Num`t`id5 ]hg Nrt nim [k`gikg jc Nssurnikg

Gidndgfgits
Cjurtggith Kninm`ni Gm`t`ji

Khnptgr 7
Num`t Gv`mgikg

Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7->

Agnri`id Jbegkt`vgs (> jc 2)
>. Wimgrstnim thg purpjsg nim typgs jc num`t gv`mgikg
usgm thrjudhjut thg num`t prjkgss.
2. Gxpan`i thg thrgg num`t gv`mgikg mgk`s`jis tj mgtgrf`ig
whnt gv`mgikg tj dnthgr nim hjw fukh tj nkkufuantg.
;. Wsg prjcgss`jina eumdfgit
eumdfgit tj mgtgrf`ig thg mgdrgg jc
pgrsuns`vgigss jc num`t gv`mgikg.

=. Mgskr`bg thg typgs jc num`t prjkgmurgs cjr dnthgr`id num`t

gv`mgikg.
7. Mgs`di ninayt`kna prjkgmurgs cjr thg vnr`jus phnsgs jc
thg num`t prjkgss.

Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7-2

Agnri`id Jbegkt`vgs (2 jc 2)
3. Wimgrstnim thg purpjsgs jc num`t mjkufgitnt`ji.
4. _rgpnrg jrdni`zgm num`t mjkufgitnt`
mjkufgitnt`ji.
ji.

8. Gxpan`i hjw tgkhijajdy nccgkts num`t gv`mgikg nim num`t

mjkufgitnt`ji.
0. Wimgrstnim nim nppay prjcgss`jina slgpt`k`sf tj
gv`mgikg nim mjkufgitnt`ji mgk`s`jis.

Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7-;

Inturg jc Num`t Gv`mgikg (> jc 2)
• Fjst jc thg num`tjr‐s
num`tjr ‐s wjrl `i cjrf`id ni num`t jp`i`ji
kjis`sts jc jbtn`i`id nim gvnaunt`id num`t gv`mgikg.
• KN[ 7?? gxpan`is thnt num`t gv`mgikg `s `icjrfnt`ji usgm
by thg num`tjr `i nrr`v`id nt kjikaus`jis upji wh`kh thg
num`t jp`i`ji `s bnsgm.
• Gv`mgikg `ikaumgs `icjrfnt`ji dgigrntgm by thg num`tjr,

th`rm pnrt`gs, nim thg ka`git.

Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7-=

Inturg jc Num`t Gv`mgikg (2 jc 2)
• Gv`mgikg `s bjth `icjrfnt`ji thnt suppjrts nim
kjrrjbjrntgs fnindgfgit‐s nssgrt`jis, nim niy
`icjrfnt`ji thnt kjitrnm`kts thjsg nssgrt`jis. @i sjfg

knsgs, `t kni
fnindgfgit
fnindgfg itbg thg nbsgikg
rgcusgs jc `icjrfnt`ji
tj prjv`mg n rgqugstgm(cjr gxnfpag,
mjkufgit).
• Ns `aaustrntgm
`aaustrntgm `i C`durg
C`durg 7->, gv`mgikg
gv`mgikg `s dnthg
dnthgrgm
rgm `i naa
phnsgs jc thg num`t prjkgss nim `s usgm tj hgap thg num`tjr
fnlg n vnr`gty jc eumdfgits nim kjikaus`jis”tj nkkgpt
(jr kjit`iug w`th) thg ka`git, tj pani thg num`t, tj mgk`mg
whgrg thgrg kjuam bg n r`sl jc fntgr`na f`sstntgfgit `i thg
c`inik`na stntgfgits, tj mgvgajp ni npprjpr`ntg r`sl
rgspjisg, nim tj kjikaumg ns tj thg typg jc num`t rgpjrt.
Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7-7
Num`t Gv`mgikg Mgk`s`jis
• IN]WRG5 Th`kh num`t prjkgmurgs tj usg:
• GX]GI]5 Th`kh `tgfs tj sgagkt cjr tgst`id:

• ]@F@ID5 Thgi tj pgrcjrf thg prjkgmurgs:

Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7-3

Num`t _rjkgmurgs (> jc 2)
• Thnt `s ni num`t prjkgmurg:
prjkgmurg:
‛ @t `s thg mgtn`agm `istrukt`jis cjr thg kjaagkt`ji jc num`t gv`mgikg”
thnt `s, thg nktuna tnsl thnt yju mj tj jbtn`i gv`mgikg w`th rgspgkt
tj ni num`t jbegkt`vg5 cjr gxnfpag, whgi yju kjuit `ivgitjry,
`ivgitjry, yju
ijtg thg qunit`ty, ajknt`ji, tnd iufbgr, ui`t jc fgnsurg nim
kjim`t`ji.

Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7-4

Num`t _rjkgmurgs (2 jc 2)
• Num`t prjkgmurgs
prjkgmurgs cnaa `itj
`itj thrgg kntgdjr`gs5
kntgdjr`gs5
‛ R`sl nssgssfgit prjkgmurgs”pgrcjrfgm
prjkgmurgs”pgrcjrfgm by thg num`tjr tj
jbtn`i `icjrfnt`ji cjr `mgit`cy`id nim nssgss`id r`sls jc fntgr`na
f`sstntgfgit `i thg c`inik`na stntgfgits mug tj crnum jjrr grrjr
grrjr..
‛ ]gsts jc kjitrjas”mgs`digm
kjitrjas”mgs`digm tj gvnauntg thg jpgrnt`id
gccgkt`vgigss jc kjitrjas nt thg nssgrt`ji agvga.
‛ [ubstnit`vg prjkgmurgs”mgs`digm
prjkgmurgs”mgs`digm tj mgtgkt fntgr`na
f`sstntgfgits `i nkkjuits wh`kh `ikaumg tgsts jc mgtn`as nim
ninayt`kna prjkgmurgs.

Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7-8

 R`sl Nssgssfgit
Num`t _rjkgmurgs5 R`sl
_rjkgmurgs
• Cjr gnkh fnejr kanss jc trnisnkt`jis nim fntgr`na dgigrna
agmdgr nkkjuit, thg num`tjr ajjls nt thg pjtgit`na cjr thg
r`sl jc fntgr`na f`sstntgfgit
f`sstntgfgit..

• Lijwagmdg jc thg ka`git‐s `imustry, bus`igss, nim jpgrnt`id

prjkgmurgs w`aa hgap thg num`tjr kjfpagtg th`s nssgssfgit.

Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7-0

 Thy, Thnt, Hjw
]hg Thy, Hjw,, nim Thgi jc Num`t
_rjkgmurgs5 Th`kh @tgfs tj [gagkt
• Jikg ni num`t prjkgmurg `s mgtgrf`igm, thg num`tjr fny5
(>) sgagkt naa `tgfs `i thg pjpuant`ji,
(2) sgagkt spgk`c`k `tgfs `i thg pjpuant`ji,

(;) usg num`t snfpa`id, jr

(=) usg sjfg kjfb`int`ji jc c`rst thrgg.

• ]hg khj`kg mgpgims upji thg purpjsg jc thg tgst nim thg
pnrt`kuanrr k`rkufstnikgs, sukh ns thg gxtgit jc nutjfntgm
pnrt`kuan
kjitrjas nim thg r`sl jc fntgr`na f`sstntgfgit, ns wgaa ns
thg prnkt`kna`ty nim/jr gcc`k`giky jc thg prjkgmurg.

Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - >?

 Thy, Thnt, Hjw
]hg Thy, Hjw,, nim Thgi jc Num`t
_rjkgmurgs5
_rjkgmurgs5 ]`f`id
• Ni num`t jc
jc c`inik`na stntgfgits us
usunaay
unaay kjvgrs
kjvgrs n pgr`jm
sukh ns n ygnr, nim ni num`t `s jctgi ijt kjfpagtgm uit`a
sgvgrna wggls jr fjiths nctgr thg gim jc thg c`skna pgr`jm.

• ]hg t`f`id jc num`t prjkgmurgs kni vnry crjf gnray `i thg

nkkjuit`id pgr`jm tj ajid nctgr `t hns gimgm.

Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - >>

Num`t _rjdrnf
• ]hg sgt jc mgtn`agm `istrukt`jis cjr thg git`rg kjaagkt`ji jc
gv`mgikg cjr ni num`t nrgn `s knaagm ni num`t prjdrnf.
prjdrnf.
• ]hgrg fny bg sgvgrna num`t prjdrnfs cjr n kykag,
mgpgim`id upji thg iufbgr jc stncc nss`digm tj ni
gidndgfgit.
• Ijrfnaay, thgrg `s ni num`t prjdrnf cjr gnkh kjfpjigit jc

thg num`t
jc ni (g.d.,
num`t nkkjuits
prjdrnf rgkg`vnbag
thnt `ikaumgs nimprjkgmurgs
num`t snags).urgs
prjkgm Ni gxkgrpt
(thg
hjw), snfpag s`zg, `tgfs tj sgagkt (thg whnt), t`f`id (thg
whgi) nim thg rgagvnit nssgrt`ji (thg why) `s d`vgi `i
7-;.
]nbag 7-;.

Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - >2

Npprjpr`ntgigss5 [nfpag [`zg
• ]hg snfpag s`zg w`aa vnry mgpgim`id upji5
‛ ]hg snfpa`id fgthjm khjsgi<
‛ ]hg typg jc tgst (g.d., tgst jc kjitrjas vs. tgst jc mgtn`as)< nim

‛ ]hg nfjuit jc nssurnikg tj bg jbtn`igm crjf thg tgst.

• ]hg snfpa`id fgthjm nim s`zg `s rgantgm tj nssgssgm
r`sls.

Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - >;

_grsuns`vgigss jc Gv`mgikg
• [ucc`k`git npprjpr`ntg gv`mgikg `s pgrsuns`vg rnthgr thni
kjikaus`vg..
kjikaus`vg
• _grsuns`vg gv`mgikg `s5
‛ Npprjpr`ntg”rgagvnit nim rga`nbag
‛ [ucc`k`git”gijudh gv`mgikg
‛ ]`fgay”
]`fgay”kjvgr`id
kjvgr`id thg npprjpr`ntg t`fg pgr`jm

Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - >=

Npprjpr`ntgigss
• Npprjpr`ntgigss
Npprjpr`ntgigss jr suc
succ`k`git
c`k`git npprjp
npprjpr`ntg
r`ntg gv`mgikg
gv`mgikg fgnis
thnt thgrg `s gijudh gv`mgikg jc n h`dh gijudh quna`ty tj
cjrf n kjikaus`ji w`th rgspgkt tj thg num`t jp`i`ji.

