1 × 3 3 + 2 × 3 2 + 2 × 3 1 + 1 × 3 0, whih is equal to 52. Obviously, in the base-3 (also alled ternary) systemweneed only three digits,

Épreuve de setion européenne

Balaned Ternary

Weusually representnumbersin the base-10positional system, where onehundredtwenty-

one iswritten 139,beauseitisequalto

1 × 10 ² + 3 × 10 ¹ + 9 × 10 ⁰

. Usingthesamepriniple in abase-3system,1221representsthedeimalnumber

1 × 3 ³ + 2 × 3 ² + 2 × 3 ¹ + 1 × 3 ⁰

, whih is equal to 52. Obviously, in the base-3 (also alled ternary) systemweneed only three digits,

namely0,1and2,andithasbeenprovedthatthissystemisalmostalwaysthemosteonomial

one,asfarasomputationaleienyisonerned.

In balaned ternary,eah digitan beanegativeunit (denoted

N

^{), azero(}

0

), orapositive unit(

1

). Theyarebalanedbeausetheyarearrangedsymmetriallyaboutzero.

Asanexample,thedeimalnumber19iswritten

1 N 01

inbalanedternary,andthisnumeral isinterpretedasfollows:

1 × 3 ³ − 1 × 3 ² + 0 × 3 ¹ + 1 × 3 ⁰

orinotherwords

27 − 9 + 0 + 1

. Every number,bothpositiveandnegative,anberepresentedinthissheme,andeahnumberhasonly

onesuhrepresentation. Thebalanedternaryountingsequenebegins: 0,1,

1 N

^,10,11,

1 N N

^.

Going inthe opposite diretion,therstfew negativenumbersare

N

^,

N 1

,

N 0

,

N N

^{. Notethat}

negativevaluesareeasytoreognizebeausetheleadingdigitisalwaysnegative.

What makes balaned ternary so pretty? It is a notation in whih everything seems easy.

Positiveandnegativenumbersareunitedinonesystem,withoutthebotherofseparatesignbits.

Arithmetiisnearlyassimpleasitiswithbinarynumbers;inpartiular,themultipliationtableis

easilybuilt. Additionandsubtrationareessentiallythesameoperation: Justnegateonenumber

andthenadd. Negationitself isalsoeortless: Changeevery

N

^{intoa1,andvieversa.}

InspiredbyGlusker,Hogan&Vass,TheternaryalulatingmahineofThomasFowler,

IEEEAnnals ofthe Historyof Computing,27-3,2005andothersoures.

Questions

1. Explain the balaned ternary ounting sequene and ontinue writing it up to the 10th

numeral. Inthesamemanner,explainthenegativenumbersequeneandwriteittothe8th

negativenumeral.

2. Writethebalanedternarynumber

1 N 10

inbase-10.

3. Writethedeimalnumber52inbalanedternary.

4. Whih propertyofbalanedternaryisespeiallyusefulformoneyomputations?

5. Build the additionand multipliationtables in balaned ternarynumerationsystem(hint:

hekthat

1 + 1

is

1 N

^;

N + N

^is

N 1

).

6. Usethesetables todothefollowingoperations:

1 N 01 + 11 N 0

;and

1 N 11 × 1 N 0

.