\input style
\chapno=4\subchno=2\subsubchno=1\chapnotrue
bOLXSHINSTVO PUBLIKACIJ O DETALYAH PROGRAMM DLYA RABOTY V SISTEME S PLAVAYUSHCHEJ 
TOCHKOJ RASSEYANY PO "TEHNICHESKIM PAMYATNYM ZAPISKAM", RASPROSTRANYAEMYM 
RAZLICHNYMI PROIZVODITELYAMI evm, NO SLUCHALISX I PUBLIKACII |TIH PROGRAMM V 
OTKRYTOJ LITERATURE. pOMIMO PRIVEDENNYH VYSHE RABOT SM.~R.~H.~Stark,
D.~v.~McMillan, {\sl Math.~Comp.,\/} {\bf 5} (1951), 86--92, GDE 
OPISANA PROGRAMMA DLYA REALIZACII NA PANELXNO-SHTEKERNOM USTROJSTVE;
D.~McCracken, Digital Computer Programming, New York, Wiley, 1957, 121--131;  
J.~W.~Carr~III, {\sl CACM,\/} {\bf 2} (May, 1959), 10--15;
W.~G.~Wadey, {\sl JACM,\/} {\bf 7} (1960), 129--139; D.~E.~Knuth, {\sl JACM,\/} 
{\bf 8} (1961), 119--128; O.~Kesner, {\sl CACM,\/} {\bf 5} (1962), 269--271; 
F.~r.~Brooks, K.~E.~Iverson, Automatic data processing, New York, Wiley, 1963,
184--199. oBSUZHDENIE ARIFMETICHESKIH OPERACIJ V SISTEME S PLAVAYUSHCHEJ TOCHKOJ 
S TOCHKI ZRENIYA INZHENERA-|LEKTRONSHCHIKA MOZHNO NAJTI V STATXE 
s.~g.~k|MPBELLA "Floating-point operation" V SBORNIKE "Planning a Computer 
System", [ed.~by~W.~Buchholz, New York, McGrow-Hill, 1962,92--121]. 
dOPOLNITELXNYE SSYLKI NA LITERATURU V~P.~4.2.2.

\excercises
\ex[10] kAK BUDUT VYGLYADETX CHISLO aVOGADRO I POSTOYANNAYA pLANKA, ESLI IH 
PREDSTAVITX V VIDE CHETYREHRAZRYADNYH CHISEL S PLAVAYUSHCHEJ TOCHKOJ PO 
OSNOVANIYU~$100$ S IZBYTKOM~$50$? (iMENNO TAKOVO BYLO BY PREDSTAVLENIE V 
MASHINE~\MIX{} (KAK V~\eqref[5]), ESLI BY RAZMER BAJTA RAVNYALSYA~$100$.)

\ex[12] pREDPOLOZHIM, CHTO POKAZATELX~$e$ LEZHIT V INTERVALE~$0\le e \le E$; 
KAKOVY NAIBOLXSHEE I NAIMENXSHEE POLOZHITELXNYE ZNACHENIYA, KOTORYE MOGUT 
BYTX ZAPISANY KAK $p\hbox{-RAZRYADNYE}$ CHISLA S PLAVAYUSHCHEJ TOCHKOJ PO 
OSNOVANIYU~$b$ S IZBYTKOM~$q$? kAKOVY NAIBOLXSHEE I NAIMENXSHEE 
POLOZHITELXNYE ZNACHENIYA, KOTORYE MOGUT BYTX PREDSTAVLENY V VIDE 
\emph{NORMALIZOVANNYH} TAKIH CHISEL?

\ex[20] pOKAZHITE, CHTO ESLI MY RABOTAEM S NORMALIZOVANNYMI 
DVOICHNYMI CHISLAMI S PLAVAYUSHCHEJ TOCHKOJ, TO SUSHCHESTVUET SPOSOB NEMNOGO 
UVELICHITX TOCHNOSTX BEZ UVELICHENIYA OB�EMA ISPOLXZUEMOJ PAMYATI: 
$p\hbox{-RAZRYADNUYU}$ DROBNUYU CHASTX MOZHNO PREDSTAVLYATX PRI POMOSHCHI VSEGO 
LISHX $p-1$~RAZRYADOV MASHINNOGO SLOVA, ESLI CHUTX-CHUTX UMENXSHITX INTERVAL 
ZNACHENIJ POKAZATELYA. 

\rex[15] pUSTX~$b=10$, $p=8$. kAKOJ REZULXTAT DAST ALGORITM~A DLYA 
OPERACII~$(50, +.98765432) \oplus (49, +.33333333)$? 
dLYA OPERACII~$(53, -.99987654) \oplus (54, +.10000000)$? 
dLYA OPERACII~$(45, -.50000001) \oplus (54, +.10000000)$?

\ex[M23] dOKAZHITE, CHTO MEZHDU SHAGAMI~A5 I~A6 ALGORITMA~A MOZHNO, NE IZMENYAYA 
REZULXTATA, POMESTITX SLEDUYUSHCHUYU OPERACIYU: ESLI~$f_u$ I~$f_v$ IMEYUT ODINAKOVYJ 
ZNAK, TO ZAMENITX~$f_v$ NA~$\sign(f_v)b^{-p-2}\floor{b^{p+2}\abs{f_v}}$; 
ESLI ZNAKI~$f_u$ I~$f_v$ PROTIVOPOLOZHNY, TO ZAMENITX~$f_v$ 
NA~$\sign(f_v)b^{-p-2}\ceil{b^{p+2}\abs{f_v}}$. (eFFEKT |TOJ OPERACII 
SOSTOIT V "UREZANII"~$f_v$ DO $p+2$~RAZRYADOV, CHTOBY MINIMIZIROVATX DLINU 
REGISTRA, NEOBHODIMUYU DLYA VYPOLNENIYA SLOZHENIYA V SHAGE~A6.)

\ex[22] uDACHNA LI IDEYA ZAMENITX STROKI~38--40 PROGRAMMY~A 
ODNOJ-EDINSTVENNOJ KOMANDOJ~"|SLC 5|"?