• ]hg num`tjr fust bg pgrsunmgm thnt h`s jr hgr jp`i`ji `s

kjrrgkt w`th n h`dh agvga jc nssurnikg.
• ]hg snfpag s`zg w`aa vnry mgpgim`id upji5

‛ ]hg snfpa`id fgthjm khjsgi<

‛ ]hg typg jc tgst (g.d., tgst jc kjitrjas vs. tgst jc mgtn`as)< nim
‛ ]hg nfjuit jc nssurnikg tj bg jbtn`igm crjf thg tgst.

• ]hg snfpa`id fgthjm nim s`zg `s rgantgm tj nssgssgm r`sls.

Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - >7

Rgagvnikg
• Rgagvnikg jc gv`mgikg mgnas w`th thg ajd`kna kjiigkt`ji
w`th thg mgs`di jc thg num`t prjkgmurg nim thg nssgrt`ji jr
kjitrja thnt `s bg`id tgstgm.

• Rgagvnikg kni bg kjis`mgrgm jiay `i tgrfs jc spgk`c`k

num`t jbegkt`vgs.

Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - >3

Rga`nb`a`ty jc Gv`mgikg (> jc ;)
• ]hgrg nrg s`x cnktjrs tj kjis`mgr whgi nssgss`id thg
rga`nb`a`ty jc gv`mgikg5
‛ Gv`mgikg jbtn`igm m`rgktay by num`tjr ”gv`mgikg jbtn`igm

m`rgktay by thg num`tjr

rgpgrcjrfnikg nim/jr thrjudh phys`kna gxnf`int`ji, jbsgrvnt`ji,
`ispgkt`ji.
‛ @imgpgimgikg jc sjurkg (`g., thg `icjrfnt`ji prjv`mgr)
prjv`mgr)” ”
gv`mgikg jbtn`igm crjf n sjurkg juts`mg thg git`ty `s fjrg rga`nbag
thni thnt jbtn`igm crjf w`th`i.

Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - >4

 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 >4

Rga`nb`a`ty jc Gv`mgikg (2 jc ;)
• ]hgrg nrg s`x cnktjrs tj kjis`mgr whgi nssgss`id thg
rga`nb`a`ty jc gv`mgikg5
‛ ^una`c`knt`jis jc sjurkg (`g., thg `icjrfnt`ji prjv`mgr)”
prjv`mgr)”

`im`v`munas
quna`c`gm tj prjv`m`id
mj sj. `icjrfnt`ji shjuam bg kjfpgtgit nim
‛ Kjis`stgiky crjf fuat`pag sjurkgs”gv`mgikg
sjurkgs”gv`mgikg w`th naa sjurkgs
kjis`stgit `s fjrg rga`nbag thni `ikjis`stgit gv`mgikg.
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 >8

Rga`nb`a`ty jc Gv`mgikg (; jc ;)
• ]hgrg nrg s`x cnktjrs tj kjis`mgr whgi nssgss`id thg
rga`nb`a`ty jc gv`mgikg5
‛ ]hg gccgkt`vgigss jc ka`git‐s
ka`git‐s `itgrina kjitrjas”whgi
kjitrjas”whgi n ka`git‐s

`itgrina kjitrjas
rga`nbag thni whginrgkjitrjas
gccgkt`vg,
nrggv`mgikg
wgnl. jbtn`igm `itgrinaay `s fjrg
‛ Mgdrgg jc jbegkt`v`ty”gv`mgikg
jbegkt`v`ty”gv`mgikg thnt rgqu`rgs thg usg jc n h`dh
agvga jc eumdfgit `s agss jbegkt`vg.
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 >0

[ucc`k`giky
• ]hg qunit`ty jc gv`mgikg jbtn`igm mgtgrf`igs `ts
succ`k`giky..
succ`k`giky
• ^unit`ty `s fgnsurgm pr`fnr`ay by thg snfpag s`zg thg
num`tjr sgagkts.
• ]hg fjst `fpjrtnit cnktjrs mgtgrf`i`id thg npprjpr`ntg
snfpag s`zg nrg5
‛ ]hg num`tjr‐s gxpgktnt`ji jc grrjrs
‛ ]hg gccgkt`vgigss jc thg ka`git‐s `itgrina kjitrjas.
k jitrjas.
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 2?

]`fga`igss
• ]hg t`fga`igss jc num`t gv`mgikg kni rgcgr tj g`thgr whgi
`t wns nkkufuantgm jr thg pgr`jm kjvgrgm by thg num`t.
‛ Cjr bnanikg shggt nkkjuits, gv`mgikg `s fjrg pgrsuns`vg whgi `t

`s jbtn`igm ns kajsg tj thg bnanikg shggt mntg ns pjss`bag.

