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TECHNICAl NOTE NO. 1113 FEBRUARY 1957 .

....• SUMMARY DISCUSSION ON PERFORMING BINARY MUlTIPLICATION WITH

THE FEWEST POSS.I BlE ADDITIONS

. George W. Reitwiesner

TECHNT' 'L L!BRAR1 . Al!XBR-LB (Bldg. 3051 -"'-. . . .$HliDEEN. PROVING GRO\IliD • II:D • •-

Deportment of the Army Project No .• 580306002 Ordnance Resec;nch and Development Project No. TB3..C007

BALLISTIC RESEARCH LABORATORIES.

ABERDEEN PROVING GROUND, MARYLAND

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. SUMWUIT DISCUSSIO~~ · Oil FERFORMl:Im :Bil't-'IRY MULTIPLICATIOlq · '·liTH· THE. FEHEST POSSIBI.ll: ADDIT".CONS

George W. Rei·Mesner

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TECHNICAL LIBRARY AMXBR-LB (Bldg. 305)· ABERDEEn PB,OVING GROUND, . MD. 21000

· Department of the Amy Project No.5BQ306002 Or@e.nc:e Research and Developmen'G PrOject Ho. 'fB3-0007 . . ' .

ABERDEEN PROVING GROUN~ MARYLA.ND

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. ' . . SllilMJ\RY DISC""<JSSION . . Oit PEnFOPJ.JLNG B:ill!\R"l N'JL'ffPL.tCATION ·· ~liTH THE FmJES'J! ross:mr;ro: J\DDI'l'I0~13

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; . Under conventional biila.ry multiplication procedures an. e.G:l:i.'.;ii;m (OJ:,

. equiva.lent:cy, a subtraction) is perf'ol'il!ild ~Or each non-zm.'O ditr~'~ ·~ ·;;lle: · . . .. ·'~'· ·'

· _wll.tipller or its absoJ.ute value, and the statisticelJ.y e:;:pec'Ged nUiill.'iilr· of'

. additions pe:~.· multiplica:tionJ.s one-halZ the nu.1iber of these C..'.g-J:ts. . . . . '

. This dincuas:l.on develops .Boolean f'ullctiona f'or.the recurziV<< den.!li:i;ion

of substitu.te sets o'f: multiplier digits; for. vhich tll.e nUlllbers o-?· uon··>.aJ.·os

' ax:e .:i.n-eclucible wi'Gh statistic~ ex;pec·ted va~ues ve1.7 near cue-·t;!:~.:hll ·&he

itJmoor of digits ~~Mcll express the si$lled muitipliel) a.r!d applies ·<;her.e

· functions to the tl:u-ee 1motixi: i.j'ii13i,ffeyresentat1on8: 2°s co=lc;a~nt .. l•s .. ~ . - - . _. . . - '· . : •. ; ~: .- . i·· .- . .-·-. '. - . . . •

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~(i , . . , . T.his lJ.O'~ s).!llml!lrlzea orie'fJ.y e. for.na.l . study ~~hicb llli$ . beguu e~r~y ·iri ·' ·

. Jlil~ 1l~57 ··.alld·.·.for ,"<rb±oh' ·i;lfu ·. dl.~e.Z:t' 6-z, a 'i'oi~~ ~;poi~t ~rns l1

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This no'ca is es~en·~i:il.;l.iy a kaf."G o:l,'· a

cue-<JJW.:L"teJ.• hour d<ll"ation 1~~tch 1957 meatizi3 of tll.a Associs!G:I.ro

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,.,., ·. C<)nvel,l'liiona.Jiy1 b'ikcy !ll';,J;tiplication i'ii ped'orm:~d by tecursively . ,. . . .. !' . ''~ .d,. I·· ~xecutuO. chii:t:> of' an :l.nterm...~a.te prodV.ci, ·. :in'cerspel"se:l.·.;i'~):l eO.d:ciic.:~s or ·.·. · i :,J

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.'(\+ .suh~~-s.ct1oil3 •. ~e choice be:j;~leen lihet1ler or no'.- an sddi tion or ct~iY~r.;.c:i;ioli · . !.: ,Jcf

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"'. ;> :. · !1-lld ·sUbtracting is goveJ."ried bY ccnsid.Eii'a.tionsc o.f' ::!=1:-el' 'L-epi:csent:i::.iq:l o.aa. ·, • ·. - ~ . • . :< .. ~ ' ~ r 11 . :1 .. mul.:(i~lier sign;. . . : ' "'( ..

