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Anormal Systems of Numeration Author(s): A. J. Kempner Source: The American Mathematical Monthly, Vol. 43, No. 10 (Dec., 1936), pp. 610-617 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2300532 . Accessed: 30/11/2014 08:15 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. http://www.jstor.org This content downloaded from 149.150.51.237 on Sun, 30 Nov 2014 08:15:50 AM All use subject to JSTOR Terms and Conditions

Anormal Systems of Numeration

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Anormal Systems of NumerationAuthor(s): A. J. KempnerSource: The American Mathematical Monthly, Vol. 43, No. 10 (Dec., 1936), pp. 610-617Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2300532 .

Accessed: 30/11/2014 08:15

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

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Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access toThe American Mathematical Monthly.

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610 ANORMAL SYSTEMS OF NUMERATION [December,

ANORMAL SYSTEMS OF NUMERATION*

By A. J. KEMPNER, University of Colorado

I. We admit as base of a number system any real number a> 1, not an integer.

By following exactly the method of procedure for the representation of numbers when the base a is an integer > 1, we are led to the following

ALGORITHM: For a base a> 1, not integral, we represent a number c >0 by the following chain of operations:

an+l > c > an;

C = ana + o3 0 < an [a], 0 O< On < a n = an-,a na + n-17 0 ? an-1 -< [a], 0? O_n-1 < a nX

* . . . . . . . . . . . . . . . . . . . . . . . . . . .

31= aoa? + o, 0 < ?ao < [a], 0 < go < a,

Oo = a-ia-1 + j5_ 0 < a-, < [a], < ,-i < a-',

_-1 = a-2a-2 + I3_2, 0 < O a-2 _ [a], O < 0-2 < a-2

* . . . . . . . . . . . . . . . . . . . . . . . . . . .

This process may terminate (if fi =0, all subsequent O's and a's are 0), or it may continue indefinitely. The representation is called canonical, and is designated by

C = aCn .. ao!a-a-2 ' (a) (can),

with or without the subscript (a).

Thus 2=1.(3/2)+1/2, 1/2=0.(3/2)0+1/2, 1/2=0 (3/2)-'+1/2, 1/2=1 *(3/2)-2+ 1/18, 1/18=0 (3/2)-3+1/18, , 1/18=0.(3/2)-7+1/18, 1/18=1 *(3/2) -8 +217/(2.3 8) , etc.; and hence

2 = 10.01000001 ... (3/2)(can).

A negative number is characterized by prefixing a minus sign.t

THEOREM: Every real number can be represented canonically to any base a > 1, and this canonical representation is unique.

The proof of this theorem is obvious. For an integral base a > 1 the canonical representation is the only one possi-

ble if the digits are restricted to 0 - ai <a, except for the ambiguity represented * Presented to the American Mathematical Society, Ann Arbor, September, 1935. This

paper is the answer to a question raised by one of my students, Joy Gilder, who wanted to know whether the number e could be used as the base of a number system.

t The numbers 0 and 1 are clearly excluded as bases. Positive numbers less than 1 and nega- tive numbers may be used as bases with slight modifications of the process and suitable restrictions on the set of digits employed.

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1936] ANORMAL SYSTEMS OF NUMERATION 611

by 1 =.999 . We shall now show that for a non-integral base a> 1 the restriction of the digits to this range is in no sense sufficient to secure unique- ness. This is illustrated by

2 = (3/2)-' + (3/2)-2 + = 111 (3/2)

which differs from the canonical representation given above; or by

1 (3/2)=(can),

= 1. (3/2)-l + 1 * (3/2)-3 + 1 (3/2)-9 ? = .101000001 . . . (3/2) .

In order to have a representation of c to base a > 1, non-integral, it is clear that the monotonically decreasing sequence On, nl*, * must approach 0, i.e.,

lim 1n-k = 0; k-boo

(this may include the case where all but a finite number of ,B are zero). In the di- visions involved in 2 = 10.010 (3/2) (can), 2 = .111 . (3/2), 1 = .1010 * (3/2),

this condition is satisfied. But if we attempt to represent 2 by starting .01 . (3/2), the divisions can be carried out, and the A form a decreasing sequence, yet they do not approach 0. This is obvious, since the largest number which can be represented to base 3/2 starting .01 . is

(3/2)-2 + (3,/2)-3 + * = .0111 . .. = 4/3 < 2.