‛ Cjr `ikjfg stntgfgit nkkjuits, gv`mgikg `s fjrg pgrsuns`vg `c `t
kjvgrs thg git`rg pgr`jm uimgr num`t rnthgr thni jiay n pnrt jc thg
pgr`jm.
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 2>

Kjfb`igm Gccgkt (> jc 2)

• ]hg num`tjr iggms tj kjis`mgr naa jc thg gv`mgikg cnktjrs
tjdgthgr, `.g., thg kjfb`igm gccgkts jc succ`k`giky,
succ`k`giky,
npprjpr`ntgigss nim t`fga`igss `i thg kjitgxt jc thg r`sl jc
fntgr`na f`sstntgfgit
f`sstntgfgit cjr thnt num`t jbegkt`vg.
jbegkt`vg.
• _nrt jc thg gv`mgikg dnthgr`id mgk`s`ji `s rgantgm tj thg
trnmgjcc jc thg kjst jc dnthgr`id gv`mgikg vgrsus `ts
pgrsuns`vgigss.
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - 22

Kjfb`igm Gccgkt (2 jc 2)
]nbag 7-> Rgant`jish`ps Nfjid Gv`mgikg Mgk`s`jis nim
_grsuns`vgigss
Num`t Gv`mgikg Mgk`s`jis ^una`ty Nccgkt`id _grsuns`vgigss jc Gv`mgikg

Inturg jc num`t prjkgmurgs Npprjpr`ntgigss

Rgagvnikg
Rga`nb`a`ty
• Jbtn`i
Jbtn`igm gm m`rg
m`rgktay
ktay by thg
thg num`
num`tjr
tjr
• @i
@img
mgpgpgimimgi
gikg
kg jjcc sjur
sjurkgkg
• ^una
^una`c `c`k
`knt
nt`j
`jis
is jc sjur
sjurkgkg
• Kjis
Kjis`s `stgi
tgiky
ky jc
jc gv`m
gv`mgi
gikgkg
• Gc
Gccgkt
cgkt`vg
`vgigs
igss s jc `i
`itgri
tgrina
na kjitr
kjitrjas
jas
• Jbeg
Jbegkt`kt`v`
v`ty
ty jc
jc gv`mg
gv`mgikgikg
Gxtgit jc ]gst`id [ucc`k`giky
• Nmgq
Nmgqun
untg
tg snfp
snfpag
ag s`
s`zg
zg
• [gagkt`
[gagkt`ji
ji jc prjpg
prjpgrr pjpuan
pjpuant`j
t`ji
i `tg
`tgfs
fs
]`f`id ]`fga`igss
• Thgi
Thgi prjkg
prjkgmur
murgs
gs nrg
nrg pgrcjrf
pgrcjrfgm
gm
• _jrt`j
_jrt`ji
i jc pgr`jm
pgr`jm bg`id
bg`id num`
num`tgm
tgm
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - 2;

[pgk`c`k ]ypgs jc Num`t _rjkgmur

_rjkgmurgs
gs (> jc 7)
• _nrndrnphs N>? tj N27 jc KN[ 7?? sgt jut sgvgi ‑wnys‖
tj kjaagkt gv`mgikg5
(>) @ispgkt`ji”a`tgrnaay
@ispgkt`ji”a`tgrnaay thg num`tjr‐s `ispgkt`ji jr kjuit jc n

tnid`bag
fjst nssgtnim
rga`nbag jr mjkufgit. @ispgkt`ji
usgcua typgs `s rgdnrmgm ns jig jc thg
jc num`t gv`mgikg.
(2) Jbsgrvnt`ji”jbsgrvnt`ji
Jbsgrvnt`ji”jbsgrvnt`ji `s thg usg jc jig‐s sgisgs tj nssgss
kgrtn`i nkt`v`t`gs. @t `s igvgr succ`k`git by `tsgac, nim thg num`tjr
fust cjaajw up `i`t`na `fprgss`jis w`th jthgr l`ims jc
kjrrjbjrnt`vg gv`mgikg.
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - 2=

@ispgkt`ji5 @itgrina Sgrsus

Sgrsus Gxtgrina
Gxtgrina
Mjkufgitnt`ji
• Ni `itgrina mjkufgit `s jig thnt hns bggi prgpnrg nim
usgm w`th`i thg ka`git‐s jrdni`znt`ji nim `s rgtn`igm w`thjut
gvgr dj`id tj ni juts`mg pnrty sukh ns n kustjfgr jr n
vgimjr.
• Ni gxtgrina mjkufgit `s jig thnt hns bggi `i thg hnims
jc sjfgjig juts`mg thg ka`git‐s jrdni`znt`ji whj `s n pnrty
tj thg trnisnkt`ji bg`id mjkufgitgm.
• Gxtgrina mjkufgits nrg rgdnrmgm ns fjrg rga`nbag thni
thjsg thnt nrg `itgrina.
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - 27

@ispgkt`ji5 Sjukh`id nim ]rnk`id

• Sjukh`id `s thg usg jc mjkufgitnt`ji tj suppjrt rgkjrmgm
trnisnkt`jis jr nfjuits.
• ]rnk`id `s thg usg jc mjkufgitnt`ji tj mgtgrf`ig `c
trnisnkt`jis jr nfjuits nrg `ikaumgm `i thg nkkjuit`id
rgkjrms.
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - 23

[pgk`c`k ]ypgs jc Num`t _rjkgmur

_rjkgmurgs
gs (2 jc 7)
• _nrndrnphs N>? tj N27 jc KN[ 7?? sgt jut sgvgi ‑wnys‖
tj kjaagkt gv`mgikg5
(;) Gxtgrina kjic`rfnt`ji”]hg
kjic`rfnt`ji”]hg rgkg`pt jc n wr`ttgi jr jrna

rgspjisg
jc crjf Num`tjrs
`icjrfnt`ji. ni
N `imgpgimgit
um`tjrs typ`knaayth`rm pnrty
jbtn`i vgr`cy`id
wr`ttgi thg nkkurnky
rgspjisgs rnthgr
thni jrna jigs. Kjic`rfnt`jis nrg h`dhay rgdnrmgm nim jctgi
usgm< hjwgvgr,
hjwgvgr, thgy nrg kjstay tj jbtn`i.
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - 24

