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7/23/2019 MTE3101: MENGENAL NOMBOR - Rekreasi Nombor
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82
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
7/23/2019 MTE3101: MENGENAL NOMBOR - Rekreasi Nombor
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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/
7/23/2019 MTE3101: MENGENAL NOMBOR - Rekreasi Nombor
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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
7/23/2019 MTE3101: MENGENAL NOMBOR - Rekreasi Nombor
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8:
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
0
7
3
1
>
:
6
8
5
0
7
3
1
>
:
6
8
7/23/2019 MTE3101: MENGENAL NOMBOR - Rekreasi Nombor
5/24
8>
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
7/23/2019 MTE3101: MENGENAL NOMBOR - Rekreasi Nombor
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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/
7/23/2019 MTE3101: MENGENAL NOMBOR - Rekreasi Nombor
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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///
7/23/2019 MTE3101: MENGENAL NOMBOR - Rekreasi Nombor
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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%
7/23/2019 MTE3101: MENGENAL NOMBOR - Rekreasi Nombor
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01
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
7/23/2019 MTE3101: MENGENAL NOMBOR - Rekreasi Nombor
17/24
03
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.gif7/23/2019 MTE3101: MENGENAL NOMBOR - Rekreasi Nombor
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07
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/
7/23/2019 MTE3101: MENGENAL NOMBOR - Rekreasi Nombor
19/24
06
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
7/23/2019 MTE3101: MENGENAL NOMBOR - Rekreasi Nombor
20/24
05
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%/
7/23/2019 MTE3101: MENGENAL NOMBOR - Rekreasi Nombor
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:2
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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0/8/> 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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