Skaitmeninio skaičiavimo ir matematikos teorijos elementai

(1)

Kmsns knfmojmo mgmfme knfmojmo ymoj aiodmgn gmsmr tirkiotuboym knfmojmo fmno gmfma sumtu snstia Kmsns knfmojmo mgmfme knfmojmo ymoj aiodmgn gmsmr tirkiotuboym knfmojmo fmno gmfma sumtu snstia knfmojmo. \ifmam non bntm aiojiomf knfmojmo kmsns =8, ymntu =, ;, <, 7, 9, 0, :, 5, gmo >. Non knfmojmo. \ifmam non bntm aiojiomf knfmojmo kmsns =8, ymntu =, ;, <, 7, 9, 0, :, 5, gmo >. Non airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm mgmfme =8, ymntu 8, =, ;, <, 7, 9, 0, :, 5, airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm mgmfme =8, ymntu 8, =, ;, <, 7, 9, 0, :, 5, gmo >.

gmo >.

\itnmp knfmojmo gisnamf ntu aiapuoymn kiotub4

mo.=8_o + …. + m<.=8_< + m;.=8_; + m=.=8_= + m8.=8_8 mo.=8_o + …. + m<.=8_< + m;.=8_; + m=.=8_= + m8.=8_8 Giojmo ‖

Giojmo ‖momo‑ mgmfme knfmojmo lmlme ymoj burmoj gmrn =8.‑ mgmfme knfmojmo lmlme ymoj burmoj gmrn =8.

Ansmf, =; knsm gntufns 4 =; 1 (=.=8_=) + (;.=8_8) 1 (=.=8) + (;.=) 1 =8 + ; 1 =;

Kmjmnamom giojmo knfmojmo4 =, =8, ==, =88, ==8, …3 Kmjmnamom giojmo knfmojmo4 =, =8, ==, =88, ==8, …3

Non airupmbmo knfmojmo kmsns ; giojmo mojbm tirkismr mgmfme =. Xmgm knfmojmo kmsns ; kmoymboym Non airupmbmo knfmojmo kmsns ; giojmo mojbm tirkismr mgmfme =. Xmgm knfmojmo kmsns ; kmoymboym mojjctm mgmfme ;, ymntu 8 gmo =.

mojjctm mgmfme ;, ymntu 8 gmo =.

Bitirmojmo4 Bitirmojmo4

=gum 1 =

=8gum 1 ;

==gum 1 <

=88gum 1 7

=8=gum 1 9

Aioufnsoym4 =8gum mrtnoym =8 gn gmfma kmsns ;. Dnbm tngmb gntufns ambm gnmojjmp kmsns =8. Kijntu Aioufnsoym4 =8gum mrtnoym =8 gn gmfma kmsns ;. Dnbm tngmb gntufns ambm gnmojjmp kmsns =8. Kijntu puf

pufm m uotuotub ub kmskmsns ns ymoymoj j fmnfmnooyooym, m, aiaioufoufnsonsoym ym =8g=8gum, um, =8t=8tnjmnjm, , =8i=8iapmapmt, t, =8f=8fnamnam, , =8i=8iomaoma, , =8t=8tudueudue,,

=8gifmpmo, =8siaknfmo, gst.

(Yotub aioufnsoym auojbno knsm giojmo lmrm ymoj fmno, sipirtn =8S;, =8(;), gff. Witmpn bmfn non ymoj (Yotub aioufnsoym auojbno knsm giojmo lmrm ymoj fmno, sipirtn =8S;, =8(;), gff. Witmpn bmfn non ymoj gnjuombmo mgmfme pioufnsmo sipirtn gn mtms).

gnjuombmo mgmfme pioufnsmo sipirtn gn mtms).

Yotub knfmojmo sifmno gisnamf (knfmojmo kmsns ; enojjm =0) bntm piremtnbmo urmnmo kirnbut4 Yotub knfmojmo sifmno gisnamf (knfmojmo kmsns ; enojjm =0) bntm piremtnbmo urmnmo kirnbut4 Knfmojmo Kmsns ; (Knoir)

Knfmojmo Kmsns ; (Knoir)

Knfmojmo kmsns ; airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm ;, ymntu 8 gmo =.