\ex[M21] kAKIE IZMENENIYA NEOBHODIMO PROIZVESTI V ALGORITME~A, CHTOBY ON 
OBESPECHIVAL VYDACHU PRAVILXNO OKRUGLENNOGO NORMALIZOVANNOGO REZULXTATA,
DAZHE ESLI VHODNYE DANNYE NE NORMALIZOVANY? (sLOVA "PRAVILXNO OKRUGLENNYJ" 
OZNACHAYUT, CHTO REZULXTAT IMEET NAIBOLXSHUYU VOZMOZHNUYU TOCHNOSTX V $p$~RAZRYADAH 
V PREDPOLOZHENII, CHTO VHODNYE DANNYE~$u$ I~$v$ V TOCHNOSTI 
RAVNY~$f_u\times b^{e_u-q}$
%% 242
I~$f_v\times b^{e_v-q}$ SOOTVETSTVENNO, HOTYA, BYTX MOZHET, I NE 
NORMALIZOVANY. v CHASTNOSTI, $u$ ILI~$v$ MOGUT IMETX NULEVUYU DROBNUYU CHASTX 
S OCHENX BOLXSHIM POKAZATELEM, I TAKOJ OPERAND V KONTEKSTE DANNOGO 
UPRAZHNENIYA VOSPRINIMALSYA BY KAK RAVNYJ NULYU.)

\ex[25] pRIVEDITE PRIMERY VHODNYH ZNACHENIJ, DLYA KOTORYH 
PODPROGRAMMA~|FADD|  V PROGRAMME~A NE OBESPECHIVAET POLUCHENIYA "PRAVILXNO 
OKRUGLENNOGO NORMALIZOVANNOGO OTVETA" V SMYSLE UPR.~7.

\ex[m24] (u.~kAHAN.) pREDPOLOZHIM, CHTO ISCHEZNOVENIE POKAZATELYA PRIVODIT K 
PRISVOENIYU REZULXTATU ZNACHENIYA NULX BEZ KAKOGO-LIBO UKAZANIYA OB OSHIBKE. 
iSPOLXZUYA VOSXMIRAZRYADNYE DESYATICHNYE CHISLA S PLAVAYUSHCHEJ TOCHKOJ S IZBYTKOM 
NULX I POKAZATELEM~$e$  V INTERVALE~$-50\le e < 50$, NAJDITE TAKIE 
POLOZHITELXNYE ZNACHENIYA~$a$, $b$, $c$, $d$ I~$y$, DLYA KOTORYH VYPOLNYAYUTSYA 
SOOTNOSHENIYA~\eqref[11].

\ex[M15] pRIVEDITE PRIMER NORMALIZOVANNYH VOSXMIRAZRYADNYH DESYATICHNYH CHISEL 
S PLAVAYUSHCHEJ TOCHKOJ~$u$ I~$v$, V PROCESSE SLOZHENIYA KOTORYH PROISHODIT 
PEREPOLNENIE PRI OKRUGLENII.

\rex[M20] dAJTE PRIMER NORMALIZOVANNYH VOSXMIRAZRYADNYH DESYATICHNYH  CHISEL S 
PLAVAYUSHCHEJ TOCHKOJ~$u$ I~$v$, V PROCESSE UMNOZHENIYA KOTORYH PROISHODIT 
PEREPOLNENIE PRI OKRUGLENII.

\ex[M25] dOKAZHITE, CHTO PEREPOLNENIE PRI OKRUGLENII NE MOZHET PROISHODITX V 
HODE VYPOLNENIYA FAZY NORMALIZACII PRI DELENII CHISEL S PLAVAYUSHCHEJ TOCHKOJ.

\rex[m23] mISTER sMYSHL�NYJ SLEDUYUSHCHIM OBRAZOM MODIFICIROVAL PRIEM, 
PRIVODYASHCHIJ K~\eqref[13] S TEM, CHTOBY PROIZVODITX \emph{UREZANIE} 
POLOZHITELXNYH CHISEL~$u$ S PLAVAYUSHCHEJ TOCHKOJ DO CELYH CHISEL~$\floor{u}$ S 
FIKSIROVANNOJ TOCHKOJ V PREDPOLOZHENII, CHTO~$0<u<b^3$:
\EQ{
  |FSUB~ONEHALF;|\quad |FADD~FUDGE|;\quad |SUB~FUDGE.|
}
(zDESX |ONEHALF|---PREDSTAVLENIE~$1/2$ V VIDE CHISLA S PLAVAYUSHCHEJ 
TOCHKOJ.) eGO IDEYA OSNOVYVALASX NA TOM NABLYUDENII, CHTO $u-1/2$ OKRUGLYAETSYA 
DO~$\floor{u}$. oDNAKO POZZHE ON OBNARUZHIL, CHTO |TA IDEYA RABOTAET NE VSEGDA.
v CHEM BYLA OSHIBKA? nET LI DRUGOGO METODA, KOTORYM BY ON MOG VOSPOLXZOVATXSYA?

\ex[23] nAPISHITE \MIX-PODPROGRAMMU, KOTORAYA, BUDUCHI PRIMENENA K 
PROIZVOLXNOMU CHISLU S PLAVAYUSHCHEJ TOCHKOJ, RASPOLOZHENNOMU V REGISTRE~|A| I NE 
OBYAZATELXNO NORMALIZOVANNOMU, PERERABATYVAET EGO V BLIZHAJSHEE CELOE CHISLO, 
PREDSTAVLENNOE V FORME S FIKSIROVANNOJ TOCHKOJ (ILI USTANAVLIVAET, CHTO |TO 
CHISLO SLISHKOM VELIKO PO ABSOLYUTNOJ VELICHINE, CHTOBY TAKOE PREOBRAZOVANIE 
BYLO VOZMOZHNO).

\rex[28] nAPISHITE \MIX-PODPROGRAMMU, KOTORUYU MOZHNO BYLO BY ISPOLXZOVATX V 
SOCHETANII S DRUGIMI PODPROGRAMMAMI |TOGO PARAGRAFA DLYA 
VYCHISLENIYA~$u\fmod{1}$, T.~E.~DLYA OPREDELENIYA~$u-\floor{u}$,  V SLUCHAE 
KOGDA CHISLO~$u$ ZADANO V FORME S PLAVAYUSHCHEJ TOCHKOJ. nUZHNO, CHTOBY POLUCHILSYA 
PRAVILXNO OKRUGLENNYJ REZULXTAT (V SMYSLE UPR.~7); SLEDOVATELXNO, 
ESLI~$u$---OCHENX MALOE OTRICATELXNOE CHISLO, TO $u\fmod{1}$~BUDET OKRUGLENO 
TAKIM OBRAZOM, CHTO REZULXTAT OKAZHETSYA RAVNYM~$1$ (HOTYA PO OPREDELENIYU~$u\fmod{1}$ 
ESTX VESHCHESTVENNOE CHISLO, VSEGDA \emph{MENXSHEE} EDINICY).