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. \ This cl.iscuss:tori brief'Ji. suim:::-i--:tzl?:s a rigqrous and JilDre. 1ep;;'~by a::'(>\'Jo.b:at

:~ :w-hibh -a_el'iv~s-_ fo3:. eacli. ot-: tb.e· ·tiU:·ca_ !m0t-?n xap_rosentlitioliS for .. o.~nf.,ry de:~ . ' ' ·c2.•s\~~li:im~nt;. ·l's-·cor.nle"!~nt, and nazm_tuil.e i-rl·l;h app~c1ed si.~1)' D. set or

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· '\ .. · · .1. repllj.~em=t'·i:m.1.-l;iplim• digi:Gt:: \rhich .a.dlllit. .. the three :values. -1, o, =d 'I'll.. and .. 1

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' for which the numper Of ncm-zeros is fr:reducibli:. :~ : .. -/'

.···:[ , . , Iimr.e<lia~ely a;Jjal·.mt is t'.ie application .to performing 'bh'rrj. m<.D:G::'::J:>Uca.ticn

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not coljcerned'1rith 'r<!d.Ucing ~ .numl:ler of r.3q11U--ed sM·f"t;s; in~r'lTit-h ~fis(li~ .. I . - . ,;. . . . .. . . . . . ::. ·. .. "·. . . . .·. . . . ... ;, in·terpla.y of shii'ti.ilg ·ac'tiQll <iit!l aao.i tiOll (!ad sui:r&rac-l;ion,: ' ~ ,;. ;~. t· ' '.~ .. : . '.'' . ' . .. ' .. -.·~~.

•: (; •' .. iiu.ov.gooll-~; this a.isct~s:ton the ·tbree-valued die;its &-e>refei"l-ed ~ as'·/ 1ff ·; "~oeffi~ients" tc>;dis)t~l:lh.~emf.r&t ~-valued 1'dig:tts", ;~:· .' ,,

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. ·\i -,. [hat; .tile coemcierl-l;s' admit 'th.ree ·Va.J.:Ues izltroduces~ PEn: Set no 'iei'lo\lC

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. .. · , ci~oign probler.1, ·for the s.Ssoeia:l>iOl:l of sign •:ith e:ny coe:fftCierit, . . . . . '

' ,;w;ttfre mUltiplier itself; cotiespoilds siq>l~/ :to !;j. ,selilcti(!n betTieen adc:-;,b,e .. ·. . . . . . .. . ". . . ·. ·••imd sti.btrac'~:Lng during the mul:tipl.ication 'pioces~.

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'Fheilce t.he;_hi@!Ci!~t i1!:1.d~ed dii3,L'c a:rrd the two bigucst. inileAe(J, co~icflct~;nts are', . ·· .'• : .. ,:;i~· .1. .r- :):~. :_:,.-,. ·:.: .• -r . , . .. : .. : · ,-. · · · ·.- · -;.; · ·r

·.· rcilt1•ic'~ed in 'i;heir Vll~UcS as·. sho;:;i. '. l, : . i ' . . '

Sillc·c the: setS Of co~ffic:t~to -~rhieh. d.ef:tc.e :tuq .ir . .1t.~c;ers oJ~ eq~..U. Ji~Sni-' .

tl.lcl.e end op~osite s:te£ are J;e:i.e.t<:O. to eo.ch o·Gher si;:wl;i b/ th-3 ilwc,;:sj . .;,ll of . • ' < ·; ' ' •• -'I ·, .•

-,the :s;l8US Qi, the ... ~J.On-z~:ro's-, -~- iosS' ct ~e..11::l~Q.i:t.ty ct. O.Jjpi1¢ai:!~o~.1 .is ot:casiouec1

.by restr:tct:ins the .;_r(lli=nt fu~·ctany to non-negxt:!.ve :(C1llti:pl:l.cr<:; ~me' since.

the lfmit~~ int'.ex n is EJJ:bi ti-~J.r.Y. ·i;he l"azJ.ge ~strtc·ticna also oc6>.s:l."n no

.. :":: loss:·~:9~ -~~rs~ity. '·-.,~.

J ;..c~~l!'c:>.tcly, for each of·.~e 'Ghree rep1·e~~hte.uo~s, ·~ho. limi·'-L•z L!uclc n . - .;;·:·

'· will ._'be related to the %lumber oZ digi.'~s by which the sie;aed r.:uJ:~:1.pliel' is

· ', .o.ei':l.ned, ··SJia ·t;he 8.;~ wU~ be.'rfua.ted to the non-sien mul:ti:plier digltf;; u:tth . . : . '· .. . . - "·· .