Lcotce 4 Lcotce 4

=gum 1 (=.;_8) 1 =

=8gum 1 (=.;_=) + (8.;_8) 1 ; + 8 1 ;

==gum 1 (=.;_=) + (=.;_8) 1 ; + = 1 <

=88gum 1 (=.;_;) + (8.;_=) + (8.;_8) 1 7 + 8 + 8 1

=88gum 1 (=.;_;) + (8.;_=) + (8.;_8) 1 7 + 8 + 8 1 77

=8=gum 1 (=.;_;) + (8.;_=) + (=.;_8) 1 7 + 8 + = 1

=8=gum 1 (=.;_;) + (8.;_=) + (=.;_8) 1 7 + 8 + = 1 99 Knfmojmo Kmsns <

Knfmojmo Kmsns <

Knfmojmo kmsns < airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm <, ymntu 8, = gmo ;.

Lcotce 4 Lcotce 4

=tnjm 1 (=.<_8) 1 =

;tnjm 1 (;.<_8) 1 ;

=8tnjm 1 (=.<_=) + (8.<_8) 1 < + 8 1 <

==tnjm 1 (=.<_=) + (=.<_8) 1 < + = 1 7

=;tnjm 1 (=.<_=) + (;.<_8) 1 < + ; 1 9

Knfmojmo Kmsns 7 Knfmojmo Kmsns 7

Knfmojmo kmsns 7 airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm 7, ymntu 8, =, ; gmo <.

(2)

Lcotce 4 Lcotce 4

=iapmt 1 (=.7_8) 1 =

<iapmt 1 (<.7_8) 1 <

=8iapmt 1 (=.7_=) + (8.7_8) 1 7 + 8 1 7

=;iapmt 1 (=.7_=) + (;.7_8) 1 7 + ; 1 0

=<iapmt 1 (=.7_=) + (<.7_8) 1 7 + < 1 :

=<iapmt 1 (=.7_=) + (<.7_8) 1 7 + < 1 : Knfmojmo Kmsns 9

Knfmojmo Kmsns 9

Knfmojmo kmsns 9 airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm 9, ymntu 8, =, ;, < gmo Knfmojmo kmsns 9 airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm 9, ymntu 8, =, ;, < gmo 7.

7.

Lcotce 4 Lcotce 4

=fnam 1 (=.9_8) 1 =

=fnam 1 (=.9_8) 1 = 7fnam 1 (7.9_8) 1 7 7fnam 1 (7.9_8) 1 7

=8fnam 1 (=.9_=) + (8.9_8) 1 9 + 8 1 9

=;fnam 1 (=.9_=) + (;.9_8) 1 9 + ; 1 :

;<fnam 1 (;.9_=) + (<.9_8) 1 =8 + < 1 =<

Knfmojmo Kmsns 0 Knfmojmo Kmsns 0

Knfmojmo kmsns 0 airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm 0, ymntu 8, =, ;, < , 7 Knfmojmo kmsns 0 airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm 0, ymntu 8, =, ;, < , 7 gmo 9.

gmo 9.

Lcotce 4 Lcotce 4

=ioma 1 (=.0_8) 1 =

=ioma 1 (=.0_8) 1 = 9ioma 1 (9.0_8) 1 9 9ioma 1 (9.0_8) 1 9

=8ioma 1 (=.0_=) + (8.0_8) 1 0 + 8 1 0

=9ioma 1 (=.0_=) + (9.0_8) 1 0 + 9 1 ==

<9ioma 1 (<.0_=) + (9.0_8) 1 =5 + 9 1 ;<

Knfmojmo Kmsns : Knfmojmo Kmsns :

Knfmojmo kmsns : airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm :, ymntu 8, =, ;, <, 7, 9 Knfmojmo kmsns : airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm :, ymntu 8, =, ;, <, 7, 9 gmo 0.

gmo 0.

Lcotce 4 Lcotce 4

=tudue 1 (=.:_8) 1 =

=tudue 1 (=.:_8) 1 = 0tudue 1 (0.:_8) 1 0 0tudue 1 (0.:_8) 1 0

=8tudue 1 (=.:_=) + (8.:_8) 1 : + 8 1 :

=9tudue 1 (=.:_=) + (9.:_8) 1 : + 9 1 =;

;0tudue 1 (;.:_=) + (0.:_8) 1 =7 + 0 1 ;8

;0tudue 1 (;.:_=) + (0.:_8) 1 =7 + 0 1 ;8 Knfmojmo Kmsns 5

Knfmojmo Kmsns 5

Knfmojmo kmsns 5 airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm 5, ymntu 8, =, ;, <, 7, 9, Knfmojmo kmsns 5 airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm 5, ymntu 8, =, ;, <, 7, 9, 0 gmo :.

