\input style , 1941~. , , . 1946~.\ .~.~Ḑ, X.~X.~㎋ ~.~~텉 ~$2$ [.\ John von~Neumann, Collected Works, Vol.~5, 41--65]. .  .~ᓕ " ?"\note{1}% { , .---{\sl . .\/}} [{\sl CACM,\/} {\bf 2} (December, 1959), 3--11]. ⛗ ~\MIX, , , , . 荒 , \MIX- , , , ,--- . 蒀, , . (瀑 , , "" , .~7; . ~4.5.2B.) 茅 \emph{} , . . ᓄ ~\MIX{} ; 10~ , : \EQ[2]{ -12345\,67890. } \dfn{ }. 򀊎 , . ⎇ , , , ; . , , , %% 212 \dfn{ }. ⛗~$1$ ~$00000\,00000$, ~$99999\,99999$; , , \emph{ ~$10^{10}$}. ~$-12345\,67890$ : \EQ[3]{ 87654\,32110. } , ~$5$, $6$, $7$, $8$ ~$9$, ~\eqref[3], , ~$+87654\,32110$.  " ". ㋀ , ~$10$; , ~$-11=\ldots{}99989$ ~$\ldots{}99998=-2$ ( , ~"$9$"). ~$x$, , ~$\floor{x/10}$ , ~$x$ .  ; , $p$~, $500\ldots{}0$, $p\hbox{-}$ . 򀊈 , , (~$x$ ~$-x$) . 噅 , ,--- \emph{ }. ~$-12345\,67890$ \EQ[4]{ 87654\,32109. } ꀆ ~$-x$ ~$9$ ~$x$. 텒 , ; ~$10^{10}-1$, , (.~.~3.2.1.1). 񍎂 , ~$99999\,99999$ ~$00000\,00000$ . 򎋜 , ~$10$, ~$2$, \emph{ }, \emph{} \emph{ }. ~\MIX{} %% 213 , ; , , . ᎋ , , . ; , , , , - . 퀏, ~\MIX{} , , ; . 녃 , ~$b$ ~$b^k$: \EQ[5]{ (\ldots a_3 a_2 a_1 a_0.a_{-1}a_{-2}\ldots)_b= (\ldots A_3 A_2 A_1 A_0.A_{-1} A_{-2}\ldots)_{b^k}, } \EQ{ A_j=(a_{kj+k-1}\ldots a_{kj+1}a_{kj})_b; } .~.~8. 򀊈 , " " , , . 茅 , $b\hbox{-}$ , . 퀏, ~$(-10)$, \EQ{ \eqalign{ (\ldots a_3 a_2 a_1 a_0 . a_{-1} a_{-2} \ldots)_{-10}&=\cr &=\ldots+a_3(-10)^3+a_2(-10)^2+a_1(-10)^1+a_0+\ldots=\cr &=\ldots-1000a_3+100a_2-10a_1+a_0-{1\over 10}a_{-1}+{1\over 100}a_{-2}-\ldots\,.\cr } } 焅, , ~$a_k$ ~$0\le a_k \le 9$. ~$12345\,67890$ "-" \EQ[6]{ (1\,93755\,73910)_{-10}, } ~$10305070900-9070503010$. 荒 , , ~$-12345\,67890$, \EQ[7]{ (28466\,48290)_{-10}, } %% 214 \emph{ , , ~$-10$ .} 񈑒 , -, .~ ~.~  [{\sl Bulletin de l'Academie Polonaise des Sciences,\/} Classe~III, {\bf 5} (1957), 233--236; S\'erie des sciences techniques, {\bf 7} (1959), 713--721] .~󝉄 [{\sl IRE Transactions,\/} {\bf EC-6} (1957), 123]. 䀋 {\sl IEEE Transactions\/} [{\bf EC-12} (1963), 274--276] %% fixed: (May, (1967) ---> (May, 1967) {\sl Computer Design\/} [{\bf 6} (May, 1967), 52--63]. (茅 , .) .~.~񎍃, .~.~, ~$-2$ ( , 1956~.). 񈑒 1955~.\ .~.~ꍓ , " " ; --- . , , .~.~~ᐸ [{\sl Publ. Math. Debrecen,\/} {\bf 1} (1950), 232--242, .~.~240], . ⛁ ~$2i$ , "-" ( ""\note{1}% { "": "quaternary"---"quater-imaginary".---{\sl . .\/}}, \emph{ $0$, $1$, $2$ ~$3$, , , )}. [.~{\sl CACM,\/} {\bf 3}, (1960), 245--247.] 퀏, %% !!! . , , %% - \EQ{ (11210.31)_{2i}=1\cdot 16+1\cdot (-8i)+2\cdot (-4)+1\cdot (2i)+3\cdot\left(-{1\over2}i\right)+1\left(-{1\over4}\right)=7{3\over4}-7{1\over2}i. } ~$(a_{2n}\ldots{} a_1 a_0.a_{-1}\ldots{} a_{-2k})_{2i}$ \EQ{ (a_{2n}\ldots a_2 a_0 . a_{-2}\ldots{} a_{-2k})_{-4}+2i(a_{2n-1}\ldots a_3 a_1.a_{-1}\ldots{} a_{-2k+1})_{-4}, } - "-" . 