MTE3101: MENGENAL NOMBOR - Rekreasi Nombor

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    0/8 Fqpheho ^ao~hog Paipahze Olk`lp

    0/8/8 Ra~hi Hmhe` &Khgej Zxqhpa%

    Ra~hi hmhe` ehdhf ~apnepe nhperhnh zage akrh~ zhkh uhog `apra~hi+ra~hi ne

    khoh ~qpq~ho olk`lp ne nhdhkouh nezqzqo zafeoggh fhzed ~hk`hf olk`lp+olk`lp

    ~apza`q~ kaogeiq~ `hpez$ dhmqp$ nho raraomqpqouh hnhdhf zhkh &Kli Zllo Zhog$

    8557%/

    : v : zad > v > zad 1 v 1 zad

    > 5 0

    : 1 7

    6 8 3

    &Fhzed ~hk`hf olk`lp+olk`lp rhnh dhmqp$ `hpez nho raraomqpq hnhdhf zhkh/%

    87 0> 8 6 81

    0: 1 7 8> 83

    > 3 8: 02 00

    82 80 85 08 :

    88 86 01 0 5

    83 : 0 8:

    1 82 88 6

    5 3 7 80

    > 81 8> 8

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    ZAMHPHF RA^HI HMHE@

    Hfde kh~akh~ei Jeoh ~adhf rqo kakrapiaohdiho ra~hi hfmhe` eoe zakaomhi

    312 ~hfqo za`adqk Khzefe/ Kaoqpq~ dhgaonh$ ~apnhrh~ouh zaohzihf `qiq Jeoh

    uhog neohkhiho Dl Zfq &@qiq Zqoghe Dl% kaojape~hiho iezhf zalphog Khfhphmh

    Jeoh ~adhf ~apmqkrh naogho zaailp iqph+iqph uhog kakrqouhe jlphi uhog radei

    rhnh jhogiaphogouh/ Raoakqho ~apza`q~ ~adhf kaouadhkh~iho ihwhzho

    zaie~hpouh nhperhnh nekqzohfiho ldaf `homep rhnh ia~eih e~q iaphoh phiuh~ Jeoh

    ia~eih e~q ~adhf kaoggqohiho jlphi uhog ne~akqe e~q `hge kaogh~hze khzhdhf

    `homep/

    Hfde kh~akh~ei Ezdhk rqdh ~adhf nerapiaohdiho kaogaohe ra~hi hmhe` eoe nhpe

    Eoneh nho za~apqzouh$ kapaih ~adhf `apqrhuh kaogfhzediho ra~hi hmhe` uhog da`ef

    `azhp nho ilkrdaiz zqzqohoouh/ Hfde hz~plolke nhperhnh Hph` rqdh ~adhf

    kaoggqohiho rldh rhnh ra~hi hmhe` eoe `hge kadqiez flplzilr rhnh khzh e~q/

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    RAPH^QPHO KAK@EOH RA^HI HMHE@

    Ia`ehzhhoouh$ ~qpq~ho olk`lp 8$0$:$>$1/ o$ negqohiho za`hghe zqzqoho

    olk`lp nhdhk ra~hi+ra~hi zageakrh~ zhkh/ Ldaf e~q$ zageakrh~ zhkh hmhe` ~qpq~

    neza`q~ za`hghe zageakrh~ `aprapeogih~ o/ Fhzed ~hk`hf `hge olk`lp+olk`lp ne

    zarhomhog `hpez$ dhmqp$ h~hq raraomqpqouh hnhdhf `apzhkhho naogho pqkqz

    /

    Mhne$ `hge zad : v :$o

    ; :/

    Fhzed ~hk`hf zarhomhog `hpez$ dhmqp nho raraomqpq$

    ; 81

    Dhogihf+dhogihf `hge kak`eoh ra~hi hmhe` naogho zad : v :=+

    Dhogihf 8=Zqzqoiho ~qpq~ho olk`lp 8 feogg 5 nhdhk ra~hi+ra~hi zarap~e `apeiq~=+

    8 0 :