0 gmo :.

Lcotce 4 Lcotce 4

=gifmpmo 1 (=.5_8) 1 =

=gifmpmo 1 (=.5_8) 1 = :gifmpmo 1 (<.5_8) 1 : :gifmpmo 1 (<.5_8) 1 :

=8gifmpmo 1 (=.5_=) + (8.5_8) 1 5 + 8 1 5

=0gifmpmo 1 (=.5_=) + (0.5_8) 1 5 + 0 1 =7

=0gifmpmo 1 (=.5_=) + (0.5_8) 1 5 + 0 1 =7 7;gifmpmo 1 (7.5_=) + (;.5_8) 1 <; + ; 1 <7 7;gifmpmo 1 (7.5_=) + (;.5_8) 1 <; + ; 1 <7 Knfmojmo Kmsns >

Knfmojmo Kmsns >

(3)

Knfmojmo kmsns > airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm >, ymntu 8, =, ;, <, 7, 9, Knfmojmo kmsns > airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm >, ymntu 8, =, ;, <, 7, 9, 0, :, gmo 5.

0, :, gmo 5.

Lcotce 4 Lcotce 4

=siaknfmo 1 (=.>_8) 1 >

5siaknfmo 1 (5.>_8) 1 5 5siaknfmo 1 (5.>_8) 1 5

=8siaknfmo 1 (=.>_=) + (8.>_8) 1 > + 8 1 >

;;siaknfmo 1 (;.>_=) + (;.>_8) 1 =5 + ; 1 ;8

;;siaknfmo 1 (;.>_=) + (;.>_8) 1 =5 + ; 1 ;8 9=siaknfmo 1 (9.>_=) + (=.>_8) 1 79 + = 1 70 9=siaknfmo 1 (9.>_=) + (=.>_8) 1 79 + = 1 70 Knfmojmo Kmsns ==

Knfmojmo Kmsns ==

Knfmojmo kmsns == airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm ==, ymntu 8, =, ;, <, 7, Knfmojmo kmsns == airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm ==, ymntu 8, =, ;, <, 7, 9, 0, :, 5, > gmo M. (euru` M 1 =8)

9, 0, :, 5, > gmo M. (euru` M 1 =8) Lcotce 4

Lcotce 4

=sikifms 1 (=.==_8) 1 =

>sikifms 1 (>.==_8) 1 >

Msikifms 1 (M.==_8) 1 (=8.

Msikifms 1 (M.==_8) 1 (=8.==_8) 1 =8 (nojmt! M 1 =8)==_8) 1 =8 (nojmt! M 1 =8)

=8sikifms 1 (=.==_=) + (8.==_8) 1 == + 8 1 ==

=Msikifms 1 (=.==_=) + (M.==_8) 1 (=.==_=) + (=8.==_8) 1 == + =8 1 ;= (nojmt! M 1 =8)

=Msikifms 1 (=.==_=) + (M.==_8) 1 (=.==_=) + (=8.==_8) 1 == + =8 1 ;= (nojmt! M 1 =8) Knfmojmo Kmsns =;

Knfmojmo Kmsns =;

Knfmojmo kmsns =; airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm =;, ymntu 8, =, ;, <, 7, Knfmojmo kmsns =; airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm =;, ymntu 8, =, ;, <, 7, 9, 0, :, 5, >, M gmo K. (euru` M 1 =8, K 1 ==)

9, 0, :, 5, >, M gmo K. (euru` M 1 =8, K 1 ==) Lcotce 4

Lcotce 4

=gumkifms 1 (=.=;_8) 1 =

>gumkifms 1 (>.=;_8) 1 >

Mgumkifms 1 (M.=;_8) 1 (=8.=;_8) 1 =8

Mgumkifms 1 (M.=;_8) 1 (=8.=;_8) 1 =8 (nojmt! M 1 =8)(nojmt! M 1 =8) Kgumkifms 1 (K.=;_8) 1 (==.=;_8) 1 == (nojmt! K 1 ==) Kgumkifms 1 (K.=;_8) 1 (==.=;_8) 1 == (nojmt! K 1 ==)