\ex[vm21] (rOBERT l.~sMIT.) rAZRABOTAJTE ALGORITM VYCHISLENIYA VESHCHESTVENNOJ 
I MNIMOJ CHASTEJ KOMPLEKSNOGO CHISLA~$(a+bi)/(c+di)$ PO ZADANNYM V FORME S 
PLAVAYUSHCHEJ TOCHKOJ VESHCHESTVENNYM ZNACHENIYAM~$a$, $b$, $c$ I~$d$. oBOJDITESX 
BEZ VYCHISLENIYA~$c^2+d^2$, TAK KAK PRI |TOM MOZHET VOZNIKNUTX PEREPOLNENIE, 
ESLI~$\abs{c}$ ILI~$\abs{d}$ PRIMERNO RAVNO KVADRATNOMU KORNYU IZ 
MAKSIMALXNOGO DOPUSTIMOGO V FORME S PLAVAYUSHCHEJ TOCHKOJ ZNACHENIYA.

\ex[40] (dZHON kOKI.) rASSMOTRITE SLEDUYUSHCHUYU IDEYU RASSHIRENIYA INTERVALA 
IZMENENIYA CHISEL S PLAVAYUSHCHEJ TOCHKOJ: ISPOLXZUETSYA ODNOSLOVNOE PREDSTAVLENIE,
PRI KOTOROM TOCHNOSTX DROBNOJ CHASTI UMENXSHAETSYA, KOGDA VELICHINA POKAZATELYA 
VOZRASTAET.

%% 243

\subsubchap{tOCHNOSTX VYPOLNENIYA ARIFMETICHESKIH DEJSTVIJ V SISTEME S PLAVAYUSHCHEJ TOCHKOJ} % 4.2.2
vYCHISLENIYA NAD CHISLAMI V FORME S PLAVAYUSHCHEJ TOCHKOJ NETOCHNY PO SAMOJ SVOEJ 
PRIRODE, I NETRUDNO STOLX NEUDACHNO PROVODITX IH, CHTOBY VYCHISLENNYE 
OTVETY POCHTI CELIKOM SOSTOYALI IZ "SHUMA". oDNA IZ GLAVNYH PROBLEM 
CHISLENNOGO ANALIZA SOSTOIT V OPREDELENII STEPENI TOCHNOSTI REZULXTATOV TEH 
ILI INYH CHISLENNYH METODOV; SYUDA VKLYUCHAETSYA I PROBLEMA "STEPENI DOVERIYA": 
MY NE ZNAEM, DO KAKOJ STEPENI MOZHNO VERITX REZULXTATAM VYCHISLENIJ NA evm. 
pOLXZOVATELI-NOVICHKI RESHAYUT |TU PROBLEMU, DOVERYAYA KOMPXYUTERU KAK 
NEPOGRESHIMOMU AVTORITETU; ONI SKLONNY SCHITATX, CHTO VSE CIFRY 
NAPECHATANNOGO OTVETA YAVLYAYUTSYA ZNACHASHCHIMI. u POLXZOVATELEJ evm, LISHENNYH 
ILLYUZIJ, PODHOD PRYAMO PROTIVOPOLOZHNYJ: ONI NEIZMENNO OPASAYUTSYA, CHTO IH 
OTVETY POCHTI BESSMYSLENNY. mNOGIE IZ SERXEZNYH MATEMATIKOV PYTALISX 
STROGO PROANALIZIROVATX POSLEDOVATELXNOSTX OPERACIJ S PLAVAYUSHCHEJ TOCHKOJ, NO,
OBNARUZHIV, CHTO ZADACHA SLISHKOM TRUDNA, KONCHALI TEM, CHTO UDOVLETVORYALISX 
PRAVDOPODOBNYMI RASSUZHDENIYAMI.

pOLNOE ISSLEDOVANIE METODOV ANALIZA OSHIBOK VYHODIT, RAZUMEETSYA, ZA RAMKI 
NASTOYASHCHEJ KNIGI, ODNAKO NEKOTORYE IZ HARAKTERISTIK OSHIBOK, VOZNIKAYUSHCHIH 
PRI VYCHISLENIYAH V SISTEME S PLAVAYUSHCHEJ TOCHKOJ, MY V |TOM PUNKTE RASSMOTRIM. 
nASHA CELX---VYYASNITX, KAK VYPOLNYATX OPERACII S PLAVAYUSHCHEJ TOCHKOJ TAKIM 
OBRAZOM, CHTOBY OBLEGCHITX, NASKOLXKO |TO VOZMOZHNO, RAZUMNYJ ANALIZ 
VELICHINY OSHIBKI.

pROSTOJ (NO DOSTATOCHNO POLEZNYJ) SPOSOB OHARAKTERIZOVATX POVEDENIE 
OPERACIJ PLAVAYUSHCHEJ ARIFMETIKI OSNOVAN NA PONYATII "ZNACHASHCHIH CIFR" ILI 
\dfn{OTNOSITELXNOJ OSHIBKI.} eSLI MY PREDSTAVLYAEM TOCHNOE VESHCHESTVENNOE 
CHISLO~$x$ V evm POSREDSTVOM, PRIBLIZHENIYA~$\widehat x=x(1+\varepsilon)$, TO 
VELICHINA~$\varepsilon=(\widehat x-x)/x$ NAZYVAETSYA OTNOSITELXNOJ OSHIBKOJ 
PRIBLIZHENIYA. gRUBO GOVORYA, V SISTEME S PLAVAYUSHCHEJ TOCHKOJ OPERACII UMNOZHENIYA 
I DELENIYA NE SLISHKOM UVELICHIVAYUT OTNOSITELXNUYU OSHIBKU, NO VYCHITANIE 
POCHTI RAVNYH VELICHIN (I SLOZHENIE~$u\oplus v$, GDE~$u$ POCHTI RAVNO~$-v$) 
MOZHET UVELICHITX EE ZNACHITELXNO. iTAK, U NAS ESTX OBSHCHEE 
"|MPIRICHESKOE" PRAVILO: SUSHCHESTVENNOJ POTERI TOCHNOSTI MOZHNO OZHIDATX OT 
SLOZHENIYA I VYCHITANIYA UKAZANNOGO VIDA, NO NE OT UMNOZHENIYA I DELENIYA.