'!;he e.I>P,areni vacancy ·.in, ihe di,'Sit a21

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. 1 pp:.t~~illir,';·~;sgcl,' AJ.6· ·!. Rli), <sli~tmic,or?~l~!~~o~ctJ · o. ~c:z~.-~iyc coc.7f'ici<mt• ,:

.:_:· . ---/':!~bC ·p:t0~d~.i1:c~;~ is. -#~iy- ths:~· of rCJ]-~9v~~Y: :.\].)pl.Yi1.1g·_·~~e:7_itTc.~_;.-c:~.~~:;~. -~:l ___ #;;a.e_ 1

1', '1m1est iu:a(;,.~"'d. Cl!t.l.in oi' ttp 'W ;uore conse~tp';i~e .non-zeros 'h,1j1 i~ Fo lo!~ger

:( c~m 'b~ · D.DpJ.ied. .._ .... ,

<cpp'J.1-etl. "to th<:. cl1s.in • 1. i' 1. :Y':!.eld:log b2 = :1 anCi. o4 = +l; ne:rc'G it is applied to th<:! ciwO::.n ~' !?,0 , .•::l.eJ_d:!.Jlg

1 .. b = -t and ·b-- = -r1 · :it. ;i_s~ si:f?D.~i'ic~_.:t?-t :'l~J.lO.t b9 t.hen co:d:~i.t.~e~ ··.-rl:tp s-,10 :o~-t.a ell .•. 7 . . •.. . . . 9 _, . to· :form the. ;a~r'~. Cij~{:n a~ected.J e.u4 hcire -t-h~ identity l·evcii-oes ·~l::.c si;_:,;;;:. cit

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:ty.nctions by >Jhich the 'o •s are r.ecursi vely def.i.:oed l.n teJ:ms of·

~r~sbentiall;y'·cilll.ytne a's. :rue qi.lar<titie~ :~;_and/ a,re ·tro.ns'itiou ·9""'"'""'te•·s ·· :\: .. · ' ; ~ '- . -·'. ·_ . .' . '. . ' . . ' ·. '. .. .'' . ~-- - ' ' ,. _-·-.,. . . . . . . ·'· ' . .· . ' . ' ' ' '· . .i.~hicli ail.rru;t, the :,y~tiesc0 .:-i\id .l, Si;!d 1 cormuellCine.;;li:!:tll ·the· initial COOJ,'litl:i.Oll:l · : ) .. ~ · -::··· .- · · :. • -. ·.. ,.-; ·:: _ : : _ "·r- f,;.. .._ . . ___ . .- :" r·~ . ·" . ';.: -· -_.~;: -. -' _ · · _ -_, . · :· _ · .__ i ·. , . - -: •. _ · · ·-.- --. ·· :·:· . · · · ·

· · 'ahown,'ccq.rz::Y ,tlie lrecursion ,t!:froucn ·· COllS<;!tilt:i.vely incre:ising1. valti(!s of ·(;Jle.' · ·

·· ,: fn~~x/J; · i~ue~d, . 1 : i~ th~ ·absolute va~~ o; i;h~ .do:t:ricie~t b , , 3ncl. t~e . J .. . .. I . . . . . ' .. o .

"sign o:f' :tj is f'~:ed by "the rs:cto•· (l • ~J+l) .. ··The s.'~ipUlation, an+r ·'" 0

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;;• ';,.·

'.;· ",<.

. 7. 8 9

10 11

.·- ; . .· . :' ~i~ .... .. '

=·· 000110 .· oobll1

=•· 001000 ~ ·: 00100l. = ~.001010

'·• 0010J_1 .. 12,· '=· ··001100 13. = ~ . ooJ.lo1 ·

' 14 { . ·. 001110 15 . = 001111 .

'16 . ~.;; . 010000. .17 = . 010001

. 18 = . . 010010

. 19 =/ ' 0100ll 201, . ~;.·:-: 010100 21 ... i=-;. 010101 22 .... ,,, 010110'

· , 23 = OlOlll: .. 24 = .. 011000. : 25 ,;;'- .. 011.001 : 26 ~=; :\, Oll010 27 "' . ' 011011 2§ ... :i' oilloo 29 , .. '.i'l• .; OlllO:l.

. J;O · o. ·011110

.31 =·. ·onill

:_;

. ,

: '

.. -.-.

-~·-.. ·.,,_ .

;;o_; I

.,,_ -;~{~;: <

.;: .-.. .;·. ,,. • .• ~,< .

'·!' .:

., I . ~.

.:·

·(.

~"-

.- .. ; ,., (·--

-f·. -·

·-:~-:

_•.;;

1 i

·-:·,·· '1

.,·

' '

'·

. ~1 '' .• 1·

:L '1

~l~ ' 1. ''·l:

i .• l

, .

1 1

'•.\ ... ~

.-1 -:J. -1 -,1 -1

·:.' . ~-

-.1 . ·:1:'

,_, .