=8gumkifms 1 (=.=;_=) + (8.=;_8) 1 =; + 8 1 =;

=Mgumkifms 1 (=.=;_=) + (M.=;_8) 1 (=.=;_=) + (=8.=;_8) 1 =; + =8 1 ;; (nojmt! M 1 =8)

;Kgumkifms 1 (;.=;_=) + (K.=;_8) 1 (;.=;_=) + (==.=;_8) 1 ;7 + == 1 <9 (nojmt! K 1 ==)

;Kgumkifms 1 (;.=;_=) + (K.=;_8) 1 (;.=;_=) + (==.=;_8) 1 ;7 + == 1 <9 (nojmt! K 1 ==) Knfmojmo Kmsns =<

Knfmojmo Kmsns =<

Knfmojmo kmsns =< airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm =<, ymntu 8, =, ;, <, 7, Knfmojmo kmsns =< airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm =<, ymntu 8, =, ;, <, 7, 9, 0, :, 5, >, M, K gmo L. (euru` M 1 =8, K 1 ==, L 1 =;)

9, 0, :, 5, >, M, K gmo L. (euru` M 1 =8, K 1 ==, L 1 =;) Lcotce 4

Lcotce 4

=tnjmkifms 1 (=.=<_8) 1 =

>tnjmkifms 1 (>.=<_8) 1 >

Ltnjmkifms 1 (L.=<_8) 1 (=;.=<_8) 1 =; (nojmt! L 1 =;) Ltnjmkifms 1 (L.=<_8) 1 (=;.=<_8) 1 =; (nojmt! L 1 =;)

=8tnjmkifms 1 (=.=<_=) + (8.=<_8) 1 =< + 8 1 =<

=Mtnjmkifms 1 (=.=<_=) + (M.=<_8) 1 (=.=<_=) + (=8.=<_8) 1 =< + =8 1 ;< (nojmt! M 1 =8)

KLtnjmkifms 1 (K.=<_=) + (L.=<_8) 1 (==.=<_=) + (=;.=<_8) 1 =7< + =; 1 =99 (nojmt! K 1 ==, L 1 =;) KLtnjmkifms 1 (K.=<_=) + (L.=<_8) 1 (==.=<_=) + (=;.=<_8) 1 =7< + =; 1 =99 (nojmt! K 1 ==, L 1 =;) Knfmojmo Kmsns =7

Knfmojmo Kmsns =7

Knfmojmo kmsns =7 airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm =7, ymntu 8, =, ;, <, 7, Knfmojmo kmsns =7 airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm =7, ymntu 8, =, ;, <, 7, 9, 0, :, 5, >, M, K, L gmo G. (euru` M 1 =8, K 1 ==, L 1 =;, G 1 =<)

9, 0, :, 5, >, M, K, L gmo G. (euru` M 1 =8, K 1 ==, L 1 =;, G 1 =<) Lcotce 4

Lcotce 4

=iapmtkifms 1 (=.=7_8) 1 =

>iapmtkifms 1 (>.=7_8) 1 >

(4)

Giapmtkifms 1 (G.=7_8) 1 (=<.=7_8) 1 =< (nojmt! G 1 =<) Giapmtkifms 1 (G.=7_8) 1 (=<.=7_8) 1 =< (nojmt! G 1 =<)

=8iapmtkifms 1 (=.=7_=) + (8.=7_8) 1 =7 + 8 1 =7

=Kiapmtkifms 1 (=.=7_=) + (K.=7_8) 1 (=.=7_=) + (==.=7_8) 1 =7 + == 1 ;9 (nojmt! K 1 ==)

=Kiapmtkifms 1 (=.=7_=) + (K.=7_8) 1 (=.=7_=) + (==.=7_8) 1 =7 + == 1 ;9 (nojmt! K 1 ==) MGiapmtkifms 1

MGiapmtkifms 1 (M.=7_=) + (M.=7_=) + (G.=7_8) 1 (G.=7_8) 1 (=8.=7_=) + (=8.=7_=) + (=<.=7_8) 1 (=<.=7_8) 1 =78 + =78 + =< 1 =< 1 =9< (nojmt! =9< (nojmt! M 1 M 1 =8, G =8, G 11

