Aggur:Tusnakt

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Uṭṭun n imnnitn g waggur ad : 46
Gr tiwallin nk f taggayyin n waggur ad

50

Amgrad n ɣassa

Alugaritm agaman nɣ Alugaritm anipiri (s tanglizt : Natural logarithm) tga yat tasɣnt nɣ twwuri tettuysan bahra ɣ tusnakt, ar tsnfal afaris s timrnit zund akkʷ tisɣnin ilugaritm. Ar tt ntara s ɣmk-ad : $\ln()$ . Ar nttini mas-d alugaritm agaman nɣ anipiri iga s uzadur n $e$ acku $ln(e)=1$ , iga alugaritm agaman ula tamnzut n tasɣnt : $x\longmapsto 1/x$ ɣ uzilal $]0;+\infty [$ .

Alugaritm ad dars yan yism yaḍni abla "agaman", iga-tt "anipiri". Ism ad yuckad zɣ John Napir nna igan yan umusnak askatlandi, nttat ad akkʷ izwarn ay-skr taflwit talugaritm ( ur trwas xtad lli nssn ɣila nkkni ).

Tettyawskar tazmilt n ulugaritm nna f nsawal nkkni ɣila s ifassn n Grégoire de Saint-Vincent d Alphonse Antonio de Sarasa ɣ usggʷas n 1649

Ad tg $a\in ]0;+\infty [$ , nzḍar ad nini mas-d asnml n tasɣnt talugaritm tga-tt unrar lli illan ɣ izddar n tasɣnt $x\mapsto 1/x$ gr $a$ d 1.

Ɣikad ast nttara :

\ln a=\int _{1}^{a}{\frac {1}{x}}\,dx.

... (Ɣr uggar...)

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Tawlaft n ɣassa

Asktiɣmr tga yan ugmiw ɣ tusnakt nna itxdamn s izdayn gr izarutn d tiɣmriwin ɣ ammas n imkṛaḍn d tisɣan tisktuɣmirin zund ukan sinus, kusinus d tanjunt.

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Tamyagart n ɣassa

Ɣ tusnakt, Targadda nɣ tamyagart n Cauchy–Schwarz, tettawsan ula s yism n targadda n Cauchy–Bunyakovsky–Schwarz, tga yat targadda bahra istawhman ɣ tusnakt kullut, ar sis nswurri ɣ aljibr imzirg, taslṭ, tiẓri n tsqqart d kigan n tɣawsiwin yaḍni. Targadda ad issufɣt-id yan umusnak afransis iga s yism Augustin-Louis Cauchy ɣ usggwas n 1821, sliɣ yufa Viktor Bunyakovsky yat targadda trwast akk ɣ mad izdin d aɣrd ɣ usggwas n 1859, mn bɛd yufatt daɣ yan umusnak almani Hermann Amandus Schwarz ɣ usggwas n 1888.

Mad aɣ tettini

Ar ttini targadda n Cauchy–Schwarz mas i akk sin imawayn ${\textstyle u}$ d ${\textstyle v}$ n yat tallunt gis afaris agnsan hann rad darnɣ tili

$|\langle \mathbf {u} ,\mathbf {v} \rangle |^{2}\leq \langle \mathbf {u} ,\mathbf {u} \rangle \cdot \langle \mathbf {v} ,\mathbf {v} \rangle$

maɣ $\langle \cdot ,\cdot \rangle$ iga yan ufaris agnsan. S umdya afaris agnsan gis afaris afsnan n ilawn d ismlaln. S ɣmkad, iɣ nusi aẓur uzmir-sin ɣ tsgiwin s snat, d iɣ asn nga alugn. Ar nttafa targadda ad :

|\langle \mathbf {u} ,\mathbf {v} \rangle |\leq \|\mathbf {u} \|\|\mathbf {v} \|.