荒 , %% 215 . 퀏, , , " ": ~$4$, ~$4$ ""~$-1$ , , ~$4$ ""~$+1$ . , : \ctable{ \strut$#$\bskip&&\bskip$#$\bskip\cr & & & & 1 & 2 & 2 & 3 & 1 & \quad [9-10i]\hfil\cr & & & & 1 & 2 & 2 & 3 & 1 & \quad [9-10i]\hfil\cr \multispan{4} & \multispan{5}\hrulefill\cr & & & & 1 & 2 & 2 & 3 & 1 \cr 1 & 0 & 3 & 2 & 0 & 2 & 1 & 3\cr & & 1 & 3 & 0 & 2 & 2\cr & 1 & 3 & 0 & 2 & 2\cr 1 & 2 & 2 & 3 & 1\cr \multispan{9}\hrulefill\cr 0 & 2 & 1 & 3 & 3 & 3 & 1 & 2 & 1 & \quad [-19-180i]\hfil\cr } 쎆 ~$\sqrt{2}i$, ~$0$ ~$1$, ~$i$ . "ል" , ~$i-1$, .~ [{\sl JACM,\/} {\bf 12} (1965), 247--248]: \EQ{ (\ldots{}a_4a_3a_2a_1a_0.a_{-1}\ldots)_{i-1} =\ldots-4a_4+(2+2i)a_3-2ia_2+(i-1)a_1+a_0-{1\over 2}(i+1)a_{-1}+\ldots\,. } ~$0$ ~$1$.  , , ~$S$, .~1\note{1}% { ~$S$ ~$i-1$ " ", .~. 瀌 䎍 ꍓ (., , .~.~‑ .~.~㓒 2- "ꂀ" ~1970~., 36--46).---{\sl . .\/}}; , ~$\sum_{k\ge 1}a_k(i-1)^{-k}$ $a_1$, $a_2$, $a_3$,~\dots{} . , ~$S$ 256~, ~$(1/16)S$; %% 216 , ~$S$ ~$135^\circ$, , , ~$(1/\sqrt{2})S$ (~$(i-1)S=S\cup (S+1)$). \picture{.~1. 썎~$S$.} , $S$~ , .~.~18. ( ~$S$ ""; .~1 .) \def\ternary{\bgroup\catcode`\!=\active} \def\endternary{\egroup} \catcode`!=\active \def!{\overline{1}} \catcode`\!=12 ᛒ , \emph{ } --- ~$3$, ~$0$, $1$, $2$ ""\note{1}% { "".---{\sl . .\/}} $-1$, $0$, $+1$. 呋 %% 217 ~$-1$ ~\ternary$!$\endternary, , : \ternary\ctable{ \strut\hfil$#$&#&$#$\hfil&\hfil$\quad#$&$#$\hfil\cr \multispan{3} 󐀂 & \multispan{2} 䅑 \cr \multispan{3} & \multispan{2} \cr 10!& & & 8\cr 11!0&.&!! & 32& {5\over 9}\cr !!10&.&11 & -32&{5\over 9}\cr !!10&.& & -33\cr 0&.&11111\ldots & &{1\over 2}\cr }\endternary 󐀂 : {\medskip\narrower \item{a)} ~$1$ ~\ternary$!$\endternary. \item{b)}獀 , , , , , . \item{c)} (.~.\ , ). \medskip} 񊋀 , \ternary\ctable{ \strut\hfil$#$\bskip&&\bskip\hfil$#$\bskip\cr ! & ! & ! & ! & ! & ! & ! & ! & ! & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \cr ! & ! & ! & 0 & 0 & 0 & 1 & 1 & 1 & ! & ! & ! & 0 & 0 & 0 & 1 & 1 & 1 & ! & ! & ! & 0 & 0 & 0 & 1 & 1 & 1 \cr ! & 0 & 1 & ! & 0 & 1 & ! & 0 & 1 & ! & 0 & 1 & ! & 0 & 1 & ! & 0 & 1 & ! & 0 & 1 & ! & 0 & 1 & ! & 0 & 1 \cr \noalign{\hrule} !0 & !1 & ! & !1 & ! & 0 & ! & 0 & 1 & !1 & ! & 0 & ! & 0 & 1 & 0 & 1 & 1! & ! & 0 & 1 & 0 & 1! & 1 & 1! & 10 \cr }\endternary ( --- .) ⛗ , , ; , : \ternary\ctable{ $#$\bskip&&\bskip$#$\bskip\cr & & &1&!&0&!& \quad [17]\cr & & &1&!&0&!& \quad [17]\cr & & & \multispan{4}\hrulefill\cr & & &!&1&0&1\cr &!&1&0&1&0\cr 1&!&0&!