It is therefore not permissible to say that we obtain non-canonical representa- tions of c to the base a by omitting in our algorithm the third column. Instead, remembering that

[a](ar-l + ar-2 + . ) [a] [a a-1

we have the

THEOREM: Necessary and sufficient for the representation of c>O to the base a>1, non-integral, is

C = anan + On , 0 < a -< [a], 0 ? 13n < [a]a n/(a - 1),

gn = an-lan-1 + ,B 1 0 _ a\n-1 _ [a], O < OBn-1 -< [a]a n l/(a -1

. *** * * * * * . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . ... . . . . . . . . *****.

go = a-,a-, + B-1, 0 < a-, _ [a], 0-< A1 < [a]a-1/(a- 1) * . * * * * * * * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . ....... . *.******

If in addition the / satisfy the stronger inequalities in the third column of the algorithm, the representation is canonical.

The notation

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612 ANORMAL SYSTEMS OF NUMERATION [December,

C = anan-1 . . . ao * a.la-2 . . . (a)

indicates that the representation may be either canonical or non-canonical. If a- = a-2 = . =0, we say that c is represented as a "whole number" to the base a (in distinction to ordinary "integers"). Finite decimals, infinite decimals, periodic decimals, pure and mixed, are defined in the obvious manner. The numerical illustrations already given show: a whole number to the base a is not necessarily an integer; an integer is not necessarily a whole number; a whole number may equal an infinite decimal (1(3/2) =.10100 . . . (3/2)); the sum of two whole numbers is not necessarily a whole number (1(3/2)+1(3/2) = 10.010 . . . (3/2)); the product of two whole numbers is not necessarily a whole number [(5/2) - (5/2) = 11(3/2) 11(3/2) = 10001.00001 * * * (3/2)]. On ac- count of these properties, the notion of a "whole number" cannot be funda- mental in such number systems, since it possesses no closure properties. However, it was obvious from the start that only the representation of the individual number in the system, and the abstract structure of the system, can interest us, not its use for arithmetic.

From the definition of canonical representation it follows immediately that: (a) If a.,a,,-, * ao-a . . . is canonical, any displacement of the decimal

point gives a canonical representation of the new number; and (b) If a,n?a-l . . . ao0a0 . . . is canonical, all numbers obtained by omitting

the first X digits from the left, X = 1, 2, , are canonical. The canonical form

C = anan-1 * aO a1-i2 ... (a) (can)

is characterized as follows: For finite decimals or "whole numbers,"

a.-la1 + an-2an2 + + ara < an

a?-2a 2 + * + a,rar < an-1

. . . . . . . . . . . . . . . . . . .

arar < ar+i

where a, is the last non-vanishing digit (left to right). This finite chain of inequalities, each involving a finite number of terms, is replaced in the case of an infinite decimal by the infinite set of inequalities, each involving an infinite number of terms:

an _a n1 + an-2an-2 + < an

an-2a n2 + ... < an-I

Therefore, for a whole number or a finite decimal to the base a, a finite number of steps suffices to decide whether a given representation is canonical or not. For an infinite decimal the decision generally offers a difficult problem.

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1936] ANORMAL SYSTEMS OF NUMERATION 613

However, some special results for a > 1 are very simple.

THEOREM: Given a number to the base a in canonicalform,

C = ? nacn-1 ... * * a 101-2 * (can),

and the same number in some non-canonical form,

C = 3m1m-1 * **00*,-1/3-2 ...

then n ?m. If n =m, then a. j n; if n =m, an =On then a?n-, >-iOn; etc.

THEOREM: If cl > C2, and

Cl= ?an?in-1 ... aO*CY-la-2 ... (can) and

C2 = 8m,m-1... * 0*, -1 -I2 ... (can),

then the same relations as in the preceding theorem hold between the a and 3. In other words, canonical numbers in increasing order of magnitude are lexicographi- cally arranged to the base a.

The root of the ambiguities in our representations lies in the following

LEMMA: For a> 1, not integral, there exist a positive integer k and k + 1 in- tegral coefficients axi, 0 < ai < [a], such that

anan + an-an-,1 + ...+ cankan-k > an+1.

Proof: We will show that if we take for each a its greatest possible value [a], the required inequality will hold for a sufficiently large k.

Since [a]>a-1, we have

an+1 [a] = [a](an + + +a'+ a-2 + .)> an+'.

a1

Hence, for k sufficiently large,

[a]an + [a]an-I + * + [a]an-k > an+l.