oDNIM IZ SLEDSTVIJ O VOZMOZHNOJ NENADEZHNOSTI SLOZHENIYA V SISTEME S PLAVAYUSHCHEJ 
TOCHKOJ YAVLYAETSYA NARUSHENIE ZAKONA ASSOCIATIVNOSTI:
\EQ[1]{
 (u\oplus v)\oplus w \ne u \oplus (v \oplus w) \rem{DLYA NEKOTORYH~$u$, $v$, $w$.}
}
%% 244
nAPRIMER:
\EQ{
 \eqalign{
   (11111113.\oplus -11111111.)\oplus 7.5111111  & = 2.0000000 \oplus 7.5111111\cr
                                                 & = 9.5111111;\cr
  11111113. \oplus (-11111111. \oplus 7.5111111) & = 11111113. \oplus -11111103.\cr
                                                 & = 10.000000.\cr
 }
}
(vSE PRIMERY |TOGO PUNKTA PRIVODYATSYA V VOSXMIRAZRYADNOJ DESYATICHNOJ 
PLAVAYUSHCHEJ SISTEME S PREDSTAVLENIEM POKAZATELEJ POSREDSTVOM PRYAMOGO 
UKAZANIYA MESTA RASPOLOZHENIYA DESYATICHNOJ TOCHKI. nAPOMNIM, CHTO, KAK I V 
P.~4.2.1, SIMVOLY~$\oplus$, $\ominus$, $\otimes$, $\oslash$ ISPOLXZUYUTSYA 
DLYA OBOZNACHENIYA OPERACIJ V SISTEME S PLAVAYUSHCHEJ TOCHKOJ, SOOTVETSTVUYUSHCHIH 
TOCHNYM OPERACIYAM~$+$, $-$, $\times$, $/$.)

v SVETE VOZMOZHNOJ UTRATY ZAKONA ASSOCIATIVNOSTI PRIVEDENNOE V NACHALE 
|TOJ GLAVY ZAMECHANIE GOSPOZHI lA tUSH [VZYATOE IZ~{\sl Math.~Gazette,\/} {\bf 12} 
(1924), 95], ESLI EGO OTNESTI K ARIFMETIKE V SISTEME S PLAVAYUSHCHEJ 
TOCHKOJ, NESET V SEBE BOLXSHUYU DOLYU ZDRAVOGO SMYSLA. mATEMATICHESKIE 
OBOZNACHENIYA TIPA~"$a_1+a_2+a_3$" ILI~"$\sum_{1\le k \le n} a_k$" PO SAMOMU 
SVOEMU SUSHCHESTVU OSNOVANY NA PREDPOLOZHENII OB ASSOCIATIVNOSTI, TAK CHTO 
PROGRAMMIST DOLZHEN BYTX OSOBENNO BDITELEN NA TOT SCHET, NE PREDPOLAGAET LI 
ON NEYAVNO SPRAVEDLIVOSTX ZAKONA ASSOCIATIVNOSTI.

\section{a.~aKSIOMATICHESKIJ PODHOD}. hOTYA ZAKON ASSOCIATIVNOSTI I NE VEREN,
ZAKON KOMMUTATIVNOSTI
\EQ[2]{
 u \oplus v = v \oplus u
}
DOLZHEN VYPOLNYATXSYA, I |TOT ZAKON MOZHET SLUZHITX SERXEZNYM KONCEPTUALXNYM 
PODSPORXEM PRI PROGRAMMIROVANII I PRI ANALIZE PROGRAMM. eTO SOOBRAZHENIE 
PODSKAZYVAET NAM, CHTO SLEDUET ISKATX NAIBOLEE SUSHCHESTVENNYE ZAKONY, KOTORYM 
UDOVLETVORYAYUT OPERACII~$\oplus$, $\ominus$, $\otimes$ I~$\oslash$; DALEE, 
NE BUDET BEZRASSUDNYM TAKOE VYSKAZYVANIE: \emph{PROGRAMMY ARIFMETICHESKIH 
OPERACIJ V SISTEME S PLAVAYUSHCHEJ TOCHKOJ SLEDUET SOSTAVLYATX TAKIM OBRAZOM, 
CHTOBY SOHRANYALOSX MAKSIMALXNO VOZMOZHNOE CHISLO STANDARTNYH MATEMATICHESKIH
ZAKONOV.}

rASSMOTRIM TEPERX NEKOTORYE IZ DRUGIH OSNOVNYH ZAKONOV, KOTORYE 
SOHRANYAYUTSYA DLYA NORMALIZOVANNYH OPERACIJ S PLAVAYUSHCHEJ TOCHKOJ, OPISANNYH V 
PREDYDUSHCHEM PUNKTE. pREZHDE VSEGO MY IMEEM
\EQ{
 \displaylinesno{
    u \ominus v = u \oplus - v ; & (3) \cr
  -(u \oplus v) = - u \oplus -v; & (4) \cr
     u \oplus v = 0 \hbox{~TOGDA I TOLXKO TOGDA, KOGDA~$v=-u$;}&(5)\cr
     u \oplus 0 = u. & (6) \cr
 }
}
%% 245
iZ |TIH ZAKONOV MOZHNO POLUCHATX I DALXNEJSHIE TOZHDESTVA; NAPRIMER,
\EQ[7]{
 u \ominus v = - (v \ominus u)
}
(SM.~UPR.~1). eTI ZAKONY \emph{NE} VYPOLNYALISX BY, ESLI BY V SISTEME S 
PLAVAYUSHCHEJ TOCHKOJ DLYA DROBNYH CHASTEJ OPERANDOV VMESTO PRYAMOGO KODA 
ISPOLXZOVALSYA DOPOLNITELXNYJ KOD; SM.~UPR.~11. s |TOJ TOCHKI ZRENIYA PRYAMOJ 
KOD DLYA PREDSTAVLENIYA CHISEL S PLAVAYUSHCHEJ TOCHKOJ IMEET NEKOTOROE 
TEORETICHESKOE PREIMUSHCHESTVO.

tOZHDESTVA~\eqref[2]--\eqref[6] SLEDUYUT NEPOSREDSTVENNO IZ 
ALGORITMOV P.~4.2.1. sLEDUYUSHCHEE PRAVILO CHUTX MENEE OCHEVIDNO:
\EQ[8]{
 \hbox{ESLI~$u \le v$, TO~$u \oplus w \le v \oplus w$.}
}
chTOBY DOKAZATX EGO, POLOZHIM PO OPREDELENIYU~$\round(x, p)={}$"$x$,  
OKRUGLENNOE DO $p$~RAZRYADOV"${}={}$
\EQ[9]{
 \cases{
  b^{e-p}\floor{b^{p-e}x+{1\over2}}, & ESLI~$b^{e-1}\le x < b^e$,\cr
                                  0, & ESLI~$x=0$, \cr
   b^{e-p}\ceil{b^{p-e}x-{1\over2}}, & ESLI~$b^{e-1}\le -x < b^e$, \cr
 }
}
GDE~$x$---VESHCHESTVENNOE CHISLO, A~$p$---CELOE POLOZHITELXNOE. 
(sR.~S~ALGORITMOM~4.2.1N, SHAG~N5.) zAMETIM, CHTO SPRAVEDLIVY RAVENSTVA
\EQ[10]{
 \round (-x, p) = - \round (x, p);\qquad \round(bx, p)=b \round (x, p);
}
I, DALEE, V SILU OPREDELENIJ~P.~4.2.1, IMEYUT MESTO VAZHNYE SOOTNOSHENIYA
\EQ{
 \eqalignno{
   u \oplus  v &= \round(u+v, p),       & (11)\cr
   u \ominus v &= \round(u-v, p),       & (12)\cr
   u \otimes v &= \round(u\times v, p), & (13)\cr
   u \oslash v &= \round(u/v, p)        & (14)\cr
 }
}
\emph{PRI USLOVII,} CHTO NE PROISHODIT PEREPOLNENIYA ILI ISCHEZNOVENIYA 
POKAZATELEJ, T.~E.~PRI USLOVII, CHTO CHISLA~$u+v$, $u-v$, $u \times v$ 
I~$u/v$ LEZHAT V DOPUSTIMYH INTERVALAH. pOLXZUYASX |TIM NABLYUDENIEM, 
MOZHNO DATX PROSTOE DOKAZATELXSTVO VSEH PERECHISLENNYH VYSHE TOZHDESTV,
VKLYUCHAYA I TOZHDESTVO~\eqref[8], YAVLYAYUSHCHEESYA SLEDSTVIEM TOGO FAKTA, 
CHTO FUNKCIYA~$\round(x, p)$ NE UBYVAET PRI VOZRASTANII~$x$.