.. -1· 1

, .. -1

, .... _. .. .. .. _·y J•

;•,·

.. :-

..... \

·, ~

::

!~' ' ';-·

: .- :·- -:~:

''.

.. ,_

· .......

~- . --~-

:r· ·f .

,_.

-.,_. . . ·:~s_;.

._. .. ~

'·',

··.r •;, -;..-·

"·'·.

. i. '

'r· · l

. '

~ . !

. -~-i

·I I

i . , .. ,._. I ·'·"' '. ' ~-- . •' .

'· ··,::- \

: -t .-::" I . ;. . ~- -! J :·.~.'j ·:

: .. I

•·' f .··1 !

I

., ;_:r~.;·=

I

I -~- ~

·····.: .. {11

;!,

; . . ( . .. ~--

. .. . .

''·· ,.·;, .,.,

. ,. i 1

I 1 I

' '

,.

•

.-

··'

,. ' '

. .'

"

'

1:: ...

. )

.

.

·'·

., ·. ;:

'.v. ·,. ;>: ~! :i· -., t.

··;· .. '· ··"

:.,- . i ,_,-.:

Iu. concrate illustration· ·chel·~ is e"'1ibitei:l in Fi($. 4, for tba :Jo.l'"l::!.cule:t

~~se n .;= 5,_. th~ enti~·e fe_.m.iiy- ti·f-:·25 ~ 52 -G~ts o'£ o~.s Hh_ich c:0l"l'C8J!0~!:1 t6 the

25 "' 32:pos:iible s~to' of, a's. . . ·f .

:;:r,. . :.•·)- -~-.'~::~:~t~~~- _·'_·_·.;_ '.: ·: __ : .. ·. <~- .. -. ·. · ... ~ _ .. ' .·: :: --. ,· ... _ .. ··:·· I. -~~e~.~J.~l\}A~~ _.}~p~r 4~:~ -~~ :j~~1·'?. :;,.f:t.gu1·~~: "~~f.i~-~-t~ · ·ttii.e · s~f:Y~ i~+t~?r.~:-a-";;j:9~ .. 'for

. D.\=''~; '._;tl:{G u~pdi~ :~iw.r:ter~- :for ri~!,jj. C:ll!d So Jn .. Ana ·);>y: 8-ll:-0..6~,: -~he ·f:l.&~':e· is ; ·' ·, -~;- . . . . . . ' . . . . . . ' ' . . . ' . . . (

~~teris:!,ble in ·'Ghe othe1·. d:trec',;ion to ,e.ccctilaqd.a.te M.gher: val\i.<>s c:i: n. ·-,. . _·':. . : . . ' . . . .

a.re d.~~incC~:·;t?~ a.lt;0r.c.e:,~ing digi:i;s 0 e.tir1 1 ar1:c:ag~ tb.~;, a ts, The nwrib,ers· Hj these. arrays defib.~ the. tr!ilica:i;ed b:l,ll-3.ry fractiuns )../5 alJ.d 2/3 >?!;ell' .,,h:e radiX

....

poin·i:.. i~ loca''l;ed .; . ,, . .'

?d~~:C0.'Q'~ -i~ . ~ ~J~·.' ...

Fo~ n = 5' the coefficierrt b" has uon-ze:;,-o value for o.nd ollly i~ol· th<val.ues

of'. the ~nteeer. A5 trhic:h lie betH~en H5

and 25 -l; !'l.ud ·~he coefficiaut ,b4 has . ' _, ~ . _, ' , . "' t' ' ' h'.' '. ' ' non-zero v1..1....1.ue .:t.Cl' and.. OJ.LJ~Y fol."· ·i:i.ne va_ue.s .. o"- . ne. i_n.;eger .L~ .. 5 .~'- J..cn ~:te t)!j'Gt·teen

H4 Emd• H5

• Except ~y t6r· si~ changes, the table Npe&:i;-6 • itsel'~ in hdves,

•:J.Uarters, etc. i'ol· .the succeeding J.ot·ier inde:ted ·. b ~ s, and th.o~·efol·c ·i;Le _l1L1;;1ber . . . . .

of non-zeros for _each of' the b' s io e:;pressible . as a fl.IIlctio:O: of t];;;, cor:7C··

spondingly indexed.H's. . t .•

A ~ol•:;,•espom1i~. argUment applies for any n •. · -~ . . ' . . .

' ~'roD\; the v~ue~ of' .~he numbel"3 Hj (a:pp~·ox~:i;ely

radi.'t point i.s loc~'ce! adjacent,to .. <;} 1 the n=b;r of'

'the b's is close to .. 2, /3.,

-.

' ..

.•·.

·' ·;.>'

.;.

··'

.,,

. '

·:·. '•.

