d i are thedigits ofthe number. Usually, ris a

Épreuve de setion européenne

The mosteientinteger base.

Numeralsinvariousbasesmaylookdierent,butthenumberstheyrepresentarethesame. In

deimalnotation,thenumeral19isshorthandforthisexpression:

1 × 10 ¹ + 9 × 10 ⁰

. Likewisethe binary(orbase-2)numeral10011isunderstoodtomean:

1 × 2 ⁴ + 0 × 2 ³ + 0 × 2 ² + 1 × 2 ¹ + 1× 2 ⁰

, whihaddsupto thesamevalue. Sodoestheternary(that is: base-3)version,201. Thegeneral

formulaforanintegerinanypositionalnotationgoeslikethis:

d ^k r ^k + . . . + d 3 r ³ + d 2 r ² + d 1 r ¹ + d 0 r ⁰

^,

where

r

^{isthe baseor radixandthe oeients}

d ⁱ

^{are thedigits ofthe number. Usually,}

r

^{is a}

positiveintegerandthedigitsareintegersin therangefrom

0

to

r − 1

.

To say that all bases represent the same numbers, however, is not to say that all numeri

representations are equally good for all purposes. The ultural preferene for base 10 and the

engineeringadvantagesofbase 2havenothingtodo withanyintrinsi propertiesof thedeimal

and binary numbering systems. Base 3, on the other hand, does havea genuine mathematial

distintioninitsfavor:byoneplausiblemeasure,itoersthemosteonomialwayofrepresenting

numbers.

Inordertominimizetheost ofwritingnumbersinaertainbase,weneedtooptimizesome

joint measure of a number's width (howmany digitsit has) and its depth (how many dierent

symbolsanoupyeahdigitposition). Anobviousstrategyistominimizetheprodutofthese

twoquantities.Inotherwords,if

r

^{istheradixand}

w

^{isthewidthindigits,wewanttominimize}

rw

whileholding

r ^w

^{onstant. Forexample,let'sonsideragainthetaskof}representingallnumbers from 0throughdeimal999,999: inbase 10 thisobviouslyrequiresawidth ofsix digits,so that

rw = 60

; binary does better: 20 binary digits sue to over the same range of numbers, so that

rw = 40

. Butternaryisbetter: theternaryrepresentationhasawidth of13digits, sothat

rw = 39

.

If wewantto solvethe problemin general,it is easiertotreat

r

^and

w

^{asontinuous rather}

than integer variables. Then it turns out that the optimum radix is

e

^{, the base of the natural}

logarithms,withanumerialvalueofabout

2 . 718

. Beause3istheintegerlosestto

e

^,itisalmost

alwaysthemosteonomialintegerradix.

AdaptedfromBrianHayes'sThird-Base,SientiAmerian 89-6,Deember2001.

Questions

1. Whatisthedierenebetweennumeralsandnumbers?

2. Verify that the ternary number 201means the samequantity asthe deimal number19.

Whatwouldbeitsexpressioninbase 4? Inwhihbase woulditbeexpressedas23?

3. Explaintheexpressionsulturalprefereneforbase10;engineeringadvantagesofbase2.

4. Explain theresultsfor

rw

^{inthethree examplesgivenattheendofparagraph3.}

5. General proof: let's onsider

r ^w = a

^{, where}

a

^{is a onstant positive real number. Using}

the natural logarithm

ln

, express

w

^{as a funtion of}

r

^{. Study the variations of funtion}

f : r 7→ rw

^{and deduethat}

e

^{isthe valueof}

r

^{that minimizes}

rw

^{asit is assertedin the}

text.