9. Real Numbers
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Transcript of 9. Real Numbers
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COEFW@NYCOX
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COR@ICORFGW
?
XIOJ EYCHIXW
FERXKBYGRFKE
Mkb movi us t`i eoturoj euchir, ojj ijsi fs t`i wkrn ka coe. Ft wos ixgjofcib hy J. Nrkeignir (?087-?06?). t`i
riputib Mircoe Cot`icotfgfoe. R`fs stoticiet riviojs fe o eut s`ijj t`i sfmefafgoet rkji ka t`i uefvirsi ka euchirs pjoyib fe t`i ivkjutfke ka `ucoe t`kum`.
E 4 R`i sit ka eoturoj euchir,
] 4 R`i sit ka w`kji euchirs,
T 4 R`i sir ka Fetimirs ,
_ 4 R`i sit ka rotfkeojis,
X 4 R`i sit ka Xioj Euchirs.
@FWRKXFGOJ AOGRW
Bibinfeb wos t`i afrst ckbire cot`icotfgfoe tk puhjfs` fe ?0=8 t`i
cot`icotfgojjy rfmkrkus biafeftfke ka frrotfkeoj euchirs. @i movi
ixpjoeotfke ka t`ifr pjogi fe t`i rioj Euchirs Wystic. @i wos ohji tk
bickestroti t`i gkcpjitieiss ka t`i rioj euchir jfei. @i afjjib fe t`i
`kjis fe t`i systic ka Xotfkeoj euchirs wft` frrotfkeoj Euchirs.
R`fs feekvotfke t`i cobi Xfg`orb Bibinfeb oe fcckrtoj afmuri fe t`i
`fstkry ka Cot`icotfgs.
Wrfefvoso Xocoeudoe (?00=-?68
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XIGOJJ
Fe kur boy tk jfai, wi bioj wft` bfaairiet typis ka euchirs w`fg` goe hi hrkobjy gjossfafib os akjjkws.
GJOWWFAFGORFKE KA EYCHIXW
(f) Eoturoj euchirs (E) 4 E + {?, 8, 7, 2 }Xicorn 4 (f) R`i sit E fs feafefti f.i. ft `os uejfcftib cichirs.
(ff) E `os t`i scojjist ijiciet eocijy ?.(fff) E `os ek jormist ijiciet. f.i., mfvi ci oey eoturoj euchir, wi goe afeb t`i hfmmir euchir
arkc t`i mfvie euchir.(fv) E bkis ekt gketofe
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(fv) Xotfkeoj euchirs 4- R`isi ori rioj euchirs w`fg` goe hi ixprissib fe t`i akrc ka q p
, w`iri p oeb q ori fetimirs
oeb .?
)(7>
?)(8
>?
)(e
oheo
eoh
oe
oho
eoh
oe
oho
(vff) Iviry rotfkeoj euchir goe hi riprisietib ift`ir os o tircfeotfem bigfcoj kr o eke-tircfeotfke ripiotfem (rigurrfem) bigfcojs.
(vfff) Rypis ka rotfkeoj euchirs 4- (o) Rircfeotfem bigfcoj euchirs oeb(h) Eke-tircfeotfke ripiotfem (rigurrfem) bigfcoj euchirs
(v) Frrotfkeoj euchirs 4- O euchir fs gojjib frrotfkeoj euchir , fa ft goe ekt hi wrfttie fe t`i akrcq p
, w`iri p & q ori
fetimirs oeb q
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I.x (f) 7
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Fa pkssfhji, jit us ossuci t`ot 8 fs o rotfkeoj euchir.
R`ieho
8 w`iri o, h ori fetimirs `ovfem ek gkccke aogtkr kt`ir t`oe ?.
8
88
h
o(squorfem hkt` sfbis)
8
8
8ho
88 8ho 8 bfvfbis o8 8 bfvfbis o
R`iriakri jit o 3 8g akr skci fetimir g. o8 3 2g 8. 8h 8 3 2g 8 h8 3 8g 8
8 bfvfbis h 8 8 bfvfbis hR`us, 8 fs o gkccke aogtkr ka o oeb h.Hut, ft gketrobfgts kur ossucptfke t`ot o oeb h `ovi ek gkccke aogtkr kt`ir t`oe ?.
Wk, kur ossucptfke t`ot 8 fs o rotfkeoj, fs wrkem.
@iegi, 8 fs frrotfkeoj.
Ix.8 \rkvi t`ot 77 fs frrotfkeoj .
Wkj. Jit 77 hi rotfkeoj 3q p
, w`iri p oeb q T oeb p, q `ovi ek gkccke aogtkr ixgipt ? ojsk q 5 ?.
77q p
Guhfem hkt` sfbis
777
q p
Cujtfpjy hkt` sfbis hy q 8
87
7qq
p , Gjiorjy [email protected] fs rotfkeoj sfegi p, q `ovi ek gkccke aogtkr.
p7, q ojsk `ovi ek gkccke aogtkr w`fji [email protected]. fs oe fetimir .
[email protected]. [email protected]. w`fg` gketrobfgts kur ossucptfke t`ot 77 fs Frrotfkeoj .
Ix. 7 \rkvi t`ot 78 fs frrotfkeoj . ^Wocpji popir (GHWI) 8
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Ix.2 \rkvi t`ot 78 fs frrotfkeoj.Wkj. Jit 78 hi rotfkeoj euchir soy x 78 x
1898.78788 x
8
91189
88 x x
Os x, 9 oeb 8 ori rotfkeojis8
98 xfs o rotfkeoj euchir .
89
18 x
fs o rotfkeoj euchir
]`fg` fs gketrobfgtfke ka t`i aogt t`ot 1 fs o frrotfkeoj euchir.
@iegi kur suppksftfke fs wrkem 78 fs oe frrotfkeoj euchir .IYGJFBW BFQFWFKE JICCO KX IYGJFBW BFQFWFKE OJMKXFR@CAkr oey twk pksftfvi fetimirs o oeb h t`iri ixfst uefqui fetimirs q oeb r sug` t`ot
O 3 hq + r, w`iri < r : h.Jit us gkesfbir o 3 8?=, h 3 9 oeb coni t`i bfvfsfke ka 8?= hy 9 os uebir 4
Bfvfeib
Bfvfskr 9) 8?=(27 _uktfiet8