mY MOZHEM TEPERX VYPISATX ESHCHE NESKOLXKO TOZHDESTV, VYTEKAYUSHCHIH IZ UKAZANNYH 
VYSHE SOOTNOSHENIJ:
\EQ{
 \displaylines{
   u \otimes v = v \otimes u, (-u)\otimes v  = -(u\otimes v), 1\otimes v = v;\cr
   u \otimes v = 0 \hbox{ TOGDA I TOLXKO TOGDA, KOGDA~$u=0$ ILI~$v=0$;}\cr
 (-u)\oslash v = u \oslash (-v) = -(u \oslash v),  0 \oslash v =0,\cr
   u \oslash 1 = u, u \oslash u =1;\cr
 \hbox{ESLI~$u\le v$, $w>0$, TO~$u \otimes w \le v \otimes w$, $u \oslash w  \le v \oslash w$, $w \oslash u \ge w \oslash v$.}\cr
 }
}
%% 246
iTAK, ESLI OPERACII V SISTEME S PLAVAYUSHCHEJ TOCHKOJ USTANOVLENY V 
SOOTVETSTVII S OPREDELENNYMI USLOVIYAMI, TO, NESMOTRYA NA NETOCHNOSTX SAMIH 
OPERACIJ, SOHRANYAETSYA NEMALAYA REGULYARNOSTX.

v PRIVEDENNOJ VYSHE KOLLEKCII TOZHDESTV, RAZUMEETSYA, VSE ZHE BROSAETSYA V 
GLAZA OTSUTSTVIE NESKOLXKIH IZVESTNYH ZAKONOV ALGEBRY; ZAKON 
ASSOCIATIVNOSTI DLYA UMNOZHENIYA V SISTEME S PLAVAYUSHCHEJ TOCHKOJ VYPOLNYAETSYA NE 
VPOLNE TOCHNO, KAK |TO BUDET VIDNO IZ UPR.~3, CHTO ZHE KASAETSYA ZAKONA 
DISTRIBUTIVNOSTI, SVYAZYVAYUSHCHEGO OPERACII~$\otimes$ I~$\oplus$), TO ON MOZHET 
NARUSHATXSYA, I PRI |TOM DOVOLXNO ZNACHITELXNO. pUSTX, NAPRIMER, 
$u=20000.000$, $v=-6.0000000$ I~$w=6.0000003$, TOGDA
\EQ{
 \eqalign{
  (u\otimes v) \oplus (u\otimes w) &= 120000.00 \oplus 120000.01 =.010000000,\cr
             u \otimes (v\oplus w) &= 20000.000 \otimes .00000030000000 = .0060000000,\cr
 }
}
TAK CHTO
\EQ[15]{
 u\otimes (v\oplus w) \ne (u\otimes v) \oplus (u \otimes w).
}

aNALOGICHNO, NETRUDNO UKAZATX PRIMERY, KOGDA~$2(u^2\oplus v^2)<(u\oplus v)^2$; 
ZAPROGRAMMIROVANNOE VYCHISLENIE SREDNEGO KVADRATICHNOGO OTKLONENIYA 
DLYA RYADA NABLYUDENIJ PO FORMULE
\EQ{
 \sigma =  {1\over n}\sqrt{n \sum_{1\le k \le n} x_k^2-\left(\sum_{1\le k \le n} x_k\right)^2}
}
MOZHET PRIVESTI K IZVLECHENIYU KVADRATNOGO KORNYA IZ OTRICATELXNOGO CHISLA!

dAZHE ESLI ALGEBRAICHESKIE ZAKONY VYPOLNYAYUTSYA NE VPOLNE STROGO, MY MOZHEM 
ISPOLXZOVATX NASHI METODY DLYA OPREDELENIYA TOGO, S KAKOJ STEPENXYU TOCHNOSTI 
VYPOLNYAETSYA ZAKON. iZ OPREDELENIYA~$\round(x, p)$ SLEDUET, CHTO
\EQ[16]{
 \round(x, p)=x(1+\sigma_p(x)),
}
GDE
\EQ[17]{
 \abs{\sigma_p(x)}\le {1\over 2}b^{1-p}.
}
sLEDOVATELXNO, MY VSEGDA MOZHEM ZAPISATX
\EQ{
 \eqalignno{
   a\oplus b &= (a+b)(1+\delta_p(a+b)),              & (18)\cr
  a\ominus b &=(a-b)(1+\delta_p(a-b)),               & (19)\cr
  a\otimes b &= (a\times a) (1+\delta_p(a\times b)), & (20)\cr
  a\oslash b &=(a/b)(1+\delta_p(a/b)).               & (21)\cr
 }
}
zDESX DOVOLXNO PROSTYM SPOSOBOM MOZHNO OCENITX OTNOSITELXNUYU OSHIBKU 
NORMALIZOVANNYH VYCHISLENIJ V SISTEME S PLAVAYUSHCHEJ TOCHKOJ. 
fORMULY~\eqref[18]--\eqref[21] SLUZHAT GLAVNYM INSTRUMENTOM DLYA OCENKI 
OSHIBOK V ARIFMETIKE NORMALIZOVANNYH CHISEL S PLAVAYUSHCHEJ TOCHKOJ.

