }$* s tmerna n 0
• Yettwasexdem aṭas deg tussniwin n uselkim

Imḍanen imassaɣen ${\displaystyle \mathbb {Z} }$

• ${\displaystyle \mathrm {Z} +=\mathbb {N} ={0,1,2,3,...}}$ (imassaɣen imuṭṭiyen )
• ${\displaystyle \mathrm {Z} -={...,-3,-2,-1}}$ (imassaɣen izelmaḍen)
• ${\displaystyle \mathrm {Z} =\mathrm {Z} -\cup (0)\cup \mathrm {Z} +=\mathbb {N} ={0,1,2,3,...}}$

Imḍanen imeɣẓanen ${\displaystyle \mathbb {Q} }$

• ${\displaystyle \mathbb {Q} =\{x|x=a/b,a\in \mathrm {Z} ,b\in \mathrm {Z} ,b\neq 0\}}$
• Zemren ad ttwarun s talɣa taẓunant n sin yimḍanen imassaɣen.

Imḍanen armeɣẓanen

• D imḍanen ur nezmir ara ad naru s talɣa ${\displaystyle a/b}$
• imedyaten : ${\displaystyle {\sqrt {2}},\pi ,e}$

Imḍanen ilawen ${\displaystyle \mathbb {R} }$

• s azdukel n yimḍanen imeɣẓanen d yimḍanen armeɣẓanen

Imḍanen isemlalen ${\displaystyle \mathbb {C} }$

• ${\displaystyle \mathbb {C} =\{x+iy|x\in \mathbb {R} ,y\in \mathbb {R} \},i}$ d tigget tasugnant
• Yegber akk imḍanen ilawen d yimḍanen isugnanen.
• Yal amḍan asemlal yezmer ad yettwaru s talɣa a + bi, anda a d b d imḍanen ilawen.

Assaɣen gar tugrummiwin

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

Assaɣ-agi yemmal-d belli:

• Yal amḍan agmaw d amḍan amassaɣ.
• Yal amḍan amassaɣ d amḍan ameɣẓan.
• Yal amḍan ameɣẓan d amḍan ilaw.
• Yal amḍan ilaw d amḍan asemlal.

Deg tusnakt, llan waṭas n tugrummiwin ur nelli ara d tid n yimḍanen. Attan kra seg-sent:

• Tabadut: D tagrumma n yizamulen yettwasxedmen i usuddes n tenfaliyin.
• Amedya: Tagrumma n yisekkilen n ugemmay amaziɣ.
• Asemres: Tettwaqqen deg tjerrumt, taseftit, d tussna n uselkim.

• Tabadut: D tagrumma n yal tawuri yezmer ad yexdem yiwen unagraw.
• Amedya: Tagrumma n yinagrawen n tfelwit n uselkim Windows.
• Asemres: Tettwasexdem deg usnulfu n yinagrawen d tseddast n yinagrawen.

• Tabadut: D tagrumma n yiberdan yezmer ad ddun yimdanen neɣ timacinin seg yiwen wadeg ɣer wayeḍ.
• Amedya: Tagrumma n yiberdan gar sin yiɣerman deg uɣaras.
• Asemres: Tettwasexdem deg usnulfu n yinagrawen n unezzeh d uselkem n yiberdan iwatan.

1. Tallit taqburt: Tikti n tegrumma tella-d seg zik, maca ur telli ara s talɣa tunṣibt.
   • Imasiwen iqburen xeddmen s tiktiwin yettemcabin ɣer tid n tegrumma.
2. Lqern wis 17-18:
   • Gottfried Wilhelm Leibniz (1646-1716) yebda yettnadi ɣef "tussna n yisurdiyen".
   • Leonhard Euler (1707-1783) yesseqdec tiktiwin n tegrumma deg waṭas n yisenfaren-ines.

1. Bernard Bolzano (1781-1848):
   • Yefka-d tabadut tamezwarut n tegrumma, maca ur d-teffiɣ ara ɣer beṛṛa.
2. Georg Cantor (1845-1918):
   • Yettwassen s "Baba n tussna n tegrumma".
   • Deg 1874, yessenked-d amahil-ines amezwaru ɣef tegrummiwin.
   • Yesnulfa-d tabadut tunṣibt n tegrumma d tussna n tegrummiwin tilawin.
3. Richard Dedekind (1831-1916):
   • Yefka-d tabadut tunṣibt n tegrumma deg 1888.
   • Yexdem ɣef tugrummiwin tilawin d yimḍanen ilawin.
4. Gottlob Frege (1848-1925):
   • Yexdem ɣef llsas n tsertiyan n tegrumma.
   • Yura adlis "Grundgesetze der Arithmetik" (1893-1903).

1. Bertrand Russell (1872-1970) d Alfred North Whitehead (1861-1947):
   • Uran "Principia Mathematica" (1910-1913), adlis ameqqran ɣef llsas n tusnakt.
2. Ernst Zermelo (1871-1953) d Abraham Fraenkel (1891-1965):
   • Snulfan-d tisertiyin n Zermelo-Fraenkel (ZF) n tegrumma.
3. John von Neumann (1903-1957):
   • Yefka-d asnulfu n tegrumma s useqdec n tegrummiwin kan.
4. Nicolas Bourbaki (1935-...):
   • Isem ustiran n yiwet n tegrawt n yimusnawen n tusnakt.
   • Xedmen aṭas ɣef useqdec n tegrumma d tsertiyin-ines deg tusnakt meṛṛa.

1. Tussna n uselkim:
   • Tagrumma tettwasexdem s waṭas deg tussna n uselkim d usnulfu n tutlayin n usihel.
2. Tussna n yisefka:
   • Tikti n tegrumma tettwasexdem deg usnulfu n yidlisen n yisefka d yinagrawen n usesfer n yisalan.
3. Tasertit:
   • Tagrumma tettwasexdem deg tsertit tanefrant d tsertit n uswizi.
4. Tasnulfut:
   • Tagrumma tettwasexdem deg tasnulfut tanefrant d usnulfu n yinagrawen imaynuten.

Tussna n tegrumma tuɣal d yiwet seg tɣula tigejdanin n tusnakt tamzarugt. Seg talalit-ines tunṣibt deg tasut tis 19, tussna-agi tesnerpes ɣef waṭas n yiḥricen n tusnakt d tussniwin nniḍen. Ass-a, tagrumma d tikti tagejdant deg tusnakt, tussna n uselkim, d waṭas n yiḥricen nniḍen n tussna d tudert n yal ass.