=<)

Knfmojmo Kmsns =9 Knfmojmo Kmsns =9

Knfmojmo kmsns =9 airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm =9, ymntu 8, =, ;, <, 7, Knfmojmo kmsns =9 airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm =9, ymntu 8, =, ;, <, 7, 9, 0, :, 5, >, M, K, L, G gmo I. (euru` M 1 =8, K 1 ==, L 1 =;, G 1 =<, I 1 =7)

9, 0, :, 5, >, M, K, L, G gmo I. (euru` M 1 =8, K 1 ==, L 1 =;, G 1 =<, I 1 =7) Lcotce 4

Lcotce 4

=fnamkifms 1 (=.=9_8) 1 =

>fnamkifms 1 (>.=9_8) 1 >

Ifnamkifms 1 (I.=9_8) 1 (=7.=9_8) 1 =7 (nojmt! I 1 =7) Ifnamkifms 1 (I.=9_8) 1 (=7.=9_8) 1 =7 (nojmt! I 1 =7)

=8fnamkifms 1 (=.=9_=) + (8.=9_8) 1 =9 + 8 1 =9

=Gfnamkifms 1 (=.=9_=) + (G.=9_8) 1 (=.=9_=) + (=<.=9_8) 1 =9 + =< 1 ;5 (nojmt! G 1 =<)

=Gfnamkifms 1 (=.=9_=) + (G.=9_8) 1 (=.=9_=) + (=<.=9_8) 1 =9 + =< 1 ;5 (nojmt! G 1 =<) MIfnamkifms 1 (M.=9_=) + (I.=

MIfnamkifms 1 (M.=9_=) + (I.=9_8) 1 (=8.=9_=) + (=7.=9_8) 1 =98 + =7 1 =09_8) 1 (=8.=9_=) + (=7.=9_8) 1 =98 + =7 1 =07 (nojmt! M 1 =8, I 1 =7)7 (nojmt! M 1 =8, I 1 =7) Knfmojmo Kmsns =0 (Eixmgisnamf)

Knfmojmo Kmsns =0 (Eixmgisnamf)

Knfmojmo kmsns =0 airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm =0, ymntu 8, =, ;, <, 7, Knfmojmo kmsns =0 airupmbmo bifcapcb knfmojmo giojmo kmoymboym mojjctm =0, ymntu 8, =, ;, <, 7, 9, 0, :, 5, >, M, K, L, G, I gmo @. (euru` M 1 =8, K 1 ==, L 1 =;, G 1 =<, I 1 =7, @ 1 =0)

9, 0, :, 5, >, M, K, L, G, I gmo @. (euru` M 1 =8, K 1 ==, L 1 =;, G 1 =<, I 1 =7, @ 1 =0)

Lcotce 4 Lcotce 4

=iomakifms 1 (=.=0_8) 1 =

>iomakifms 1 (>.=0_8) 1 >

@iomakifms 1 (@.=0_8) 1 (=9.=0_8) 1 =9 (nojmt! @ 1 =9)

=8iomakifms 1 (=.=0_=) + (8.=0_8) 1 =0 + 8 1 =0

=@iomakifms 1 (=.=0_=) + (@.=0_8) 1 (=.=0_=) + (=9.=0_8) 1 =0 + =9 1 <= (nojmt! @ 1 =9)

=@iomakifms 1 (=.=0_=) + (@.=0_8) 1 (=.=0_=) + (=9.=0_8) 1 =0 + =9 1 <= (nojmt! @ 1 =9) M@iomakifms 1 (M.=0_=) + (@.=0_8) 1 (=8

M@iomakifms 1 (M.=0_=) + (@.=0_8) 1 (=8.=0_=) + (=9.=0_8) 1 =08 + =9 1 =:9 (nojm.=0_=) + (=9.=0_8) 1 =08 + =9 1 =:9 (nojmt! M 1 =8, @ 1 =9)t! M 1 =8, @ 1 =9)

AI[YKME KNFMOJMO BI KM\N\ =8

AI[YKME KNFMOJMO BI KM\N\ =8 (GI\NAMF)(GI\NAMF) Lmrm = 4 XIOJY[MNMO

Lmrm = 4 XIOJY[MNMO Lcotce 4

Lcotce 4

Ykme ;<7fnam bi kmsns =8.

