Mathematics Magazine, Vol. 31, № 2. (Nov. – Dec., 1957), pp. 98- 110.
PREAMBULA
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Sincerely yours,
Sylvia Bergman
A NUMBER SYSTEM WITH AN
l RRAT l O N A L BASE
George Bergman
The reader i s probably familiar with t h e binary system and t h e decim a l system and probably understands the basis f o r any others of t h a t
type, such as t h e t r i n a r y o r duodecimal. how eve^., 1 have developed a
system t h a t i s based, not on an integer, o r even a r a t i o n a l number, but
on the i r r a t i o n a l number r ( t a u ) , otherwise lmown as the "golden sec/2.)
tion* , approximately 1.618033989 in value, and equal t o ( 1+ ~ ' 2
I n order t o understand t h i s system, one must comprehend two peculia r i t i e s of the number r They are based on t a u ' s d i s t i n c i v e property 1
that
r" = an-1 + r n - 2
.
.
(a. ) Take any approximation (A of r Taking t h e reciprocal, we get a
number ( a 3 t h a t i s proportionately the same distance from P / r a s A I W ~ S
from 7 , but arithmetically nearer. Adding lr2
we g e t a number (A 2) t h a t
i s proportionately nearer r than a1 was t o l / r but arithmetically just
as near. Since a1 i s arithmetically nearer than A A 2 i s nearer i n b o t h
respects t o r than A l e Repeating the process of taking the reciprocal
and adding 1, we approach r Now, taking 1 as A
and expressing our
approximations of 7 ( i . e . A lrA2rA3, e t c . ) as f r a c t i o n s , we g e t
.
Taking e i t h e r the numerators o r the denominators, we g e t what is
know as t h e Fibonacci Series, e a c h t e r m o f w h i c h is formed by adding
the two previous t e ~ m 3;
s for
(We designate t h e nthterm of t h e Fibonacci S e r i e s by f n , s e t t i n g fi= 1,
f 2 = 1. This p r a c t i c e s h a l l be used throughout t h e a r t i c l e . )
(b.) Any i n t e g r a l power of r can be expressed i n t h e form 7 " = AT + B,
where A andB are i n t e g e r s and, i n f a c t , numbers i n t h e Fibonacci Series.
The explanation of t h i s s t a r t l i n g f a c t is r e a l l y rather simple:
MATHEMATICS MAGAZINE
Since
r 1 = IT
and since
r3=
T 2
+
+0
(lr
T 3 =
Ti;
and r 2
+
0)
=
+
17
(1s
+
1,
+ 1)
=
2r
+
1.
I n the same way
I n general
T n= p - 1
+ r
n-2
=(fn~lr+fn-2)+(fn-2r+fn-,)
=
( f n - 1 + fn- 2)'
=
fn'
+ fn- 1
+
( f n -2
+
fn-?)
4
(see Mote
2.)
Can t h i s be applied t o negative powers of T 7 We donst lrrlow any Fxbonacci numbers before 1, but it i s easy ro see how we can f i n d them:
Taking 1 and 1 as our first two, we can see t h a t the term before must.
be 0 , since t h a t i s the only nurnber which, when added t o 1 gives P. En
the same way, the number before t h a t must be 1, since 1 1 s the only nwiber t h a t , &en added t o 0 gives 1; and the next term must be -1, since
no other number gives 0 when added t o 1. Continu~ngt h i s process, we
Obviously. this i s a l t e r n a t e l y
g e t 0 , 1, -1, 2 , - 3 , 5, - 8 , 13, -21..
+ 1 and -1 times the corresponding Fibonacci nunhers, But can t h i s be
proved t o be t r u e i l l a l l cases? I t can by inductioml. The xule we wnnt
t o prove, expressed as an equation, i s :
..
f
Let us assume it t r u e f o r y
the Fibonacci Series:
=
"Y
=
1, 2,
(-1)Y + I f y
...
n. Now b y the b a s i s property of
The inductive proof i s completed by t h e examples already c i t e d .
Applying t h i s t o powers of r , we make a l i s t of them from r
-5
to r 5 :
NUMBER SYSTEM
Now, a t l a s t , we s h a l l get back t o our concept of a system based on r .
Like t h e binary system, it can have only two symbols: 1 and 0. But, un100 = 011 (place t h e
l i k e t h e binary system, it has t h e r u l e ( 2 ) :
decimal point anywhere - i t ' s a general r u l e ) . But how do we f i n d the
numbers? We know t h a t 1 i s r O o r 1.0. Next, looking a t t h e t a b l e of
powers of r , one notices t h a t
r l = IT + O
and
r-2 = - I T + 2 .