%% 247

v KACHESTVE PRIMERA TIPICHNOJ PROCEDURY OCENKI OSHIBKI RASSMOTRIM ZAKON 
ASSOCIATIVNOSTI UMNOZHENIYA. kAK POKAZYVAET UPR.~3, $(u\times v)\otimes w$, 
VOOBSHCHE GOVORYA, NE RAVNO~$u\otimes (v\otimes w)$; NO SITUACIYA V DANNOM 
SLUCHAE NAMNOGO LUCHSHE, CHEM V SLUCHAE ZAKONA ASSOCIATIVNOSTI 
SLOZHENIYA~\eqref[1] I ZAKONA DISTRIBUTIVNOSTI~\eqref[15]. v SAMOM DELE, 
VVIDU~\eqref[17] I~\eqref[20] IMEEM
\EQ{
 \eqalignter{
   (u\otimes v)\otimes w &= ((uv)(1+\delta_1))\otimes w &= uvw(1+\delta_1)(1+\delta_2),\cr
    u\otimes(v\otimes w) &= u\otimes((vw)(1+\delta_3))  &= uvw(1+\delta_3)(1+\delta_4)\cr
 }
}
DLYA NEKOTORYH~$\delta_1$, $\delta_2$, $\delta_3$, $\delta_4$ PRI USLOVII, 
CHTO NE PROISHODIT PEREPOLNENIYA ILI ISCHEZNOVENIYA POKAZATELYA, 
PRICHEM~$\abs{\delta_j}\le {1\over2}b^{1-p}$ DLYA KAZHDOGO~$j$.

sLEDOVATELXNO,
\EQ{
  {(u\otimes v)\otimes w \over u\otimes (v\otimes w)}
  ={(1+\delta_1)(1+\delta_2)\over (1+\delta_3)(1+\delta_4)}
  =1+\delta,
}
GDE
\EQ[22]{
  \abs{\delta}\le 2b^{1-p}/\left(1-{1\over2}b^{1-p}\right)^2.
}

tEM SAMYM MY USTANOVILI, CHTO $(u\otimes v)\otimes w$ \emph{PRIBLIZITELXNO 
RAVNO}~$u\otimes (v\otimes w)$, ZA ISKLYUCHENIEM TEH SLUCHAEV, KOGDA 
PROISHODIT ISCHEZNOVENIE ILI PEREPOLNENIE POKAZATELYA. eTA INTUITIVNAYA 
IDEYA "PRIBLIZITELXNOGO RAVENSTVA" ZASLUZHIVAET BOLEE PODROBNOGO IZUCHENIYA; 
MOZHNO LI RAZUMNYM OBRAZOM DATX BOLEE TOCHNUYU FORMULIROVKU |TOGO UTVERZHDENIYA?

pROGRAMMIST, ISPOLXZUYUSHCHIJ ARIFMETICHESKIE OPERACII V SISTEME S PLAVAYUSHCHEJ 
TOCHKOJ, POCHTI NIKOGDA NE ISPYTYVAET ZHELANIYA PROVERITX, NE VYPOLNYAETSYA LI 
RAVENSTVO~$u=v$ (ILI PO KRAJNEJ MERE ON EDVA LI KOGDA-NIBUDX PYTAETSYA 
|TO SDELATX), TAK KAK RAVENSTVO YAVLYAETSYA PREDELXNO MALOVEROYATNYM 
SOBYTIEM. nAPRIMER, ESLI ISPOLXZUETSYA REKURRENTNOE SOOTNOSHENIE
\EQ{
  x_{n+1}=f(x_n),
}
O KOTOROM TEORIYA, VZYATAYA IZ KAKOJ-TO KNIZHKI, UTVERZHDAET, 
CHTO $x_n$~STREMITSYA K NEKOTOROMU PREDELU PRI~$n\to\infty$, TO, KAK PRAVILO,
BYLO BY OSHIBKOJ PRODOLZHATX VYCHISLENIYA, POKA DLYA NEKOTOROGO~$n$ NE 
OSUSHCHESTVITSYA RAVENSTVO~$x_{n+1}=x_n$, TAK KAK POSLEDOVATELXNOSTX~$x_n$ 
MOZHET VVIDU OKRUGLENIYA PROMEZHUTOCHNYH REZULXTATOV OKAZATXSYA PERIODICHESKOJ S 
BOLXSHIM PERIODOM. rAZUMNO PRODOLZHATX VYCHISLENIYA LISHX DO TEH POR, POKA 
DLYA NEKOTOROGO PODHODYASHCHIM OBRAZOM VYBRANNOGO~$\delta$ NE STANET 
SPRAVEDLIVO NERAVENSTVO~$\abs{x_{n+1}-x_n}<\delta$; NO TAK KAK MY NE 
ZNAEM ZARANEE PORYADKA VELICHINY~$x_n$, ESHCHE BOLEE PRAVILXNO DOZHIDATXSYA 
POYAVLENIYA NERAVENSTVA
%%248
\EQ[23]{
  \abs{x_{n+1}-x_n}\le \varepsilon\abs{x_n};
}
CHISLO~$\varepsilon$ GORAZDO LEGCHE VYBRATX ZARANEE. 
sOOTNOSHENIE~\eqref[23]---|TO DRUGOJ SPOSOB VYRAZHENIYA TOGO FAKTA, CHTO 
CHISLA~$x_{n+1}$ I~$x_n$ PRIBLIZITELXNO RAVNY; I NASHE OBSUZHDENIE POKAZYVAET,
CHTO PRI RASSMOTRENII VYCHISLENIJ NAD CHISLAMI S PLAVAYUSHCHEJ TOCHKOJ 
SOOTNOSHENIE "PRIBLIZITELXNOGO RAVENSTVA" BYLO BY BOLEE POLEZNO, CHEM 
TRADICIONNOE SOOTNOSHENIE RAVENSTVA, ESLI TOLXKO NAM UDASTSYA OPREDELITX 
PERVOE SOOTNOSHENIE NADLEZHASHCHIM OBRAZOM.