Dmwmk 4 Dmwmk 4

;<7fnam 1 (;.9_;) + (<.9_=) + (7.9_8) 1 98 + =9 + 7 1 0>

Lmrm ; 4 EC[OI[

Lcotce 4 Lcotce 4

Ykme ;<7fnam bi kmsns =8.

Dmwmk 4 Dmwmk 4

| | ; ; < < 77 9

9 | | =8 =8 0909

--- --- ++

; ; =< 0>=< 0>

Dmgn dmwmkmooym 0>

AIOJYKME KNFMOJMO KM\N\ =8 (GI\NAMF)

AIOJYKME KNFMOJMO KM\N\ =8 (GI\NAMF) BI KM\N\ FMNOO^MBI KM\N\ FMNOO^M

(5)

Lmrm =. CXI[M\N XIAKMJNMO Lmrm =. CXI[M\N XIAKMJNMO

Yotub aiojukme knfmojmo gisnamf aiodmgn knfmojmo kmsns fmnooym gnjuombmo cpirmsn piakmjnmo Yotub aiojukme knfmojmo gisnamf aiodmgn knfmojmo kmsns fmnooym gnjuombmo cpirmsn piakmjnmo giojmo kmsnsoym smaknf aiapiremtnbmo snsmoym.

giojmo kmsnsoym smaknf aiapiremtnbmo snsmoym.

Lcotce 4 Lcotce 4

Ykme =98 bi kmsns 5.

Dmwmk 4 Dmwmk 4

=98 4 5 1 =5 snsm 0

=5 4 5 1 ; snsm ;

; 4 5 1 8 snsm ;octi 4 uotub aioufnsbmo kmsns 5 oym, piakmlmmo gnfmbubmo gmrn kmwme. Dmgn

; 4 5 1 8 snsm ;octi 4 uotub aioufnsbmo kmsns 5 oym, piakmlmmo gnfmbubmo gmrn kmwme. Dmgn dmwmkmooym ;;0gifmpmo

dmwmkmooym ;;0gifmpmo Lmrm ; 4 AIOLM[N JLG Lmrm ; 4 AIOLM[N JLG Lcotce 4

Lcotce 4

Dmwmk 4 Dmwmk 4

=98 1 5 x =5 + 0

=5 1 5 x ; + ;

Octi 4 uotub aioufnsbmo kmsns 5 oym, gnkmlm gmrn piakmjn tirbilnfoym ymntu ; fmfu bi snsm oym. Dmgn Octi 4 uotub aioufnsbmo kmsns 5 oym, gnkmlm gmrn piakmjn tirbilnfoym ymntu ; fmfu bi snsm oym. Dmgn dmwmkmooym ;;0gifmpmo

dmwmkmooym ;;0gifmpmo mtmu

mtmu

=98 1 5 x =5 + 0

=5 1 5 x ; + ;

; 1 5 x 8 + ; octi 4 uotub aioufnsbmo kmsns 5 oym, piremtnbmo snsmoym, fmfu piakmlmmo gnfmbubmo

; 1 5 x 8 + ; octi 4 uotub aioufnsbmo kmsns 5 oym, piremtnbmo snsmoym, fmfu piakmlmmo gnfmbubmo gmrn kmwme. Dmgn dmwmkmooym ;;0gifmpmo

gmrn kmwme. Dmgn dmwmkmooym ;;0gifmpmo Lmrm < 4 XIOJY[MNMO

Lmrm < 4 XIOJY[MNMO Lcotce 4

Lcotce 4

Dmwmk 4 Dmwmk 4

=98 1 m.5_; + k.5 +l

=98 1 =5 x 5 + 0

=98 1 (5x; + ;)5 + 0

=98 1 ;x5_; + ;x5 + 0

=98 1 ;x5_; + ;x5 + 0 dmgn gnkmlm ;;0gifmpmo dmgn gnkmlm ;;0gifmpmo

FMWNEMO N FMWNEMO N

Ykme bi kmsns =8 Ykme bi kmsns =8

=. ;8==tnjm

=. ;8==tnjm Dmwmk 4 Dmwmk 4

Lmrm = 4 XIOJYM[MNMO Lmrm = 4 XIOJYM[MNMO

;8==tnjm

1 (;.<_<) + (8.<_;) + (=.<_=) + (=.<_8) 1 (;.<_<) + (8.<_;) + (=.<_=) + (=.<_8) 1 97 + 8 + < + =