    > 1 3

    7 6 5

    Dhogihf 0=^qihpiho ianqnqiho olk`lp rhnh raraomqpq/

    5 0 7

    > 1 3

    : 6 8

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    Dhogihf := Rq~hpiho zageakrh~ zhkh e~q za`houhi >1 nho pl`lfihoouh

    zafeoggh kak`ao~qi zh~q ra~hi hmhe` naogho fhzed ~hk`hf 81/

    Dhogihf+dhogihf `hge kak`eoh ra~hi hmhe` naogho zad > v > =+

    Dhogihf 8= Zqzqoiho ~qpq~ho olk`lp 8 feoggh 83 nhdhk ra~hi+ra~hi zarap~e

    `apeiq~=+

    8 0 : >

    1 3 7 6

    5 82 88 80

    8: 8> 81 83

    > 5 0

    : 1 7

    6 8 3

    > 5 0

    : 1 7

    6 8 3

    5

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    Dhogihf 0=^qihpiho ianqnqiho olk`lp+olk`lp uhog `aphnh rhnh raraomqpq/

    83 0 : 8:

    1 3 7 6

    5 82 88 80

    > 8> 81 8

    Dhogihf := ^qihpiho mqgh raraomqpq iarhnh zageakrh~ uhog da`ef iajed rhnh

    rqzh~ il~hi ~apza`q~/

    83 0 : 8:

    1 3 7 6

    5 82 88 80

    > 8> 81 8

    HRDEIHZE RA^HI HMHE@

    Kq}ei

    Ra~hi hmhe` negqohiho nhdhk kaoao~qiho pekh kq}ei/ ^apnhrh~ouh

    `a`aphrh hfde kq}ei Eoneh uhog kaoggqohiho ra~hi hmhe` nhdhk kq}ei kapaih/

    Kapaih kaoghe~iho fq`qogho ho~hph olk`lp rqzh~ naogho fhzed ~hk`hf

    iazakqh olk`lp+olk`lp nhdhk ra~hi hmhe` nhdhk kaoao~qiho pekh kq}ei=+

    83 0 : 8:

    1 88 82 6

    5 7 3 80

    > 8> 81 8

    Fhzed ~hk`hf olk`lp nhdhk ra~hi hmhe` ; olk`lp rqzh~ v 5

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    Zqnliq

    Zqnliq ~adhf nerapiaohdiho rhnh 8575 nho kaomhne rlrqdhp ne Marqo rhnh

    8562+ho/ Zqnliq uhog za~apqzouh kaomhne zh~q rapkheoho ~aih+~aie uhog rlrqdhp

    ne Aplrhf za`aohpouh ehdhf zhdhf zh~q `ao~qi iarhnh ra~hi Dh~eo/ Ra~hi Zqnliq

    ~apnepe nhperhnh zad 5 v 5 uhog `azhp$ nho ne nhdhkouh kaoghonqoge 5 `qhf zq`+

    il~hi iajed uhog ~apnepe nhperhnh zad : v :/ Za~ehr zq`+il~hi hiho neraoqfe naogho

    olk`lp 8 feoggh 5 zaihde zhfhm/ Ldaf e~q$ za~ehr `hpez$ dhmqp nho zq`+il~hi hiho

    kaogfhzediho mqkdhf uhog zhkh zaiephouh ne~hk`hf olk`lp+olk`lpouh/

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    0/8/0 Oez`hf Akhz &Gldnao Ph~el%

    Kh~akh~ei zapeog ihde neihe~iho naogho iaeonhfho/ Zhdhf zh~q jlo~lf uhog

    ~apnaih~ zadheo nhperhnh iaeonhfho hdhk zakqdh mhne ehdhf iaeonhfho h~hq rqo

    iajho~eiho uhog `ldaf ie~h defh~ rhnh ce}eihd khoqzeh/ Fhieih~ouh$ nhuh ~hpeiho

    rhnh zech~+zech~ ce}eihd zazqh~q e~q `apgho~qog iarhnh oez`hf/ Khoqzeh hiho da`ef