\cr \multispan{7}\hrulefill\cr 0&1&1&!&!&0&1& \quad [289]\cr }\endternary .~.~4.3.1-31.  , %% 218 ; , \EQ{ 208.3=(21\,201.022002200220\ldots)_3. } ( , , , .~4.4-12.) 䀋 ~$\ldots{}11111.11111\ldots{}$; \EQ{ (\ldots{}11111210012.210121012101\ldots)_3. } 퀊, ~$\ldots{}11111.11111\ldots{}$, ; \ternary\EQ[8]{ 208.3=(10!!01.10!010!010!0\ldots)_3. }\endternary , , "", ~$\ldots{}11111.11111\ldots{}$ .  , " ဘ ", 􈁎 , ဘ . [.\ W.~Ahrens, Mathematische Unterhaltungen und Spiele, {\bf 1}, Leipzig, Teubner, 1910, \S~3.4.]  䆎 녑 [The philosophy of arithmetic, Edinburgh, 1817; .~.~33--34, 54, 64--65, 117, 150] .~ꎘ [{\sl Comptes Rendus,\/} {\bf 11} (1840), 789--798], , ~$5\times 5$. "" 녎 뀋 [{\sl Comptes Rendus,\/} {\bf 11} (1840), 903--905], . 뀋 , 쓐 1945--1946~.\ ; . 񋎆 , , , $\ln 2/\ln 3\approx 63\%$~ , .  .\ {\sl AMM\/} [{\bf 57} (1950), 90--93] "High-speed computing devices" [Engineering Research Associates, McGraw-Hill, 1950, 287--289]. %% 219 , , ( "-" "--"\note{1}% {Flip---, flap---, flop--- (\emph{.}); flip-flop--- .---{\sl . .\/}}). 䐓 --- \emph{ ,} ( \emph{ }). 呋 ~$\$ ( $k$~ ), \EQ[9]{ \left[\matrix{ \ldots, & a_3, & a_2, & a_1, & a_0; & a_{-1}, & a_{-2}, & \ldots \cr \ldots, & b_3, & b_2, & b_1, & b_0; & b_{-1}, & b_{-2}, & \ldots \cr } \right]=\ldots+a_3b_2b_1b_0+a_2b_1b_0+a_1b_0+a_0+a_{-1}/b_{-1}+a_{-2}/b_{-1}b_{-2}+\ldots\,. } : $b_0$, $b_1$, $b_2$,~\dots{} , , , , ~$a_k$ ~$0\le a_k < b_k$.  --- \emph{ ,} ~$b_k=k+2$. 葏 , \EQ[10]{ c_n n!+c_{n-1}(n-1)!+\cdots+c_22!+c_1, } ~$0\le c_k \le k$. 񈑒 ; . 퀏, " , 2 , 9 , 22 , 57 492 " \EQ{ \left[\matrix{ 3, & 2, & 9, & 22, & 57; & 492\cr & 7, & 24, & 60, & 60; & 1000\cr }\right]\hbox{ .} } "10 , 6 , " \EQ{ \left[\matrix{ 10, & 6, & 3; & 1\cr & 20, & 12; & 2\cr }\right]\hbox{ .} } , , , , (.~.~4.3.1-9).  %% 220 , . ㅎ ꀍ [{\sl Zeitschrift f\"ur Mathematik und Physik,\/} {\bf 14} (1869), 121--128). 䎏 .~26 ~29.  , , , : (.~1.2.8-35), 􈁎 (.~1.2.8-34); - (.~1.28-35), (.~ 4.3.2), 㐝 (.~7.2.1) (\S~9.1). 텊 , \emph{} , .~ [{\sl Acta Mathematica,\/} Acad. Sci. Hung., {\bf 11} (1960), 401--416]. \excercises \ex[15] ⛐ $-10$, $-9$, $-8$,~\dots, $8$, $9$, $10$ ~$-2$. \rex[24] : (a)~ ( ); (b)~- (~$-2$); (c)~ ; (d)~ ~$b=1/10$. 葏 : (i)~$-49$, (ii)~$-3{1\over7}$ ( ); (iii)~$\pi$ ( ). \ex[20] ⛐~$-49+i$ - . \ex[15] , \MIX- ~|A| , 3- 4- , ~|B|---, 2- 3- . (񀌛 ~1.) ㄅ ~|A| ~|X| $$ \hbox{a)~\mixcode LDA & A MUL & B? \endmixcode } \hbox{b)~\mixcode LDA & A SRAX & 5 DIV & B? \endmixcode } $$ \ex[00] , , . \ex[16] ꀊ $p\hbox{-}$ , (a)~ , (b)~ , (c)~ ? \ex[20] , . 񓙅 \emph{ ,} " "? 񓙅 ? \ex[10] 䎊 ~\eqref[5]. %% 221 \bye