Adding them together, one g e t s
Therefore, 2 = 10.0 1 ( i n t h i s system). Of course, because of r u l e ( 3 ,
t h i s can also be expressed a s 1.11,
10.0011, 10.001011, 1.101011,
e t c . , but 10.01 i s what I c a l l t h e simplest form ( t h a t form i n which
there a r e no two 1's i n succession, and which, therefore, cannot be
acted q o n by t h e reverse of r u l e (21, c a l l e d simplification ( 1 1 = 100).
To convert a number t o i t s simplest form, repeatedly simplify t h e l e f t most p a i r of consecutive 1's.
To continue with our U t r a n s l a t i o n " of numbers i n t o t h i s system, we
n o t i c e ( a f t e r a careful examination of t h e t a b l e ) t h a t r = IT + 1 and
= -IT + 2 ,
and adding them together r + r-2 = 3 , and so 3 i s
100.01 i n t h i s system. What about 4? Well, since r + r - 2 = 3, 7 + r -2
+ r ( 101.01) must equal 4, since r O = 1. Can t h i s method of adding 1be
used f o r other numbers? The answer i s "yes"; j u s t convert the number
i n t o t h e form i n which t h e r e i s a zero i n t h e u n i t s column and p l a c e a
1 i n it. I f t h e method of conversion i s not obvious, use t h i s method:
a. Change t o t h e simplest form.
h. If t h e r e
i s , look
cause i t
t o it) ;
i s no 1 i n t h e u n i t s column, you are finished. I f t h e r e
i n the
column ( t h e r e c a n ' t be any i n t h e
column bei s i n i t s simplest form and t h e r e i s a 1 i n t h e column next
i f there i s
C there, expand 7 the 1 i n t o t h e r -1 and r-2
c o l u m n s ( t h a t ' s all); i f t h e r e i s 1, look i n the r -4 column; i f t h e r e
i s a zero t h e r e , expand t h e 1 i n t h e r -2 column i n t o t h e r -3 and r-4
column and t h e 1 i n t h e u n i t s column i n t o t h e r-I and r'2 columns.If
t h e r e i s a 1, look i n t h e r-6 column; i f t h e r e i s a zero, expand t h e
1 i n t h e r -4 column i n t o t h e r -5 and r-6 c o l u m n s , t h e 1 i n t h e r -2
-.
MATHEMATICS MAGAZINE
101
column i n t o t h e r-jand ~ - ~ c s l u r n n and
.
t h e 1 i n t h e u n i t s column
i n t o t h e 7-I and r -2 columns; i f on t h e o t h e r hand t h e r e i s a 1 ,
look i n t h e T - ~column, etc. I f i t i s the endless fraction 1.0 10 10 101 ,
change it t o 10.000000
We can now construct a t a b l e of i n t e g e r s i n t h i s system. Here a r e
those from 0 t o 14 ( i n t h e i r simplest forms)
...
. .. .
5 - 1000.1001
10 - 10POO. 0101
0-0
1-1
6 - 1010.0001
LL - POdOP. 0 101
7'
10000.0001 12 - 100000.10B001
2 - 10.01
8 - 10001.0001
13 - 100010.001001
3 - 100. OP
4 - 101.01
9 - 10010~0101 1 4 - 100100.1P0110
Examples o f the basic processes
-
1. Change 100101.111001 (equals 16) LO t h e simplest form, Tl~c f i r s t
p a i r ( f a r t h e s t l e f t ) i s i n t h e u n i t s and 7-l column, so we simplify i t
i n t o t h e r column, giving 100110.011001, This time t h e p a i r f a r t h e s t
t o t h e l e f t i s t h e one we have j u s t created with our new 1 i n t h e r i
column ( t h e r e s u l t of our s i m p l i f i c a t i o n ) , added t o t h e P already i n
t h e r column. This we simplify i n t o a 1 i n t h e T column, which g i v e s
us 101000.011001. F i n a l l y , we change t h e l a s t remainirlg p a i r ( i n the
7 -2
and T -3 columns) i n t o a 1 i n t h e r-1 column, a r r i v i n g a t our f i n a l
answer: 101000.100001
2. Change 101.01 (4) t o a form with a zero i n the u n i t s column. ( i , e ,
1 can be added). One can see t h a t i t i s already i n i t s
simplest form. However, t h e r e i s a 9 i n che u n i t s column, and we must
remove i t . The f i r s t t h i n g we do i s l w k i n t h e 7 -2 column; sirace t h e r e
exi s a 1 t h e r e , we look i n t h e r-4 column. This i s errpty, and so
pand t h e 1 i n t h e a "2 column i n t o t h e 7
ad T - ~ COIIB~CS,
getting
l01,0011. b w t h a t t h e a
c o l m i s empky, we can ercpand {,he&u n i t int-r.