,.

9 ;.

; ..

l/5 and 2/5 \!hen ~;he non-zeros foJ... i;G.C.h Of

·.•·

,; ..

····l,

:.;.

. ;·

. . .

"j, . r '

: f :!·

l i I " --~._,....j i' ... ·•·' _,, __ ,,

• ,<"

··'.

.. , : r

.·. _ _.,

:'·

. , { i

!.f.'. . -~

fi-, !,;_:

i:·.,-:·

~ ;:,

... · '·· 1.

/i

-;_.·· _,,.

• i' .. -·· : ,;

5 .the r'~:act niimb~ of. u6~'-:zero co~fficients·'toi; any ,-:~P;;;·_.,. . '.,:r: . .

is .eXhibH~ci as: P. arid P .·J· n

.-,

..• .. ,

or

.t ·:;. :. If' ~s.ch olgit :a. ~soilmas : thei ''\-ali.ce; 0 '=i · 1 1d th indepeti'dent. p:co·beb,5.ii ty

-r;;.- _. '.· - ':.<!• l.' -.-- •, ·;:·~.- - ,;. "-. ---;· . .

. , J.'f?;,, th~n ~a.ch o:f' ,th~, :2n posGt~1: ~comb1.~U9:ns o:f: v<!J.ue.> G.lll.OU!3 the q.'s. ?"curs

; Vi." •. · .tli i>ro. :bali. :t.l;i<--· 2ifl .. ,·· '~.lie. ~~c. e i~<'r,o ... J.io!;s' 'i='l"'~'·t!ie' ;,vne""'"'d 'il.ti.n.f.oer .. s o. ;;;:;;on. ··:4ero " VJ --~~---·:i ;··;.,,,.·""'';"-~?'~-\.~..,_ __

;~~s.,;lo'se iit~iCies .a~ i~ss tllaJJ. 'c;i• eCJ.uai. tQ a :part.icule.;.· :i.Ude:c; j 61: :a, are , div~n · l>;y E . ~n~ jjj ' 1.p t~~~ t{.,;~~;~;. ·

( .. . ·.· , J n .,, :

·.:

__ ,, ·-. j

i .. ·. I

. ,: ·•

-·,i.:

'•I·

I ., ,I·.

:.,!

. .... _. '_;,

-,;

i.-- •

.. -.. ~ ..

• !•

. \. ·-,. 'f;. . -,j_-;;, ;•.\·

*,~~~--~---~~~~~~~--~~--------~---~ ;P j = (2n d -r)J .2n-:+:~)~:l ·' J~: . I

, ,, .. ~ ~1?~?.{6 .. , ,. .. 1 · ,,: • .·I.e,, ;.' • .<; , i

·. ~ \ -' . '- . . . .

=·· .. <.~+i)/3 + (l + (-l)j.2-q+l))/9

. .,

... --~

= ·~n-cl) /"5 + (1 ; ( -1)11 .2-(~+l)J/.9 -2 .. (n+l}'

.. ,

~/F~·'~-g-·._~·~5-·~~··~''+·----~~~·~·~--~~---"~--------~--------~'-· ____ I •• , _;- 'l ,-,,

·:-_,

_-... ~-- .. .--.:-,:

'·:· .

10 ,_.·· ._loi.

--.-~-,~--

·,.

•!. ,<,-.

.) ,-,/;.:.

';<

.·'.,': -~-

' · ... ,

.. ~ 4'- ·: ·. ., II ·I "

... : ~.; ~-) .,

' ·~

,_._

)'

' .. ' i· ! . i I l

i; i

.....

'. \'

., ·v .

. ~-

-:. ·' .. . j-_:·.

I •;

.. f

~- '

'

j.: .,

"

" '

1 ' 1

i ,,

·, '

; _i

i l.

1•·. ,j,

~--

i ,. i ! i

I l ' ' ;I-

i .-

.:> ;:<•

:1 r:

:.i, 1;,·

. ;._ .

;j.''

· .. :,-,

..

"

',•

:: .,..

· ...

)

'~-: ....

.. T<;n-ra.ra ._3.pi)lY·il?-g the·· l19oJ:&~.~~ · :t.lm·::ti6ns .. to tho

·.tliera ;:xe_ eJ¢.ibit~~- ~ ~~~~~- 6;:-~ilie -e.e:i:'tni·(.·~oliS ·~~cl. rri.n£~~ ~~-.. s1er~~~D. irr~crte:~·s X~·; tln:'ca r.u.u~b,.~i'· ?.~-:::::~:~~::;s~r{~~y~~-1~-~s :--··: ,~. ... ·.