For an integral base a, the greatest permissible digit is not greater than a - 1 but equal to a - 1, leading to the (trivial) ambiguities in the representation of numbers in such systems (.999 * * = 1, etc.). For a > 1, not integral, an unend- ing repetition of the greatest digit [a] cannot occur in a number in canonical form. For example,

. 111 (3/2)(non-can) = 10. 10100(3/2)(can) = 2.

The following two properties (to which Professor W. B. Carver calls my at- tention) are easily verified:

(1) Any representation an * * *o* a-1a-2 * * * is canonical if it contains no digit >a -1, i.e., if it does not contain the greatest digit [a].

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614 ANORMAL SYSTEMS OF NUMERATION [December,

(2) Any representation containing the digit [a] is canonical if it is made up of fewer than

log [a] -log { [a] -a + 1} log a

digits.

II. a= 3/2. As representative of the case a rational, we carry out in some detail the examination of the special case a = 3/2. All types of anormality occur already in this illustration, which is particularly simple to handle, because of the small numbers in numerator and denominator of a, and because the digits are restricted to 0, 1.

For simplicity of notation, but as is immediately clear, without real loss of generality, we consider fractions (to base 3/2) starting 1. -.. A shift of decimal point corresponds to multiplication throughout the whole process by some posi- tive or negative integral power of the base, and causes no difficulty.

We see that a canonical representation, base 3/2, cannot start with 1.1 (and therefore also not with .11 , 11. , etc.) This follows immediately from

1 + 1 (3/2)-1 = 5/3 > 3/2.

Therefore a combination . . . 11 . . . may not occur anywhere in a canonical representation to base 3/2.

Consider next 1.0 as first two digits of a canonical representation. Is it possible to continue 1.01? This is satisfactory (1+0 (3/2)-1+1 (3/2)-1 = 13/9 < 3/2), so that we have no conflict with the canonical character. Assume therefore as first three digits 101 . - . Then 1011 impossible, on account of the sequence 11. Try 10101 . ; now

1 + O.(3/2)-1 + 1.(3/2)-2 + 0.(3/2)-3 + 1.(3/2) -4 > 3/2,

so that this sequence may not occur, either at the beginning or anywhere else in the decimal representation. In the same way 101001, 1010001, 10100001 are all excluded, and 101000001 is the first permissible sequence starting with 101.

We start over again with 1001, which is permissible (1+8/27<3/2); 10011 is impossible; 100101 possible.

These considerations may be continued, but they can lead only to neces- sary, not sufficient, conditions for a representation to be canonical. Hence:

LEMMA: Necessary but not sufficient conditions for a number to be in canonical form to the base 3/2 are, reading digits from left to right:

1. there must be at least one 0 between two consecutive 1's, 2. the sequence 101 must be followed at the right by at least five O's: 10100000

. 1 t 0 m ; 3. the combination 1001 must be followed at the right by at least one 0: 10010

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1936] ANORMAL SYSTEMS OF NUMERATION 615

According to this lemma, 1001001 * may or may not occur in a canonical representation: it actually may, since

1 27 1 + (3/2)-3 + (3/2)-6 + . . . = -(3/2)- - < 32.

1 32)3 19

On the other hand, 1001010000100 is not canonical. If we ask what relation to the base 3/2 corresponds to 1 =.99 * to the

base 10, we find

1 = (3/2)-l + (3/2)-3 + (3/2)-8 + * = .101000010 ... (3/2),

infinite and non-canonical. It seems likely that every real number #0 admits, to base a> 1, non-inte-

gral, an infinite number of distinct non-canonical representations.

III. a =.u/v, u prime to v.

THEOREM: A necessary condition for a fraction c = r/s, r prime to s, to admit a periodic representation to the base u/v (canonical or otherwise) is that all prime factors of s must be factors of uv(uX -v"), where X is the number of digits in the period.

Proof: Let

c = an * ao a-i a-p+ia-p * a-p-A+1 * (u/v)

ax = a,,al + * + a-p+la-P+l + (a-pa-P + * + a_p_x+la-P-xl). )

a -1

or r ug uX Vr

= E: Aa-+ EaC-r - t s Va u x - vX UT

with the summations taken over appropriate finite ranges for a and r. Of course no irrational number can be periodic to a rational base a, canonical

or otherwise; and the last formula gives the class of rational numbers repre- sentable in at least one way (canonical or otherwise) in periodic form. To actu- ally determine the numbers periodic in canonical form is again a difficult problem. (On the other hand, we know that an integer or a rational number may lead to an infinite non-periodic canonical representation.) In terms of the algorithm, a necessary and sufficient condition for periodicity of a canonical representation (period of X digits starting with a,,) is

#-P = #3p_xax.