    ~ap~hpei iarhnh jepe ce}eihd eonesenq dheo zaiephouh ce}eihd eonesenq ~apza`q~ kakrqouhe

    zeka~pe nho oez`hf uhog zaek`hog/ @age~q mqgh rhnh phq~ whmhf zazalphog/

    Eonesenq uhog kakrqouhe phq~ whmhf uhog zaek`hog nho `ap~arh~ho naogho

    oez`hf hiho kqnhf kaohpei rapfh~eho lphog ne zaiadedeogouh/ Dhdq zajhph ~enhi

    zanhp$ kapaih hiho `apraonhrh~ `hfhwh eonesenq ~apza`q~ kakrqouhe rhphz pqrh

    uhog made~h/

    Hfde zheo~ez kakrapjhuhe `hfhwh naogho kakrqouhe ~q`qf `hnho uhog

    zaek`hog qiqphoouh &uhioe$ zamhmhp naogho oez`hf%$ zazalphog e~q kakedeie ~q`qf

    `hnho uhog zefh~/

    "Se~pqseho Kho" uhog nedqiez ldaf Dalohpnh nh Seoje kaoqomqiiho oez`hf akhzuhog hnh nhdhk nekaoze ~q`qf `hnho khoqzeh/

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    Dqiezho Dalohpnl nh Seoje kaoaihoiho rhnh qiqpho zihdh uhog ~apnhrh~

    rhnh ~q`qf `hnho khoqzeh/ Zihdh h~hq rqo oez`hf uhog neouh~hiho ne `hwhf

    kapqrhiho zh~q Oez`hf Akhz&Gldnao Ph~el%=

    Ihie ia rqzh~ = Rqzh~ ia Iarhdh

    @age~q mqgh `hge zaoe `eoh zazqh~q `hogqoho/ Ie~h hiho kanhrh~e zazqh~q

    `hogqoho ~apza`q~ da`ef kaohpei zaiephouh zihdh uhog negqohiho nhdhk

    paihhoouh hnhdhf `ap~arh~ho naogho Oez`hf Akhz/

    Oez`hf Akhz hnhdhf `apnhzhpiho iarhnh olk`lp Ce`lohjje$ ne khoh

    za~ehr ~qpq~ho olk`lp hnhdhf kapqrhiho fhzed ~hk`hf `hge nqh olk`lp

    za`adqkouh/

    8$ 8$ 0$ :$ 1$ 6$ 8:$ 08$ ///

    @hghekhohihf olk`lp Ce`lohjje `ldaf kak`ao~qi Oez`hf Akhz,08 ; 8/38521///

    11,:> ; 8/3873>///

    65,11 ; 8/38638///

    Meih ne~apqziho$ ie~h hiho kakrapldaf zh~q olk`lp uhog kaohpei uhog

    neiaohde za`hghe rfe nhdhk ihdhogho hfde kh~akh~ei&Gldnao Ph~el h~hq Gldnao

    Zaj~elo%=

    ; 8/386 2:: 566 7///

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    IAEONHFHO NHO IAJHO^EIHO CE]EIHD

    @hghekhohihf ie~h `ldaf kaogh~hiho uhog zazalphog e~q made~h< Ie~h hk`ed

    Mazzejh Zekrzlo za`hghe jlo~lf/ Phkhe uhog kaogh~hiho `hfhwh Mazzejh

    Zekrzlo kakrqouhe rhphz pqrh uhog jho~ei nho enahd/ Kaoghrh

    Zakaomhi nhpe Paohezzhoja zafeogghdhf iarhnh }hkho klnao$ radqiez nho hfde

    zaoe ~adhf kaogei~ephc rfe za`hghe zh~q qiqpho uhog `ldaf kak`apeiho ~hpeiho

    az~a~eih iarhnh ek`hogho nho fhpkloe zazqh~q paih `ao~qi/

    Meih ne~ade~e$ za~ehr nekaoze `hge za~ehr fqpqc ne h~hz hnhdhf nepaih `apnhzhpiho

    oez`hf rfe &oez`hf akhz%

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    0/8/: ZAGE ^EGH RHZJHD

    Zhdhf zh~q Rldh Olk`lp uhog rhdeog kaohpei hnhdhf Zage~egh Rhzjhd/ Zage ~egh

    Rhzjhd kapqrhiho za`qhf zage ~egh olk`lp h~hq zage ~egh hpe~ka~ei uhog ~apnepe nhperhnh

    raihde `elolkehd/ Zage ~egh Rhzjhd neohkhiho zakraoh @dheza Rhzjhd &830:+8330%

    zalphog hfde kh~akh~ei Raphojez/ Whdhqrqo neohkhiho za`hghe zage ~egh Rhzjhd hiho