. : • ..< ,· t :·_,_· . '· ' .· ..

i>f1 ,_ aild XA i?r_./~he t~ee r~~ti?c' .. l.iiye:_1~-~pl";:~~l?-ts.:~ions: ·..:.. r s co~•;- ... 1 ,_ . . :r_,._. -(· 1 ; e .:;;;._ ~:·.'-"····'-'.:.J.-..., u

--.-/'com.9leJp.ent~_ •·rind nl.~~ftlli"1~ 'l·ii~h·: i{p}j;.~:·.l~:~d .. siin. ·. ~... . '\'· . .. \:• ' ,,_ . ·' ' . . · .. ·. - ,': ' . ". •' -· . . ~,_

r.. . In-~- thrGe::·CaS.es ~~h¢· _·j.~:~~G~t .. ~. }: in. ;~t8f~~~- .bi·. cil:. e.:t-6·~-~-l't:!.:i~ .n1:;i:iczr~Ji ot

~j:~itS, :·;i.· \ih:tcih_>~~t -~c4~. ~~~~f--~:~l~e::; 'i~dicF.Li~i~ _th~-- ·sigil.- ~f _.tua:··h:rt~·.SGJ.~·: ·:

0 ?.:ad. 1~ ilith ·t.he· li.~[~11~·:$-G·. ii:.<'l..cc.E-0. · (.:i.i:L·t

'than' zero' :i.f 1·. The·: ~i[9J. ·qf~_-tt~ ··in .. ~e~~t< ~~-el"o ,:Ls

i~o: tr~e~~~~~~et1: r_a~~~~~~·t.D.ti~iiS·-~--.:

Zr-.·,..o -tr·· -0-- · ·~~~A ·:~--~t g"·--..n··-:e· ; .. _ '--_ . • f ""':"''"'' ........ .. a.;; •• , .. -.

.;, .;It reaiil.inS '.;o apply the :foregoing argu;n~:o:i;e: by relating ll to W and 'the

.,

,·~

•."

-~-}.'

;\

·-.•

. ~'

.•·· · .. ' .

. ~:· .

. ... ,__......:.__ ..,..------

I I

I I

I I

'X 1

X 2

= 1

..... ~ .... ,_~ \V-2 · i. ,._ (l-.,2x··'-1l.~;·) 0 x .. 2: ~1Y .!..._;;. _ , ~ _ _:_

= ""~ 1L2,. . _ ~ .>. ·2 i ' 6 0 1.xi · .. ~i-1 · .

/'.':·

·: '21-1-1 =- ~x .• - . · · u-1 ·

';, L·

0 '.-!f .. I XAI I. ', ~~-1

·2. -lx1!

· ;" 1 <:" 0 /-. L:·2:.'" -

. 1•1-1 -2 d!.. X <. ~1:1,-1 - .2

Fig. 6.

,-- . ,;,·. ,-

·-,·· .·

1l

,, ",

, .. ~ ::'·

-, ' I l I I

I I I i I l I I !

' ' I I

! I l

"

~1 -,,_

.t.

. ~ ·-

'!.

·:.:.·y

,,

'!

' .

-l •• ·-·-;;;:--· ----·-·---;--.,.--..,.-;--.-, . -----------·--·:"":::---....,...- ---,.... .. -.-·-. --- ---·-------' "''("

.--;;;

,_ ..

;.; :1.'-1

.f,''

,_): ....

t: '

·."'

··~

._.-.

·• '

·--·' ·":. ;-.

1.··

;._

-.' ~

. -~)---

'. ~!:; -~; , ..

~-.;;;_·.,.--"----.____~.,....;~.-~--'--~-

i ", -~

~ _'. ·;.=.

. ' ··' • .~, . .I

.·!''S ··· .. {:.·.~~ ... ·, -"~'-

_,._

·' 'l

,; ;·-

,. '·.

~--_

~~r~-~ .. .. ;.:·. '';-~\;

·-.· •·:: . -~

·::

,.

·."

I.

~-' ,.

·.I'

,. l

,, __

" :;,""~

!f:'

··'

x·. ;.2

= (1

=

./ __ ·r = (i

·!.

=

=

Fig. 7.

.. :

-,:._

;-_

i

2x. ) •.. 2·· ......... ·. ·.·0,{-2 ·I' ··~-1.;

)' ~-

.... - i ....... 2 J.·

·-f t-. . ~:,: :.\ . . , ... ~e.~~--- ~:_ :·:·~.:...:-...

.:;· . :;

2:) . . ' . . ,l-1,.

·._,

-~.:-

H-2 6

. . .

......... ·,~1~2 ·. }· 0 ~:.-

~ .·:

.. ,_

--~

__ ;o··

.•

·.--... ·#

-;- '•

: >-· --~ .;,. '.~-

.'. 'f-..