[pgk`c`k ]ypgs jc Num`t _rjkgmur

_rjkgmurgs
gs (; jc 7)
• _nrndrnphs N>? tj N27 jc KN[ 7?? sgt jut sgvgi ‑wnys‖
tj kjaagkt gv`mgikg5
(=) Rgknakuant`ji”`ivjavgs
Rgknakuant`ji”`ivjavgs rgkhgkl`id thg kjfputnt`jis nim

fnthgfnt`kna
uimgr num`t. wjrl kjfpagtgm by thg ka`git mur`id thg pgr`jm
(7) Rgpgrcjrfnikg”thg
Rgpgrcjrfnikg”thg rgmj`id jc iji-fnthgfnt`kna prjkgmurgs
sukh ns `itgrina kjitrjas. ]h`s kjuam `ikaumg rgkhgkl`id jc
trniscgrs jc `icjrfnt`ji. Rg-pgrcjrfnikg rga`gs hgnv`ay upji thg
usg jc kjfputgr-nss`stgm num`t tgsts.
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - 28

[pgk`c`k ]ypgs jc Num`t _rjkgmur

_rjkgmurgs
gs (= jc 7)
• _nrndrnphs N>? tj N27 jc KN[ 7?? sgt jut sgvgi ‑wnys‖
tj kjaagkt gv`mgikg5
(3) Ninayt`kna prjkgmurgs”thg
prjkgmurgs”thg usg jc kjfpnr`sjis nim

rgant`jish`ps bgtwggi c`inik`na nim iji-c`inik`na `icjrfnt`ji tj

mgtgrf`ig whgthgr nkkjuit bnanikgs nppgnr rgnsjinbag. KN[
72? stntgs thnt ninayt`kna prjkgmurgs shjuam bg usgm ji naa
num`ts tj kjimukt r`sl nssgssfgits, nim nt, jr ignr thg gim jc,
thg num`t whgi nssgss`id thg c`inik`na stntgfgits ns n whjag.
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - 20

[pgk`c`k ]ypgs jc Num`t _rjkgmur

_rjkgmurgs
gs (7 jc 7)
• _nrndrnphs N>? tj N27 jc KN[ 7?? sgt jut sgvgi ‑wnys‖
tj kjaagkt gv`mgikg5
(4) @iqu`ry”jbtn`i`id
@iqu`ry”jbtn`i`id jc wr`ttgi jr jrna `icjrfnt`ji crjf thg ka`git `i

rgspjisg tj qugst`jis crjf thg num`tjr. @t `s usunaay igkgssnry tj

jbtn`i curthgr kjrrjbjrnt`id gv`mgikg thrjudh jthgr prjkgmurgs.
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - ;?

Npprjpr`ntgigss jc ]ypgs
]ypgs jc _rjkgmur
_rjkgmurgs
gs
• ]hg gccgkt`vgigss jc `itgrina kjitrjas hns n s`di`c`knit
gccgkt ji thg rga`nb`a`ty jc fjst typgs jc gv`mgikg.
• Bjth `ispgkt`ji nim rgpgrcjrfnikg nrg a`lgay tj bg h`dhay
rga`nbag `c `itgrina kjitrjas nrg gccgkt`vg.
• [jfg typgs jc gv`mgikg nrg rnrgay succ`k`git tj prjv`mg
kjfpgtgit gv`mgikg tj snt`scy ni jbegkt`vg.

• Kjis`mgr thg kjst jc thg prjkgmurg.

 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - ;>

Rga`nb`a
Rga`nb`a`ty
`ty jc ]ypgs
]ypgs jc Num`t _rjkgmur
_rjkgmurgs
gs (> jc 2)

]nbag 7-= Rga`nb`a`ty jc ]ypgs jc Num`t _rjkgmurgs

Kr`tgr`n tj Mgtgrf`ig Rga`nb`a`ty

Gccgkt`vgigss jc
]ypg jc Num`t @imgpgimgikg jc Ka`git‐s @itgrina Num`tjr‐s M`rgkt ^una`c`knt`jis jc Jbegkt`v`ty jc
_rjkgmurg _rjv`mgr Kjitrja Lijwagmdg _rjv`mgr Gv`mgikg
@ispgkt`ji” H`dh (num`tjr Snr`gs H`dh Ijrfnaay h`dh H`dh
_hys`kna mjgs prjkgmurg) (num`tjr mjgs
Gxnf`int`ji prjkgmurg)
@ispgkt`ji” Snr`gs”gxtgrina Snr`gs Ajw Snr`gs H`dh
Mjkufgits nim mjkufgits
Rgkjrms fjrg
`imgpgimgit
thni `itgrina
mjkufgits
Jbsgrvnt`ji H`dh (num`tjr Snr`gs H`dh Ijrfnaay h`dh Fgm`uf
mjgs) (num`tjr mjgs)
@iqu`r`gs jc thg Ajw (ka`git Ijt nppa`knbag Ajw Snr`gs Snr`gs”ajw tj
ka`git prjv`mgs) h`dh
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - ;2