    ~a~hre ilozar eoe ~adhf rqo negqohiho `ap~hfqo dhkhouh ldaf hfde kh~akh~ei ~apnhfqdq

    za`adqk Rhzjhd nedhfepiho dhge/

    Qo~qi kak`eoh zage~egh$ nekqdhiho naogho "8" ne `hfhgeho h~hz$ iakqneho

    ~apqziho kada~hiiho olk`lp ne `hwhf eoe nhdhk rldh zage~egh/ @apnhzhpiho ghk`hp nhrh~nedefh~ olk`lp 8 `aphnh rhnh ianqnqiho uhog kak`ao~qi za`qhf zage ~egh/ Khohihdh

    olk`lp uhog dheo kapqrhiho fhzed ~hk`hf nqh olk`lp ne h~hzouh/ Jlo~lfouh 8 ) 8 ; 0/

    Za~ehr olk`lp fhouh nqh hogih ne h~hzouh ne~hk`hf `apzhkh+zhkh/

    Whdhq`hghekhohrqo$ olk`lp ne ra~hi ~are iazakqhouh hnhdhf 8/ Za~ehr olk`lp fhouh

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    nqh hogih ne h~hzouh uhog ne~hk`hf `apzhkh+zhkh qo~qi kaonhrh~iho olk`lp ne`hwhf/

    Kadhdqe zage ~egh Rhzjhd eoe ie~h `ldaf kaogaohdrhz~e `a`aphrh jepe+jepe qoei uhog ~apnhrh~

    rhnh zage~egh eoe/

    Za~ehr olk`lp ne ~are ehdhf 8

    Ghk`hp kaoqomqiiho `hfhwh 8 ) : ; >

    http://qran.wordpress.com/PascalTriangleAnimated2.gifhttp://qran.wordpress.com/PascalTriangleAnimated2.gif
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    Jlphi ne nhdhk Zage~egh Rhzjhd

    h% Nehglohd

    Nehglohd rap~hkh

    Nehglohd rap~hkh ehe~q ra~hi kaouaplog `epq kqnh fhouh kakrqouhe hogih "8" /

    Nehglohd Ianqh

    Nehglohd kaouaplog ianqh uhog `apwhpoh jlidh~ kakrqouhe Hogih @ap~apqzho

    &Jlqo~eog Oqk`apz% ehe~q 8$ 0$ :$ nho za~apqzouh/

    Nehglohd ia~egh

    Rhnh nehglohd ia~egh uhog `apwhpoh `epq gadhr$ nhrh~ nedefh~ `hfhwh zage~egh eoe

    kakrqouhe jepe+jepe olk`lp zage~egh/ Mqkdhf il~hi za~ehr ~eogih~ uhog ne~hk`hf hiho

    kaogfhzediho olk`lp zage~egh/

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    Nehglohd ia~egh uhog kaogfhzediho olk`lp zage~egh/

    `% Ghomed nho Gaohr

    Meih nedlpaiiho olk`lp ghomed nho olk`lp gaohr$ ehouh hiho `aphifep naogho rldh uhog

    zhkh zarap~e Zage~egh Zeapreozie/

    Zage~egh Rhzjhd uhog kakrqouhe rldh uhog zhkh zarap~e Zage~egh Zeapreozie

    8

    8)0;:

    8)0):;3

    8)0):)>;82

    8)0):)>)1;81

    8)0):)>)1)3;08

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    j% Mqkdhf Olk`lp Kadeo~hog