This follows immediately from the algorithm.

IV. a irrational. Let a = sllk (irrational), s> 1.

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616 ANORMAL SYSTEMS OF NUMERATION [December,

C = Cn 0 * a!'C-1 . .'

= OenSnlIk + in_ls(n l)/,lk + + ao + aj_ISlk +

0 < ai < [Sllk].

Multiplication by sylk is of course accomplished by shifting the decimal point 'y units to the right; but addition or multiplication of two numbers to base S/k is generally not obvious.

Making use of ak = s, the representation to base sllk (canonical or otherwise) corresponds to the (always possible) representation of c > 0 in the form

C = fo(s) + f1(s).sl/k + * * * + fk_l(S).S(k-l)/k

where fi(s) is a polynomial, or power series with a finite number of positive and a finite or infinite number of negative exponents, all with non-negative coefficients ? [sllk]. The canonical representation of a number to base a, a a quadratic irrationality, proceeds in obvious fashion. With a = N/5, for example, we have

1/2 = 1.5-1/2 + (1/2 - 5-1/2)

= 1.5-1/2 + 05-1 + (1/2 -5-1/2)

= 1.5-1/2 + 0. 5-1 + 0.5-3/2 + (1/2 - 5-112)

= 1.5-1/2 + 0. 5-1 + 0.5-31/2 + 1.5-2 + (23/50 - 5-1/2)

= 1. 5-1/2 + 0. 5-1 + 0. 5-312 + 1. 5-2 + 0. 5-5/2 + 1. 5-3 + ...

= .100101 ... (./,) (can) .

For a given irrational base necessary conditions for a representation to be canonical may be derived as for a = 3/2.

For example, for the base a = /5, consider the representation

C = a-la_2 * . 4

Try a-1=2, a-2=2, i.e., c =.22 - * * . Since 2.5-1/2+2.5-1>1, the sequence 22 cannot occur in a canonical representation. Similarly 21 impossible; 20 of course possible; 10, 11, 12, 00, 01, 02 all possible. The following sequences of three digits cannot occur: 222, 221, 220, 212, 211, 210, 202, 122. This can of course be continued to take care of sequences of any assigned length, but can clearly lead only to necessary conditions for the representation to be canonical to base V5.

Consider the case where the base a is a root of the quadratic equation

aoa2 + ala + a2 = 0,

ai integers and D = al - 4a oa2 a non-square; so that

a = h + kV\D, h, k rational.

Numbers c which can in at least one way be represented as whole numbers (canonical or otherwise) are

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1936] ANORMAL SYSTEMS OF NUMERATION 617

C = an(h + k-VD)n + an (h + kV/D)n-l + al(h + kVD) + ao 0 < ai < [a].

Since the rational part of (h+ kV/D) i and the part involving -/D are respectively

(h + kV\D) + (h-kV\D) i (h + k\ID)i - ( h-kV\D)i - ~~~~~~and -A

2 2

the numbers are c =A +BVD,

1 n

A, B = - ai[(h + kV\D)i + (h -kVD)i], O-< i < [a]. 2 i-o

Numbers permitting at least one representation as finite or periodic decimals are similarly characterized. For an algebraic number of order n >2, we may make use of the reduction

an= (alan-I + a2an-2 + + an)/ao,

without however obtaining results of particular interest.

V. a transcendental. The same methods lead to the same general type of re- sults as in the other cases. Certain (denumerable) classes of transcendental numbers connected rationally with the given base a are obtained for the totalities of numbers representable as periodic decimals, in at least one way:

c = E aiai + al - aiai, periodic decimals,

with summations taken over appropriate ranges. Thus, to the base e, the peri- odic decimal

ex -1 at-1 ... a-x ... (e) = aiei, 0 < ai < 2.

e" - 1*X

Nothing is known concerning the class of numbers permitting canonical peri- odic representation except the trivial fact that it must be a sub-class of the less restricted set.

VI. Occurrence of digits in canonical representations. For a = 3/2, with the two possible digits 0, 1, we found that the sequence 11

is excluded, that 101 must be followed by five zeros at least, that 1001 may be followed by 01. It is clear that we must have in every infinite canonical repre- sentation to base 3/2 a preponderance of O's, and that a maximum percentage of l's will occur in .1001001 . . (can), leading to the maximum density of l's as compared with O's of 1:2.

For a base a >2, the largest digit [a] seems to play in infinite canonical representations a role similar to that of the digit 1, to the base 3/2.

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