Rga`nb`a
Rga`nb`a`ty
`ty jc ]ypgs
]ypgs jc Num`t _rjkgmur
_rjkgmurgs
gs (2 jc 2)

]nbag 7-= Kjit`iugm

Kr`tgr`n tj Mgtgrf`ig Rga`nb`a`ty

Gccgkt`vgigss jc
]ypg jc Num`t @imgpgimgikg jc Ka`git‐s @itgrina Num`tjr‐s M`rgkt ^una`c`knt`jis jc Jbegkt`v`ty jc
_rjkgmurg _rjv`mgr Kjitrja Lijwagmdg _rjv`mgr Gv`mgikg
Gxtgrina H`dh Ijt nppa`knbag Ajw Snr`gs”usunaay H`dh
kjic`rfnt`ji h`dh
Rgknakuant`ji H`dh (num`tjr Snr`gs H`dh H`dh (num`tjr H`dh
mjgs) mjgs)
Rgpgrcjrfnikg H`dh (n
(num`tjr Snr`gs H`dh H`dh (num`tjr H`dh
mjgs) mjgs)
Ninayt`kna H`dh/ajw (num`tjr Snr`gs Ajw Ijrfnaay h`dh Snr`gs”mgpgims
prjkgmurgs mjgs/ka`git (num`tjr ji thg rga`nb`a`ty
rgspjims) mjgs/ka`git jc thg mntn
rgspjims)
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - ;;

]grfs, Num`t _rjkgmurgs, nim ]ypgs

]ypgs jc
Gv`mgikg (> jc ;)
]nbag 7-7 ]
]grfs,
grfs, Num`t _rjkgmurgs,
_rjkgmur gs, nim ]ypgs jc Gv`mgikg
]grf nim Mgc`i`t`ji @aaustrnt`vg Num`t _rjkgmurg ]ypg jc Num`t _rjkgmurg
Gxnf`ig”N rgnsjinbay mgtn`agm stumy jc
Gxnf`ig”N Gxnf`ig n snfpag jc vgimjrs‐ `ivj`kgs jr @ispgkt`ji
n mjkufgit jr rgkjrm tj mgtgrf`ig mntn tj mgtgrf`ig whgthgr thg djjms jr
spgk`c`k cnkts nbjut `t. sgrv`kgs rgkg`vgm nrg rgnsjinbag nim jc

thg typg ijrfnaay usgm by thg

bus`igss. th g ka`git‐s

[kni”N agss mgtn`agm gxnf`int`ji jc n

[kni”N [kni thg snags rgpjrt, ajjl`id cjr anrdg Ninayt`kna prjkgmurgs
mjkufgit jr rgkjrm tj mgtgrf`ig `c nim uiusuna trnisnkt`jis.
thgrg `s sjfgth`id uiusuna wnrrnit`id
Cjr anrdg snags h`stjry c`ags, usg num`t
curthgr `ivgst`dnt`ji.
sjctwnrg tj rui ni gxkgpt`ji rgpjrt cjr
anrdg nfjuits.
Rgnm”Ni gxnf`int`ji jc wr`ttgi Rgnm thg f`iutgs jc n bjnrm jc m`rgktjrs‐ @ispgkt`ji
`icjrfnt`ji tj mgtgrf`ig cnkts pgrt`igit fggt`id, nim suffnr`zg naa `icjrfnt`ji
tj thg num`t nim thg rgkjrm`id jc thjsg thnt `s pgrt`igit tj thg c`inik`na
cnkts `i n wjrl`id pnpgr. stntgfgits `i n wjrl`id pnpgr.
Kjfputg”N knakuant`ji mjig by thg
Kjfputg”N Kjfputg thg `ivgitjry turijvgr rnt`js, nim Ninayt`kna prjkgmurgs
num`tjr `imgpgimgit jc thg ka`git. kjfpnrg tj prgv`jus ygnrs ns n tgst jc
`ivgitjry.
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - ;=

]grfs, Num`t _rjkgmurgs, nim ]ypgs

]ypgs jc
Gv`mgikg (2 jc ;)
]nbag 7-7 Kjit`iugm
]grf nim Mgc`i`t`ji @aaustrnt`vg Num`t _rjkgmurg ]ypg jc Num`t _rjkgmurg