dRUGIMI SLOVAMI, TOT FAKT, CHTO STROGOE RAVENSTVO VELICHIN V SISTEME S 
PLAVAYUSHCHEJ TOCHKOJ IGRAET OCHENX NEBOLXSHUYU ROLX, PRIVODIT K NEOBHODIMOSTI 
VVESTI NOVUYU OPERACIYU \emph{SRAVNENIYA VELICHIN S PLAVAYUSHCHEJ TOCHKOJ,} 
PREDNAZNACHENNUYU DLYA OBLEGCHENIYA OCENOK OTNOSITELXNYH ZNACHENIJ DVUH TAKIH 
VELICHIN. pREDSTAVLYAYUTSYA PRIGODNYMI SLEDUYUSHCHIE OPREDELENIYA DLYA CHISEL S 
PLAVAYUSHCHEJ TOCHKOJ~$u=(e_n, f_n)$ I~$v=(e_v, f_v)$ PO OSNOVANIYU~$b$ S 
IZBYTKOM~$q$:
\EQ{
 \eqalignno{
  u &\prec v(\varepsilon) \hbox{ TOGDA I TOLXKO TOGDA, KOGDA~$v-u>\varepsilon\max(b^{e_u-q}, b^{e_v-q})$;}           &(24)\cr
  u &\sim v(\varepsilon) \hbox{ TOGDA I TOLXKO TOGDA, KOGDA~$\abs{v-u}\le \varepsilon \max (b^{e_u-q}, b^{e_v-q})$;} &(25)\cr
  u &\succ v(\varepsilon) \hbox{ TOGDA I TOLXKO TOGDA, KOGDA~$u-v>\varepsilon\max(b^{e_u-q}, b^{e_v-q})$;}           &(26)\cr
  u &\approx v(\varepsilon) \hbox{ TOGDA I TOLXKO TOGDA, KOGDA~$\abs{v-u}\le\varepsilon\min(b^{e_u-q}, b^{e_v-q})$.} &(27)\cr
 }
}
sOGLASNO |TIM OPREDELENIYAM, DLYA LYUBOJ DANNOJ PARY ZNACHENIJ~$u$,~$v$ MOZHET 
VYPOLNYATXSYA V TOCHNOSTI ODNO IZ SOOTNOSHENIJ~$u\prec v$ ("OPREDELENNO MENXSHE"), 
$u\sim v$ ("PRIBLIZITELXNO RAVNO") ILI~$u\succ v$  ("OPREDELENNO BOLXSHE"). 
sOOTNOSHENIE~$u\approx v$---NESKOLXKO BOLEE SILXNOE, NEZHELI~$u\sim v$, I EGO 
MOZHNO CHITATX TAK: "$u$ PO SUSHCHESTVU RAVNO~$v$". vSE |TI SOOTNOSHENIYA ZADAYUTSYA 
POSREDSTVOM POLOZHITELXNOGO CHISLA~$\varepsilon$, IZMERYAYUSHCHEGO STEPENX 
RASSMATRIVAEMOGO PRIBLIZHENIYA.

oDNIM IZ SPOSOBOV ISTOLKOVANIYA PRIVEDENNYH OPREDELENIJ SOSTOIT V TOM, 
CHTOBY LYUBOMU CHISLU~$u$ S PLAVAYUSHCHEJ TOCHKOJ POSTAVITX V SOOTVETSTVIE 
MNOZHESTVO~$S(u)\set{x \mid \abs{x-u}\le \varepsilon b^{e_u-q}}$;
MNOZHESTVO~$S(u)$ PREDSTAVLYAET SOBOJ SOVOKUPNOSTX VESHCHESTVENNYH CHISEL, 
RASPOLOZHENNYH VBLIZI~$u$, I OPREDELENO PRI POMOSHCHI POKAZATELYA~$u$. v 
TERMINAH |TIH MNOZHESTV SOOTNOSHENIE~$u\prec v$ VYPOLNYAETSYA TOGDA I TOLXKO 
TOGDA, KOGDA~$S(u)<v$ I~$u<S(v)$; $u\sim v$ TOGDA I TOLXKO TOGDA, 
KOGDA~$u\in S(v)$ ILI~$v\in S(u)$; $u\succ v$ 
%% 249
TOGDA I TOLXKO TOGDA, KOGDA~$u>S(v)$ I~$S(u)>v$; $u\approx v$ TOGDA 
I TOLXKO TOGDA, KOGDA~$u\in S(v)$ I~$v\in S(u)$. (zDESX MY PREDPOLAGAEM,
CHTO PARAMETR~$\varepsilon$, IZMERYAYUSHCHIJ STEPENX PRIBLIZHENIYA, FIKSIROVAN;
V BOLEE PODROBNOJ ZAPISI MOZHNO BYLO OTRAZITX I ZAVISIMOSTX~$S(u)$ OT~$\varepsilon$.)

vOT NEKOTORYE SLEDSTVIYA IZ PRIVEDENNYH OPREDELENIJ:
\EQ{
 \displaylinesno{
  \hbox{ESLI~$u\prec v(\varepsilon)$, TO~$v\succ u(\varepsilon)$;}                                                  & (28)\cr
  \hbox{ESLI~$u\approx v(\varepsilon)$, TO~$u\sim v(\varepsilon)$;}                                                 & (29)\cr
  u\approx u(\varepsilon);                                                                                          & (30)\cr
  \hbox{ESLI~$u\prec v(\varepsilon)$, TO~$u<v$;}                                                                    & (31)\cr
  \hbox{ESLI~$u\prec v(\varepsilon_1)$  I~$\varepsilon_1\ge \varepsilon_2$, TO~$u\prec v(\varepsilon_2)$;}          & (32)\cr
  \hbox{ESLI~$u\sim v(\varepsilon_1)$ I~$\varepsilon_1\le \varepsilon_2$, TO~$u\sim v(\varepsilon_2)$;}             & (33)\cr
  \hbox{ESLI~$u\approx v(\varepsilon_1)$  I~$\varepsilon_1\le \varepsilon_2$, TO~$u\approx v(\varepsilon_2)$;}      & (34)\cr
  \hbox{ESLI~$u\prec v(\varepsilon_1)$ I~$v\prec w(\varepsilon_2)$, TO~$u\prec w(\varepsilon_1+\varepsilon_2)$;}    & (35)\cr
  \hbox{ESLI~$u\approx v(\varepsilon_1)$ I~$v\approx w(\varepsilon_2)$, TO~$u\sim w(\varepsilon_1+\varepsilon_2)$.} & (36)\cr
 }
}
dALEE, MOZHNO BEZ TRUDA DOKAZATX, CHTO
\EQ{
 \displaylinesno{
   \hbox{$\abs{u-v}\le \varepsilon\abs{u}$ I~$\abs{u-v}\le \varepsilon\abs{v}$ VLEKUT~$u\approx v(\varepsilon)$,} & (37)\cr
   \hbox{$\abs{u-v}\le\varepsilon\abs{u}$ ILI~$\abs{u-v}\le\varepsilon\abs{v}$  VLEKUT~$u\sim v(\varepsilon)$}    & (38)\cr
 }
}
I, OBRATNO, DLYA \emph{NORMALIZOVANNYH} CHISEL~$u$ I~$v$ PRI~$\varepsilon<1$ 
\EQ{
 \displaylinesno{
   \hbox{$u\approx v(\varepsilon)$ VLECHET~$\abs{u-v}\le b\varepsilon\abs{u}$ I~$\abs{u-v}\le b\varepsilon\abs{v}$;} & (39)\cr
   \hbox{$u\sim v(\varepsilon)$  VLECHET~$\abs{u-v}\le b\varepsilon\abs{u}$ ILI~$\abs{u-v}\le b\varepsilon\abs{v}$.} & (40)\cr
 }
}