·'/"

:,_ ··'·~:

-:,:

. :,-::-.;. '_-,

( .. '

. -~

"'· '

. f.

n. ··~ 1-1

. i"-.

\.

·.·,_

_,_._

. '

_; ..

·;;'

.. ;_

·-._·· .. ·

~ .. -. ·_

;:

.. . .. ,. ·· ..

.·.·-

-;.;-. :: ·;;

'

··.-:

. '• . '---;

. ~-. ~---

~.-.! --- ... ,. I

~~;. ;

·t· 1 ' r. U f·:' II

;. r--:··!_.,:

.,

! i ! !

i

·.;: ;''o '.j

1 ,'I

,._ ~--. ·.,-

·.·-~-~-"-'~{ ·:·"'

. -~. {·., .. '··· ... -;,:

. ' )' .<; ' ;. ; ·. ',,,, .. _; r/ ·, .. : ,9_ , L7~: { . .,, ~

: 1,:-~_li;,:\''" .r £b.-, F:i.~;>'ii'r 'c~e in·~eger d~i~n::t:i:o~v.i a.re'· .:~ph1•ased i;o fon.l~ 1:::-.-o trc.c·i;ablil ·' ; '

' ·. I,; •tor 'ch~ -appltbo.tioD. of the Boolean furic'~ions.. Fol' the l •.s .. C01',;'0)..GJ::l"·'~ case ;~~· ·· .. L·1t( '·~ -~~~-.~l."e~~;e~-~~d~::~a_·}~ ~ii;ned. ·me.gp:itUC:e·, :.:~.nd.':for_ til~ 2~s -comp~eht::n-~ C'-'iSe X~--_.:i~:-

. · · _·:_ : · . ·t,_,-: ·._,·. _. ·k•·' ;.} - ·: ~ _ ._,:." _ · ... :.. _' . . , , . , . -..... ! -

i i .:.· \>• . , . ~:eexJ.)r,~c:se(i'to plAce 'che. sign dfgitjbo·l;;l:l:cilisi~e.';e:ad Oj.itsid.e of the r.u.;;.:v:·.t1.c•:l. · ~,:.';.·······,·' ... ~.·.:.~:~\~ .... :.-.J~; .. ,:,.;_.~'.·.·~.:I,' .•. ,~-- .·' ·•"i"' .' ... ·-'o•; .. , .. ,,·,< ?'•·: ''.';'': . - ,,\ +, ',•" . -~- OJ .. , < ' ' • •• ,- -' ' • • '

1

~ .. ,_1

••. ;\. · '·\:{ >:: ', ;~~?,; if~:;·i·~Y~~Gt6nsh~~~s~~~<i~~[~~· -·~· e.~~-· i;-::a:r~f ~hc~ni} . · .. r~ .. is. m~;ci,ih~ni .. ' .' ·: ·. . . ;,, , 'f sia;rlfi.cs,nt .tha(G ri is J.'elat,iVG:l;Y: one l"d.c;be:i' fOl~' ·~lJ.e 2's ·co;npli?l!'2!l·~ cc;,se '.;llE.'-\

·· f'~r ·thE> oti1er : ty1o '~ll,ses •. · ' .· l: ·· ·' ·,

. ' l. :In S.U .Co."sea ~the. a ~a are ifefiXle~i.- a_~ ·Ghe reapect.:r.ve coefficients Of ~i- in

· th~ · s~~·t;;j~o~s ·-s~ov~ (~b.~ .. i·~p~-~~~-~~- .s~i~~~:Cio~1~-- for bo·Sll .c-~illglCx~j~-~ l~~rn:'$'S~:t:;ta:~;i<!1l~) · /· t ' -~ :. ' '· ,

\':

J. i· ' 'i< I '·!f

r.· ; :'--:· ;

··:~- .

J''-~1·.·,

.1.

' t. ' ' ·!

1. ' -~ .

Fori thJ' ~~t\lde and ~iliti'' ;:-ep;?eserr~ation the !ll'll.tiplice:t.ion l1:i.'OCeti,lh'<• ' :;. . . . . ·~ : ' . . . . .

· must forin the :product of•tt-~o p0si·~·lve fa~tcrr:; .e.nd app~d the.· e.pproprj.o:i;e .s1.(;-n.

:The sign-Ot \he-prOdUct ·l.s .tri;/-;i.eily determ'ine{t ... from .. ~lie si.~s of the :?i.Ct<•; ... s_-> ,_ ·-~ -.. :' . '!'· ·'_ :_-·. . . < .. _-. )~- ' -.··,:_; . . ~

and the1:'efore ·the entire fo1•egoing argu!"..etit a.ppl.ies dil•Gdl:,y fol.· this ra-: p:rasente.tiori.. · ... · . , .. ·. ' '{ 1i . . . . . :

. . . '' . . .;ij. •f •' . . . ' . :b . ·- . . . . . . . ·.