    Meih nerapfh~eiho$ mqkdhf olk`lp kadeo~hog h~hq flpe}lo~hd kak`ao~qi zh~q rldh uhog

    kaohpei/

    Fhzed ~hk`hf olk`lp za~ehr flpe}lo~hd hnhdhf olk`lp iqhzh nqh/

    n% ^qpq~ho Ce`lohjje

    Hrh`edh ne~hk`hf olk`lp kaogeiq~ ~qpq~ho ihoho h~hz &zarap~e uhog needqz~phzeiho%

    olk`lp ~qpq~ho Ce`lohjje hiho nerapldafe/

    &^qpq~ho Ce`lohjje `apkqdh "8$ 8" nho iakqneho `ap~apqzho naogho kaohk`hf nqh

    olk`lp za`adqkouh$ kezhdouh : )1 ; 6$ khih 1 )6 ; 8:$ nho za~apqzouh%/

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    a% Zeka~pez

    Nho zage~egh Rhzjhd mqgh zeka~pez/ Hogih+hogih ne zeze iepe hnh olk`lp uhog `aprhnhoho

    zhkh ne zeze ihoho$ zarap~e ghk`hp japkeo/

    Zajhph peogihzouh$ olk`lp Rhzjhd kakrqouhe `houhi iaez~ekawhho nho jepe+jepe

    uhog qoei nho kaohpei/ Zage ~egh Rhzjhd `ehzhouh negqohiho nhdhk hdga`ph nho

    ia`hphogihdeho/ Kadhdqe zage ~egh Rhzjhd eoe mqgh ie~h nhrh~ kadefh~ `hfhwh kh~akh~ei e~q

    zazqh~q uhog qoei uhog rapdq ne~aplihe nho nechfhke/ Zage~egh Rhzjhd kapqrhiho zhdhf

    zh~q ho~hph iahmhe`ho olk`lp ne nqoeh/ Phmhf ne`hwhf kapqrhiho peogihzho uhog

    kaoqomqiiho ho~hph `a`aphrh jepe+jepe uhog ~apnhrh~ rhnh Zage~egh Rhzjhd/

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    :0

    0/8/> Rhdeonplk

    Rhdeonplka ehdhf ih~h h~hq cphzh uhog ne`hjh zhkh `hei ne narho khfqrqo

    `adhihog/ Rhdeonplk rap~hkh ne~akqe nhdhk `ao~qi ~qdezho rhnh ra~hi `h~q uhog

    ne~hohk ne Fapjqdhoaqk/ Fhzed raoakqho e~q nehogghpiho ne~hohk rhnh ~hfqo 70

    HN/ Rapih~ho uhog ne~qdez ne ra~hi `h~q e~q ehdhf Zh~lp Hparl ^aoa~ Lraph Pl~hz/

    Ehouh `ap`ao~qi zage akrh~ &pqmqi VVVV% nho ehouh `ldaf ne`hjh nhpe hphf ~aghi

    khfqrqo kadeo~hog ~a~hre ~a~hr kaogfhzediho zh~q rapih~hho uhog zhkh/

    Nhpe zqnq~ Kh~akh~ei rqdh$ rhdeonplk `apkhizqn mqkdhf olk`lp uhog

    ne`hjh zhkh ne narho nho ne`adhihog/

    L`mai~ec

    e% Qo~qi kaogqiqp ia`ldafho eonesenq nhdhk kaouadazheiho khzhdhf

    rhdeonplka/

    ee% Kaogqme keonh eonesenq nhdhk kaouadazheiho lraphze kh~akh~ei uhog hzhz

    eee% Kadh~ef eonesenq qo~qi kaogeph olk`lp naogho ~arh~

    Jhph+jhph kak`qh~ rhdeonplka=

    8% Redef zh~q olk`lp$ jlo~lf ; 63

    0% Olk`lp nhpe jlo~lf ne h~hz$63 ne~ap`hdeiiho kaomhne 36

    :% Ianqh+nqh olk`lp iakqnehoouh ne~hk`hf 63 ) 36 ; 81>

    >% Mqkdhf 81> e~q ne~ap`hdeiiho ~a~hre olk`lp uhog ~aoghf neiaihdiho/

    1% Khih$ mqkdhf 81> ne~ap`hdeiiho kaomhne >18 nho ne~hk`hf/

    3% Fhzed uhog nerapldafe ehdhf 321 nho mqkdhf eoe ne~ap`hdeiiho zaihde dhge

    kaomhne 123/7% Fhzed 321 ) 123 ; 8888

    6% Fhzed uhog nerapldafe ehdhf olk`lp uhog zhkh ne`hjh ne narho nho ne

    `adhihog/

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