v KACHESTVE PRIMERA VSEH |TIH SOOTNOSHENIJ IMEEM [KAK I V~\eqref[22]]
\EQ{
  \eqalign{
%% PROVERITX, PRAVILXNO LI POSTAVLEN PERVYJ \abs
    \abs{(u\otimes v) \otimes w - u\otimes (v\otimes w)}&=\abs{u\otimes (v\otimes w)}\cdot \abs{{(1+\delta_1)(1+\delta_2)\over (1+\delta_3)(1+\delta_4)}-1}\le \cr
                                                        &\le {2b^{1-p}\over \left(1-{1\over 2}b^{1-p}\right)^2}\abs{u\otimes(v\otimes w)},\cr
  }
}
I TO ZHE SAMOE NERAVENSTVO VYPOLNYAETSYA, ESLI POMENYATX 
MESTAMI~$(u\otimes v) \otimes w$ I~$u\otimes (v\otimes w)$. sLEDOVATELXNO, 
VVIDU~\eqref[27], SPRAVEDLIVO SOOTNOSHENIE
\EQ[41]{
%% CHTO TAKOE |TOT (\varepsilon)? zNAK PROPUSHCHEN.
  (u\otimes v) \otimes w \approx u \otimes (v\otimes w) (\varepsilon)
}
DLYA~$\varepsilon\ge 2b^{1-p}/(1-b^{1-p})^2$. nAPRIMER, PRI~$b=10$ I~$p=8$ 
MOZHNO VZYATX~$\varepsilon=0.00000021$. tAKIM OBRAZOM, U NAS IMEETSYA 
DOVOLXNO HOROSHIJ "ZAKON ASSOCIATIVNOSTI".

%% $\sim$ => $\sim$,
sOOTNOSHENIYA~$\prec$, $\sim$, $\succ$ I~$\approx$ POLEZNY DLYA CHISLENNYH 
ALGORITMOV, I PO|TOMU RAZUMNA IDEYA OBESPECHENIYA evm PROGRAMMAMI SRAVNENIYA 
CHISEL S PLAVAYUSHCHEJ TOCHKOJ NARYADU S PROGRAMMAMI VYPOLNENIYA NAD NIMI 
ARIFMETICHESKIH DEJSTVIJ.

%% 250

tEPERX VNOVX PEREKLYUCHIM NASHE VNIMANIE NA VOPROS O NAHOZHDENII 
\emph{TOCHNYH} SOOTNOSHENIJ, KOTORYM UDOVLETVORYAYUT OPERACII NAD VELICHINAMI S 
PLAVAYUSHCHEJ TOCHKOJ. iNTERESNO OTMETITX, CHTO SLOZHENIE I VYCHITANIE TAKIH 
VELICHIN NE POLNOSTXYU VYPADAYUT IZ POLYA ZRENIYA AKSIOMATIKI, TAK KAK ONI 
UDOVLETVORYAYUT NETRIVIALXNYM TOZHDESTVAM, FORMULIRUEMYM V TEOREMAH~A I~B.

\proclaim tEOREMA~A. pUSTX~$u$ I~$v$---NORMALIZOVANNYE CHISLA S PLAVAYUSHCHEJ TOCHKOJ. 
tOGDA
\EQ[42]{
  ((u\oplus v)\ominus u)+((u\oplus v)\ominus ((u\oplus v)\ominus u))=u\oplus v,
}
%% PRI USLOVII CHTO => PRI USLOVII, CHTO
PRI USLOVII, CHTO NE PROISHODIT PEREPOLNENIYA ILI ISCHEZNOVENIYA POKAZATELYA.

\emph{zAMECHANIE.} eTO DOVOLXNO GROMOZDKOE TOZHDESTVO MOZHNO PEREPISATX V 
SLEDUYUSHCHEM BOLEE PROSTOM VIDE. pOLOZHIM
\EQ[43]{
  \twocoleqalign{
     u'&=(u\oplus v)\ominus v,  & v' &=(u\oplus v)\ominus u ;\cr
    u''&=(u\oplus v)\ominus v', & v''&=(u\oplus v)\ominus u'.\cr
  }
}
iNTUITIVNO YASNO, CHTO $u'$ I~$u''$ DOLZHNY BYTX PRIBLIZHENIYAMI K~$u$, A~$v'$ 
I~$v''$---PRIBLIZHENIYAMI K~$v$. tEOREMA~A UTVERZHDAET, CHTO
\EQ[44]{
  u\oplus v = u'+v'' = u''+v'.
}
eTO BOLEE SILXNOE UTVERZHDENIE, NEZHELI TOZHDESTVO
\EQ[45]{
  u\oplus v = u' \oplus v'' = u'' \oplus v',
}
YAVLYAYUSHCHEESYA ESHCHE ODNIM SLEDSTVIEM TEOREMY~A (SM.~UPR.~12).

\proof vVIDU SIMMETRICHNOSTI NASHIH PREDPOLOZHENIJ DOSTATOCHNO USTANOVITX 
SPRAVEDLIVOSTX RAVENSTV~\eqref[44] PRI USLOVII~$u\ge\abs{v}$. v POSLEDUYUSHCHEM 
DOKAZATELXSTVE UDOBNO BUDET ISPOLXZOVATX SOKRASHCHENIYA
\EQ[46]{
  d=e_u-e_v\ge 0, \qquad w=u\oplus v \ge 0
}
I RABOTATX S CELYMI CHISLAMI VMESTO DROBEJ; ESLI MALAYA LATINSKAYA 
BUKVA~$x$ OBOZNACHAET NORMALIZOVANNUYU VELICHINU~$(e_x, f_x)$ S PLAVAYUSHCHEJ 
TOCHKOJ, TO SOOTVETSTVUYUSHCHAYA ZAGLAVNAYA BUKVA~$X$ BUDET OBOZNACHATX 
CHISLO~$b^{p+e_x-e_v}f_x$. v CHASTNOSTI, $U=b^{p+d}f_u$,  $V=b^p f_v$;  |TI 
VELICHINY, RAVNO KAK I~$U'$, $V'$, $U''$, $V''$ I~$W$, YAVLYAYUTSYA CELYMI 
CHISLAMI, DESYATICHNYE TOCHKI KOTORYH VYROVNENY TAKIM OBRAZOM, CHTO~\eqref[44] 
|KVIVALENTNO
\EQ[47]{
  W=U'+V''=U''+V'.
}
tEPERX DOKAZATELXSTVO SVODITSYA K DOVOLXNO SKUCHNOMU RASSMOTRENIYU RYADA 
CHASTNYH SLUCHAEV.

%% 251

\bye