Rgkjfputg”N knakuant`ji mjig tj

Rgkjfputg”N Rgkjfputg thg ui`t snags pr`kg t`fgs thg Rgknakuant`ji
mgtgrf`ig whgthgr n ka`git‐s knakuant`ji iufbgr jc ui`ts cjr n snfpag jc mupa`kntg
`s kjrrgkt. snags `ivj`kgs, nim kjfpnrg thg tjtnas tj
thg ka`git‐s knakuant`jis.
Cjjt”Ni nmm`t`ji jc n kjaufi jc iufbgrs Cjjt thg snags h`stjry c`ags us`id num`t Rgknakuant`ji
tj mgtgrf`ig `c thg tjtna `s thg snfg ns sjctwnrg, nim kjfpnrg naa tjtnas tj thg
thg ka`git‐s. dgigrna agmdgr.
]rnkg”Ni `istrukt`ji ijrfnaay nssjk`ntgm
]rnkg”Ni ]rnkg n snfpag jc snags trnisnkt`jis crjf Rgknakuant`ji Rgpgrcjrfnikg
w`th mjkufgitnt`ji jr rgpgrcjrfnikg. thg snags rgpjrts tj snags `ivj`kgs, nim
]hg `istrukt`ji shjuam stntg whnt thg kjfpnrg kustjfgr infg, mntg, nim thg
num`tjr `s trnk`id nim whgrg `t `s bg`id tjtna mjaanr vnaug jc thg snag.
trnkgm crjf nim tj. Crgqugitay
Crgqugitay,, ni num`t
]rnkg pjst`ids crjf thg snags rgpjrts tj
prjkgmurg thnt `ikaumgs thg tgrf ‑trnkg‖
thg dgigrna agmdgr nkkjuits.
w`aa nasj `ikaumg n sgkjim `istrukt`ji,
sukh ns ‑kjfpnrg‖ jr ‑rgkjfputg.‖
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - ;7

]grfs, Num`t _rjkgmurgs, nim ]ypgs

]ypgs jc
Gv`mgikg (; jc ;)
]nbag 7-7 Kjit`iugm
]grf nim Mgc`i`t`ji @aaustrnt`vg Num`t _rjkgmurg ]ypg jc Num`t _rjkgmurg
Kjfpnrg”N kjfpnr`sji jc `icjrfnt`ji `i
Kjfpnrg”N [gagkt n snfpag jc snags `ivj`kgs nim Rgpgrcjrfnikg
twj m`ccgrgit ajknt`jis. ]hg `istrukt`ji kjfpnrg thg ui`t sgaa`id pr`kg ns stntgm
shjuam stntg wh`kh `icjrfnt`ji `s bg`id ji thg `ivj`kg tj thg fnstgr c`ags jc ui`t

kjfpnrgm
prnkt`kna. `i ns fukh mgtn`a ns `s sgaa`id pr`kgs nuthjr`zgm by
fnindgfgit.
Kjuit”N mgtgrf`int`ji jc nssgts ji hnim Kjuit pgtty knsh ji hnim nt thg bnanikg @ispgkt`ji
nt n d`vgi t`fg. ]h`s tgrf shjuam jiay bg shggt mntg.
nssjk`ntgm w`th thg typg jc gv`mgikg
mgc`igm ns phys`kna gxnf`int`ji.
Jbsgrvg”]hg nkt jc jbsgrvnt`ji shjuam
Jbsgrvg”]hg Jbsgrvg whgthgr thg twj `ivgitjry kjuit Jbsgrvnt`ji
bg nssjk`ntgm w`th thg typg jc gv`mgikg tgnfs `imgpgimgitay kjuit nim rgkjrm
mgc`igm ns jbsgrvnt`ji. `ivgitjry qunit`t`gs.
@iqu`rg”]hg nkt jc `iqu`ry shjuam bg
@iqu`rg”]hg @iqu`rg jc fnindgfgit whgthgr thgrg `s @iqu`r`gs jc ka`git
nssjk`ntgm w`th thg typg jc gv`mgikg niy jbsjagtg `ivgitjry ji hnim nt thg
mgc`igm ns `iqu`ry. bnanikg shggt mntg.

[jurkg5 Bnsgm ji Dajvgr, [tgvgi F., nim Mjudans C. _rnw`tt. ‑Gihnik`id Num`tjr _rjcgss`jina [lgpt`k`sf5 ]hg
_rjcgss`jina [lgpt`k`sf Kjit`iuuf.‖ Kurrgit @ssugs `i Num`t`id 8.2 (2?>=)5 p>-p>?.
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - ;3

Mgs`di Ninayt`kna _rjkgmur

_rjkgmurgs
gs
• _gr KN[ 72?.=, ninayt`kna prjkgmurgs nrg gvnaunt`jis jc
c`inik`na `icjrfnt`ji thrjudh ninays`s jc panus`bag
rgant`jish`ps nfjid c`inik`na nim ijic`inik`na mntn.
mntn.
• ]hgy fny bg pgrcjrfgm thrjudhjut thg gidndgfgit5
>. @i num`t panii`id
2. Mur`id thg r`sl rgspjisg
;. Mur`id thg kjfpagt`ji
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - ;4

Cjur ]ypgs jc Ninayt`kna _rjkgmurgs

• ]hg num`tjr typ`knaay kjfpnrgs thg ka`git‐s bnanikgs nim
rnt`js w`th gxpgktgm bnanikgs nim rnt`js us`id jig jr fjrg
jc thg cjaajw`id typgs jc ninayt`kna prjkgmurgs. @i gnkh
knsg, num`tjrs kjfpnrg ka`git mntn w`th5
>. @imustry mntn.
2. [`f`anr pr`jr-pgr`jm mntn.
;. Ka`git-mgtgrf`igm gxpgktgm rgsuats.