D yat tɣawsa yaḍn, tasgiwin s snat gaddan iɣ ${\textstyle \mathbf {u} }$ d ${\textstyle \mathbf {v} }$ gan ilelliyn imzirgn (yaɛni gan imsadaɣn). Iɣ ${\textstyle u_{1},\ldots ,u_{n}\in \mathbb {C} }$ d ${\textstyle v_{1},\ldots ,v_{n}\in \mathbb {C} }$ d ufaris agnsan iga afaris agnsan asmlal anaway, hann targadda tzḍar ad ttyura zund ɣikad (maɣ tirra n taɛṛṛaḍt ar sis nmmal unaftay n ismlaln): i $\mathbf {u} ,\mathbf {v} \in \mathbb {C} ^{n}$ , darnɣ

$\left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|^{2}=\left|\sum _{k=1}^{n}u_{k}{\bar {v}}_{k}\right|^{2}\leq \langle \mathbf {u} ,\mathbf {u} \rangle \langle \mathbf {v} ,\mathbf {v} \rangle$

$=\left(\sum _{k=1}^{n}u_{k}{\bar {u}}_{k}\right)\left(\sum _{k=1}^{n}v_{k}{\bar {v}}_{k}\right)$

$=\sum _{j=1}^{n}|u_{j}|^{2}\sum _{k=1}^{n}|v_{k}|^{2}$

Ad t igan,

$|u_{1}{\bar {v}}_{1}+\cdots +u_{n}{\bar {v}}_{n}|^{2}\leq (|u_{1}|^{2}+\cdots +|u_{n}|^{2})(|v_{1}|^{2}+\cdots +|v_{n}|^{2})$

(Ɣr uggar...)

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Imgrad mqqurnin

Amḍan amnzu

Amḍan amnzu iga yan umḍan ummid agaman ilan ɣar sin inbḍayn imyallan igan ummidn gn imufrarn (ad tn igan 1 d nttan nit). S ɣmka, 1 ur igi amnzu acku ur dars abla yan unbḍay ummid amufrar; ula 0 ur t igi acku issn ad ittubḍa f imḍanen ummidn imufrarn s timmad nnsn.

Ɣ unmgal, amḍan ummid arilm igan afaris n sin imḍanen ummidn igamanen nttini as uddis. S umdya 6 d 12 d uddisn acku 6 = 2 × 3 d 12 = 3 × 4 nɣ 2 × 6, mac 11 iga amnzu acku 1 d 11 ka igan inbḍayn nns.

Imḍanen 0 d 1 ur gin imnza wala gan uddisn. kra n imusnaktn ssiḍinen zikk-lli (ar tasut tiss 19) 1 d amḍan amnzu, mac ɣ tizwuri n tasut tiss 20, issinf yan umsasa 1 zɣ imḍanen imnza

25 n imḍanen imnza imẓẓiyn f 100 :

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 d 91. (Ɣr uggar...)

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Lqist n ɣassa

Ɣ taskriɣmrt, Asaḍuf n id kusinus (ittawsan ula s yism n tanfalit n kusinus nɣ askkud n Al Kaci) tga yan usaḍuf nna ismunn tiɣziwin n tasgiwin n kra n umkṛaḍ d kusinus n yan zɣ tiɣmriwin. S uswuri n tirra zund tin Fig. 1, asaḍuf n iwankn n id kusinus

$c^{2}=a^{2}+b^{2}-2ab\cos \gamma ,$

maɣ $\gamma$ tga tiɣmrt lli illan gr tiɣziwin $a$ d $b$ d tasga tamuzzlt n $c$ . I tawlaft ann nit rad darnɣ ilin ula:

$a^{2}=b^{2}+c^{2}-2bc\cos \alpha ,$

$b^{2}=a^{2}+c^{2}-2ac\cos \beta .$

Asaḍuf n id kusinus ar tskar asmata i askkud n Pythagor, nna ur illan dar imkṛaḍn unziɣn: iɣ tga $\gamma$ tiɣmrt taɣadant (nna dar illa 90 daraja, nɣ ${\textstyle {\pi \over 2}}$ s raḍyan), hann ${\textstyle cos(\gamma )=0}$ , s ɣmk ad as iskar usaḍuf n id kusinus asmata i askkud n Pythagor:

$c^{2}=a^{2}+b^{2}.$

Ar nswurri s usaḍuf n id kusinus n id kusinus iɣ nra ad nḥasb tiɣmrt tiss kraḍt n kra n umkṛaḍ iɣ yad tawssan darnɣ snat tasgiwin d tiɣmrt nnsn iqqnn, d ula ɣ lḥsab n tiɣmriwin n kra n umkṛaḍ iɣ yad ttawssan darnɣ tasgiwin nns s kraḍ-itsnt. (Ɣr uggar...)

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