• F~r. 'both\ comp+ement ,,repre6(;)ntaticnG, ,ho;,ever, cartu'i.n aii.jw:r~m.cnts are . . '~ .. ··.- .. . . ,;-_ .. _. ; _.' ' . . . . : ~--- .. :.; ··.' . /~ . . . . '

·neCess·aa-,_ iil~.:-i:;p.e·--.. l3Q9lean. t\lnctiOns bacau.;.-;~ th~ sign o:f' ·i:.b.e ntit?.e);:·ical. data 1-. :~~f-;·--.-. ·-.1-.· -·~:----_ ... :._; _; ~J~-- ·_ . ·-r~-.. !-

pa.r-:;ic:t:i?e:'ces mare .il;i.tirGS.tely. in. the COi@Utati.Ol"..a.i. yrocedures,.. , ._. ' . . ? . . • - ,:': . . .• ) -~ ' ,, . '.; ~·-

. l,f J?b';;,the''!'s con,pJ.e!r.imt reP:tesen'GEd.;:i.on, the multiplier ho.s been rqolu::asecl' . .. ~·~·

:;as a· si~ed 'mag.a.itude; 't-rh~c'? ~i4e·· e_ni:.i:. .. r~ tO~egoiug ~g~a.-~If~_-_:becoutS!s cJ.i.x·e~~y:

. ~J?plieif).ble lThen th~,Bo~l~~ ~1

t:!.on t~hj.cb sfi;ixe.s the signs of the 1>'s tp . ·, . -- .. _· : .. _ ,-. :- ;, \'. •' . ··'.>,: adjilstCq..,to s.ccommod.-'"-'~e the sign. o.f. ·~b.e limn:i.p!ier. ·· . . .,-; ! . • • "~~ ~ • -. .' , \,. ;~f. I;~· .• __ ., .-. > w·-

.· .. :,·· Fo];- tb.e.'2is cc~'PJ.e~nt,.re:Pteseu:ri:.ation. the 'Germ -~_1 .2 in tl:t">

.·.

definitio!;l is lilOln.entarilY.· disrci~arded, and 'Ghe b•'s e.:.:e dete1~li:l.ned for tl\e .. 'l' '.

in'cee;er \;bich i·s a.e:t':tned by the. :;r,enuining sumi:0.iion. Conseq'!.entl.y one ,.,o"'"'

.-cbe:rrici,ie~·c ij'~ is d~·~gl--min~d t~ithet·e vrar~_.orig1_nally d'igii;s_ :-:,. and the'

~oei':fid.~nt ~. is associ~t~ 11i.tu the s!.~a po,rer (n) of 2 as ~s the momcntc.l·ily ·.. ' . . .. ·.;t . .· . . . . . . . . . .

di'sl'C(!,"O.i'ded dia;i t r1~~1 . . T'arQugh p. sl:l.g.lJ.t read.justmen:(; in tb.e :Boolean fu.uct.ion

'~hicb. es_-te.blishes 'cl:ie· sign of a 'par-~icu.lar one .of ·~he b's, ·i;b.e coelficient b . . . . . . ll

is· fo1·ced to ha're ·the same ve.J.ue as, aud therefore to cancel, ·l;he J2011 recor.-

, sid_ered digit Y11_1 . 'l"ae number :~f re1n.ainii;ls b•~:i' is therefore u = l-1: \rhich, ·

indeed, is eq_Ji8.,l to the 9riginal, nUl.li!Jer of d:i.gi t$ x .. . ' . l.

. For ell t:b.l•ce r::~~re~entati9ns, then, the seiu" numbel; of bQs al'O developed. ·:.•

., '· .• ·, ;J

,. ::+·

'i, ' . '(, :f· ·--~-·····-:r

.: ... -~.:.

' ., c.; :., .....

\·

'. J

....... ;c '·\

····: ,\'

. : .. t .. ,· ·.'·. '.:·.,

-~

,. .,~· .

;·:-:

'<·'

.'!' '•

._, ·-

:•

;--

' ' .. 1·

··\;'

' . . : ~~-

. ~-- '. B.re· e:;chlbiteii ' - I •

. . ~-"-· '";

··: ... :'---. ·ii '\~ ,­

•\ . ~ ' , .

t.~~e -o• s .

: P,.~e. ~r.b-reas~ti ~ .. ac~sively riS -~·t.:~O:{)..S o.Z,: the. k? s fo:.• oll -tl"'X3·e _ri..>.mibe-r·

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