1 97 + 8 + < + = 1 95

1 95

(6)

Lmrm ; 4 EC[OI[

|

| ; ; 8 8 = = ==

<

< | | 0 0 =5 9:=5 9:

--- -- ++

; ; 0 0 => => 9595 dmgn dmwmkmooym 95 dmgn dmwmkmooym 95

;. <8;=iapmt

;. <8;=iapmt Dmwmk 4

Dmwmk 4

<8;=iapmt

1 (<.7_<) + (8.7_;) + (;.7_=) + (=.7_8) 1 (<.7_<) + (8.7_;) + (;.7_=) + (=.7_8) 1 =>; + 8 + 5 + =

1 =>; + 8 + 5 + = 1 ;8=

1 ;8=

<. 7;;<fnam

<. 7;;<fnam Dmwmk 4 Dmwmk 4 7;;<fnam 7;;<fnam

1 (7.9_<) + (;.9_;) + (;.9_=) + (<.9_8) 1 (7.9_<) + (;.9_;) + (;.9_=) + (<.9_8) 1 988 + 98 + =8 + <

1 988 + 98 + =8 + <

1 90<

7. 9;9<ioma 7. 9;9<ioma Dmwmk 4 Dmwmk 4 9;9<ioma 9;9<ioma

1 (9.0_<) + (;.0_;) + (9.0_=) + (<.0_8) 1 (9.0_<) + (;.0_;) + (9.0_=) + (<.0_8) 1 =858 + :; + <8 + <

1 =858 + :; + <8 + <

1 ==59 1 ==59

9. <=<0tudue 9. <=<0tudue Dmwmk 4 Dmwmk 4

<=<0tudue

1 (<.:_<) + (=.:_;) + (<.:_=) + (0.:_8) 1 (<.:_<) + (=.:_;) + (<.:_=) + (0.:_8) 1 =8;> + 7> + ;= + 0

1 =8;> + 7> + ;= + 0 1 ==89

1 ==89

0. ;0:9gifmpmo 0. ;0:9gifmpmo Dmwmk 4

Dmwmk 4

;0:9gifmpmo

1 (;.5_<) + (0.5_;) + (:.5_=) + (9.5_8) 1 (;.5_<) + (0.5_;) + (:.5_=) + (9.5_8) 1 =8;7 + <57 + 90 + 9

1 =8;7 + <57 + 90 + 9 1 =70>

1 =70>

:. 9;7ioma :. 9;7ioma Dmwmk 4 Dmwmk 4 9;7ioma 9;7ioma

1 (9.0_;) + (;.0_=) + (7.0_8) 1 (9.0_;) + (;.0_=) + (7.0_8) 1 =58 + =; + 7

1 =58 + =; + 7 1 =>0

1 =>0

5. 7;<fnam 5. 7;<fnam Dmwmk 4 Dmwmk 4

(7)

7;<fnam 7;<fnam

1 (7.9_;) + (;.9_=) + (<.9_8) 1 (7.9_;) + (;.9_=) + (<.9_8) 1 =88 + =8 + <

1 =88 + =8 + <

1 ==<

>. =8;8<iapmt

>. =8;8<iapmt Dmwmk 4

Dmwmk 4

=8;8<iapmt1 (=.7_7) + (8.7_<) + (;.7_;) + ( 8.7_=) + (<.7_8)

=8;8<iapmt1 (=.7_7) + (8.7_<) + (;.7_;) + ( 8.7_=) + (<.7_8) 1 ;90 + 8 + <; + 8 + <

1 ;90 + 8 + <; + 8 + <

1 ;>=

(Lmrm Ecroir knsm bmfnmo lckm siognrn gn ruame) (Lmrm Ecroir knsm bmfnmo lckm siognrn gn ruame) FMWNEMO NN

FMWNEMO NN

=. Ykme 9: bi kmsns ;.