=. Num`tjr-mgtgrf`igm gxpgktgm rgsuats.

rgsuats.
• [tnt`st`kna tgkhi`qugs thnt n`m `i `itgrprgt`id rgsuats kni bg
nppa`gm tj ninayt`kna prjkgmurgs.
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - ;8

Ws`id Ninayt`kna _rjkgmurgs ns [ubstnit`vg

]gsts
• KN[ 72? mgskr`bgs thg prjkgss jc mgvgajp`id substnit`vg
ninayt`kna prjkgmurgs5
‛ Mgvgajp ni `imgpgimgit gxpgktnt`ji

‛ Mgc`ig n s`di`c`knit m`ccgrgikg

‛ Kjfputg thg m`ccgrgikg
‛ @ivgst`dntg s`di`c`knit m`ccgrgikgs

• ]hg mgdrgg jc prgk`s`ji jc thg ninayt`kna prjkgmurgs `s

mgpgimgit ji n iufbgr jc m`ccgrgit cnktjrs.
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - ;0

Mjkufgitnt`ji
• Num`t mjkufgitnt`ji `s thg wr`ttgi jr gagktrji`k rgkjrm jc
r`sl nssgssfgits,
nssgssfgits, prjkgmurgs jr tgsts pgrcjrfgm,
`icjrfnt`ji jbtn`igm, nim kjikaus`jis rgnkhgm.
• Num`t mjkufgitnt`ji
mjkufgitnt`ji `s
`s prgpnrgm nim lgpt5
‛ Ns n bns`s cjr panii`id thg num`t
num`t
‛ Ns n rgkjrm jc thg gv`mgikg nkkufuantgm
nkkufuantgm nim thg rgsuats jc thg
tgsts
‛ ]j prjv`mg suppjrt cjr mgtgrf`i`id thg prjpgr typg jc num`tjr‐s
num`tjr ‐s
rgpjrt
‛ Bns`s cjr rgv`gw by supgrv`sjry pgrsjiiga
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - =?

C`ag Nrkh`vg5
Nrkh`vg5 Tjrl`id
Tjrl`id _npgr Kjitgits nim
Jrdni`znt`ji
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - =>

_rgpnrnt`ji jc Tjrl`id
Tjrl`id _npgrs
• Tjrl`id pnpgrs shjuam pjssgss kgrtn`i khnrnktgr`st`ks5
‛ Gnkh wjrl`id pnpgr shjuam bg prjpgray `mgit`c`gm w`th sukh
`icjrfnt`ji ns thg ka`git‐s infg, thg pgr`jm kjvgrgm, n mgskr`pt`ji

jc
nimthg
nikjitgits, thg infg jc thg prgpnrgr,
`imgx kjmg. prgpnrgr, thg mntg jc prgpnrnt`ji,
‛ Tjrl`id pnpgrs shjuam bg `imgxgm nim krjss-rgcgrgikgm tj n`m `i
jrdni`z`id nim c`a`id.
‛ Kjfpagtgm wjrl`id pnpgrs fust kagnray `im`kntg thg num`t wjrl
pgrcjrfgm
‛ Gnkh wjrl`id pnpgr shjuam `ikaumg gijudh `icjrfnt`ji tj cuac`aa thg
jbegkt`vgs cjr wh`kh `t wns mgs`digm.
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - =2

Jwigrsh`p jc Tjrl`id _npgrs nim

Kjic`mgit`na`ty jc Tjrl`id _npgrs
• ]hg wjrl`id pnpgrs prgpnrgm mur`id thg gidndgfgit,
`ikaum`id thjsg prgpnrgm by thg ka`git cjr thg num`tjr, nrg
thg prjpgrty jc thg num`tjr.
• ]hg ruags jc kjimukt jc thg prjcgss`jina nkkjuit`id bjm`gs
rgqu`rg thg`r fgfbgrs ijt tj m`skajsg niy kjic`mgit`na
`icjrfnt`ji jbtn`igm `i thg kjursg jc n prjcgss`jina
gidndgfgit gxkgpt w`th thg kjisgit jc thg ka`git jr whgi
rgqu`rgm by thg kjurts jr by thg prjcgss`jina nkkjuit`id
nssjk`nt`jis.
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - =;

Gccgkt jc ]gkhijajdy
]gkhijajdy ji Num`t Gv`mgikg nim
Num`t Mjkufgitnt`ji
• Num`t sjctwnrg `s sjctwnrg usgm tj nutjfntg prgpnrnt`ji
jc num`t wjrl`id pnpgrs jr thg ninays`s jc ka`git mntn c`ags.
• Num`tjrs jctgi
jctgi usg gidndgfgit
gidndgfgit fnindgfgit
fnindgfgit sjctwnrg tj
jrdni`zg nim ninayzg num`t mjkufgitnt`ji.
• Num`tjrs nasj
nasj usg ajkna
ajkna nrgn igtwjrls
igtwjrls nim drjupshnrg
drjupshnrg
sjctwnrg prjdrnfs tj nkkgss num`t mjkufgitnt`ji
s`fuatnigjusay crjf rgfjtg ajknt`jis.
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - ==

_rjcgss`jina [lgpt`k`sf, Gv`mgikg, nim

Mjkufgitnt`ji
• ]hg num`tjr nmeusts bjth thg agvga jc gv`mgikg nim
mjkufgitnt`ji nkkjrm`id tj thg nfjuit jc slgpt`k`sf
igkgssnry bnsgm upji thg r`sl jc fntgr`na f`sstntgfgit
nim whgthgr thg gv`mgikg kjaagktgm `s kjis`stgit w`th thg
jr`d`ina r`sl nssgssfgit.
 Kjpyr`dht © 2?>0 _gnrsji Kninmn @ik. 7 - =7

Nppay`id _rjcgss`jina
[lgpt`k`sf