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:

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

## '''Tigrummiwin n yimḍanen'''

### <big>Imḍanen igmawiyen</big>
imḍanen igmawiyen ( \natnums * ) : \natnums * = {1,2,3,4,...} 

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

:

**<big>imḍanen ummiden <math>\natnums </math></big>**

\natnums = {0,1,2,3,4,5,...} 

* Yemgarad ɣef \natnums * s tmerna n 0
* Yettwasexdem aṭas deg tussniwin n uselkim

**Imḍanen imassaɣen <math>\Z
 </math>**
* \Zeta+ = \natnums = {0,1,2,3,...}  (imassaɣen imuṭṭiyen )
* \Zeta- = {...,-3,-2,-1}  (imassaɣen izelmaḍen)
* \Zeta = \Zeta- \cup  ( 0 )   \cup \Zeta+ = \natnums = {0,1,2,3,...}

**<big>Imḍanen imeɣẓanen <math>\Q
 </math></big>**

* **<big><math>\Q =  \{x|x = a/b, a \in \Zeta, b \in \Zeta, b \neq 0\} 
 </math></big>**
* Zemren ad ttwarun s talɣa taẓunant n sin yimḍanen imassaɣen.

**Imḍanen armeɣẓanen** 

* D imḍanen ur nezmir ara ad naru s talɣa **<math>a / b
 </math>**
* imedyaten : **<math>\sqrt{2}, \pi, e
 </math>**

**<big>Imḍanen ilawen <math>\R
 </math></big>**

* s azdukel n yimḍanen imeɣẓanen d yimḍanen armeɣẓanen

**<big>[[Amḍan asemlal|Imḍanen isemlalen <math>\Complex
 </math>]]</big>**

* **<math>\Complex = \{x + iy | x \in \R, y \in \R \}, i
 </math>** 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**

\N \subset \Z \subset \Q\subset \R\subset \Complex
 

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.

### '''<big>Tugrummiwin nniḍen</big>'''
Deg tusnakt, llan waṭas n tugrummiwin ur nelli ara d tid n yimḍanen. Attan kra seg-sent:

#### <big>Tigrummiwin n yizamulen</big>
* **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.

#### <big>Tigrummiwin n yinagrawen</big>
* **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.

#### <big>Tigrummiwin n yiberdan</big>
* **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.

## '''Amezruy'''

### <big>Tazwara (Send lqern wis 19)</big>
# **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.
# **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.

### <big>Tasut tis 19: Talalit n tussna n tegrumma</big>
# **Bernard Bolzano (1781-1848)**:
#* Yefka-d tabadut tamezwarut n tegrumma, maca ur d-teffiɣ ara ɣer beṛṛa.
# **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.
# **Richard Dedekind (1831-1916)**:
#* Yefka-d tabadut tunṣibt n tegrumma deg 1888.
#* Yexdem ɣef tugrummiwin tilawin d yimḍanen ilawin.
# **Gottlob Frege (1848-1925)**:
#* Yexdem ɣef llsas n tsertiyan n tegrumma.
#* Yura adlis "Grundgesetze der Arithmetik" (1893-1903).

### <big>Tasut tis 20: Asnerni d useqdec</big>
# **Bertrand Russell (1872-1970) d Alfred North Whitehead (1861-1947)**:
#* Uran "Principia Mathematica" (1910-1913), adlis ameqqran ɣef llsas n tusnakt.
# **Ernst Zermelo (1871-1953) d Abraham Fraenkel (1891-1965)**:
#* Snulfan-d tisertiyin n Zermelo-Fraenkel (ZF) n tegrumma.
# **John von Neumann (1903-1957)**:
#* Yefka-d asnulfu n tegrumma s useqdec n tegrummiwin kan.
# **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.

### <big>Tasut tis 21: Aseqdec atraru d tussniwin nniḍen</big>
# **Tussna n uselkim**:
#* Tagrumma tettwasexdem s waṭas deg tussna n uselkim d usnulfu n tutlayin n usihel.
# **Tussna n yisefka**:
#* Tikti n tegrumma tettwasexdem deg usnulfu n yidlisen n yisefka d yinagrawen n usesfer n yisalan.
# **Tasertit**:
#* Tagrumma tettwasexdem deg tsertit tanefrant d tsertit n uswizi.
# **Tasnulfut**:
#* Tagrumma tettwasexdem deg tasnulfut tanefrant d usnulfu n yinagrawen imaynuten.

### Taggrayt
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.