Dmwmk 4 Dmwmk 4

Lmrm =. CXI[M\N XIAKMJNMO Lmrm =. CXI[M\N XIAKMJNMO 9: 4 ; 1 ;5 snsm =

9: 4 ; 1 ;5 snsm =

;5 4 ; 1 =7 snsm 8

=7 4 ; 1 : snsm 8

=7 4 ; 1 : snsm 8 : 4 ; 1 < snsm = : 4 ; 1 < snsm =

< 4 ; 1 = snsm =

= 4 ; 1 8 snsm =

Kmlm snsmoym gmrn kmwme, dmgn dmwmkmooym ===88=gum Kmlm snsmoym gmrn kmwme, dmgn dmwmkmooym ===88=gum Lmrm ;. AIOLM[N JLG

Lmrm ;. AIOLM[N JLG 9: 1 ; x ;5 + =

9: 1 ; x ;5 + =

;5 1 ; x =7 + 8

=7 1 ; x : + 8

=7 1 ; x : + 8 : 1 ; x < + = : 1 ; x < + =

< 1 ; x = + =

= 1 ; x 8 + =

Kmlm snsmoym gmrn kmwme, dmgn dmwmkmooym ===88=gum Kmlm snsmoym gmrn kmwme, dmgn dmwmkmooym ===88=gum

;. Ykme ;9: bi kmsns <.

Dmwmk 4 Dmwmk 4

;9: 1 < x 59 + ;

;9: 1 < x 59 + ; 59 1 < x ;5 + = 59 1 < x ;5 + =

;5 1 < x > + =

> 1 < x < + 8

< 1 < x = + 8

= 1 < x 8 + =

\ienojjm ;9: 1 =88==;tnjm

<. Ykme 7<; bi kmsns =;

Dmwmk 4 Dmwmk 4

7;< 4 =; 1 <9 snsm <

<9 4 =; 1 ; snsm == (Nojmt! K 1 ==)

; 4 =; 1 8 snsm ;

\ienojjm 7;< 1 ;K<gumkifms

(8)

7. Ykme 0:5 bi eixmgisnamf 7. Ykme 0:5 bi eixmgisnamf Dmwmk 4

Dmwmk 4

0:5 4 =0 1 7; snsm 0 0:5 4 =0 1 7; snsm 0

7; 4 =0 1 ; snsm =8 (Nojmt! M 1 =8) 7; 4 =0 1 ; snsm =8 (Nojmt! M 1 =8)

; 4 =0 1 8 snsm ;

\ienojjm 0:5 1 ;M0iomakifms

\ienojjm 0:5 1 ;M0iomakifms 9. Ykme =8==gum gmfma kmsns 9 9. Ykme =8==gum gmfma kmsns 9 Dmwmk 4

Dmwmk 4

Ykme bi kmsns =8 gufu, Ykme bi kmsns =8 gufu,

=8==gum

1 (=.;_<) + (8.;_;) + (=.;_=) + (=.;_8) 1 (=.;_<) + (8.;_;) + (=.;_=) + (=.;_8) 1 5 + 8 + ; + = 1 ==

1 5 + 8 + ; + = 1 ==

Ykme bi kmsns 9, Ykme bi kmsns 9,

== 1 9 x ; + =

; 1 9 x 8 + ;

== 1 ;=fnam, sienojjm =8==gum 1 ;=fnam

== 1 ;=fnam, sienojjm =8==gum 1 ;=fnam 0. Ykme ==88==88=gum gmfma kmsns =<

0. Ykme ==88==88=gum gmfma kmsns =<

Dmwmk 4 Dmwmk 4

Ykme bi kmsns =8 gufu, Ykme bi kmsns =8 gufu,

==88==88=gum

1 (=.;_5) + (=.;_:) + (8.;_0) + (8.;_9) + (=.;_7) + (=.;_<) + (8.;_;) + (8.;_=) + (=.;_8) 1 (=.;_5) + (=.;_:) + (8.;_0) + (8.;_9) + (=.;_7) + (=.;_<) + (8.;_;) + (8.;_=) + (=.;_8) 1 ;90 + =;5 + 8 + 8 + =0 + 5 + 8 + 8 + =

1 ;90 + =;5 + 8 + 8 + =0 + 5 + 8 + 8 + = 1 78>

1 78>

Ykme bi kmsns =<, Ykme bi kmsns =<, 78> 1 =< x <= + 0 78> 1 =< x <= + 0

<= 1 =< x ; + 9

; 1 =< x 8 + ;

sienojjm ==88==88=gum 1 ;90tnjmkifms sienojjm ==88==88=gum 1 ;90tnjmkifms