a form t o which
-'
-'
a p a r i n t h e r-2 and r -1 cogums, gettlrag 100,3411, vJfiich has a
he ~ ~ 1 i tco1e7lEn,
.s
We c m now acid 1 LC, I.!,:
7r3ro
The Arithmetic Oj~ez-cii:~r-rs
'4-132 arithmetical o'perations, although they a r e basi.caiiy t,he s s x ~a s
i n my o t h e r system, a r e , i n pa:ac;tice, q u i t e d i f f e r e n t because of r,he
pe:;uiiarities of t h i s system. As our f i r s t s t e p i.n a l l of t!~enl, vie el.iniinate zeros, which would only hinder u s , =d shoi~tt h e plaice vel.raes o f
X's by act.ural placanent i n cdjlu~ns, For 4r2s*;mce, f o w (101.,0:) wou1.d
be
ilj 13.1 , t h e heavy l i n e represent:i.ag the "deci!na.i. p o i ~ l t i,' The
ne~essi.5.y for this s t e p results frorr, the f a c t t h a t . th3ugh i .the
~sys" .
terns Lo ~;$lich we are aca@r3ust3alrinii,
i;he s : ~eps a$dl,t~.orrare sirip:ie enough
102
NUMBER SYSTEM
t o be performed mentally, t h i s i s not so i n the t a u system; nor i s each
column non-dependent on the one t o t h e l e f t of it. It i s t h u s necessary t o have lined columns i n which t o carry out t h e work.
Now f o r the actual processes, we s h a l l s t a r t with addition. The example
would be represented by
I n t h i s set-up i t can be seen t h a t we have a p a i r , consisting of a 1 i n
the r column and one i n the r 4 column. This we simplify i n t o a 1 i n t h e
r 5 column.
Now, however, we have no obvious way t o continue. We are l e f t with two
1's i n t h e same column. We can n e i t h e r add them together t o give 2 ( a s
we would i n t h e decimal system), nor i s t h e r e any simple '\carrying*
operation. We must, therefore, change t h i s t o a form not having two 1 ' s
i n t h e same column. We will s t a r t by expanding one of t h e 1%i n t h e rf
column:
Now we can simplify the
columns:
and t h e one i n the
7-l
air
we have j u s t created i n t h e r 1 and u n i t s
and r2
columns:
We s h a l l now use t h e same type procedure f o r t h e 1's i n the
We expand one of the- 1% there:
and simplify i n t h e r -4 ard r
-5
T
-
column.
~
I'I
columns:
and express our answer i n ordinary form, writing 1's i n columns with an
un-crossed-out 1 and 0% i n the columns where a l l have been crossed out:
100101.001001 (15)
103
MATHEMATICS MAGAZINE
For g e n e r a l r u l e s as t o procedure, I b e l i e v e t h a t t h e s e w i l l do
i n most cases. (These g e n e r a l r u l e s and t h e ones f o r t h e o t h e r
processes are not the types of r u l e s t h a t , i f disobeyed, w u l d give t h e
wrong answer, but merely guides t o t h e cpickest way t o g e t the r i g h t
one ) :
a) Expand only when t h a t i s the only way t o remve a 1 from the same
column as another, regardless of whether t h i s will r e s u l t i n the same
s i t u a t i o n i n another cclumn, but only i f no more simplificat.ion can be
done.
b) Simplify whenever possible, and, i f there are two o r more p a i r s ,
always simplify the one f a r t h e s t t o t h e l e f t first. Because of this rule,
simplification should never r e s u l t i n two 1's i n t h e same column, i. e.
111111) should be simplified i n t o 111~
A l j h e t h 11
I
1
I bid.
Subtract ion
Subtraction is the next process I s h a l l describe, As i n addition, w e
s e t up t h e numbers i n columns, but here vw: s h a l l assign negative values
t o t h e 1's from the subtrahend. For instance, t o find (11-6) we s e t up
We now "cancel"
the 1 and t h e -1 i n t h e r -4 column, giving
Next, we expand the 1 i n the r
column, g e t t i n g
Pgain, we cancel, t h i s time i n the r
' column,
And again we expand, t h i s time the 1 i n t h e r
column, g e t t i n g
104
N U M B E R SYSTEM
For the t h i r d time we cancel, ( i n t h e
T
3 column) giving
This we t r e a t j u s t a s we would i f we were adding and a r r i v e d a t t h i s
stage; by expanding one of the 1's i n the u n i t s column we g e t
Next w e s i m p l i f y the p a i r i n the u n i t s and r - I columns g e t t i n g
and t h e p a i r we thereby form i n the
T
column g e t t i n g
T
and
F i n a l l y , we expand one of t h e 1's i n t h e
T - ~column, g e t t i n g
and simplify t h e r e s u l t i n g- p- a i r 6n t h e
our f i n a l answer
o r 1000.1001
1
d
T
(5)
For s u b t r a c t i o n i t i s harder t o formulate a general r u l e , but I t h i n k
i t w u l d suffice t o say: Cancel whenever p o s s i b l e and simplify o r expand
whenever t h a t would permit c a n c e l l a t i o n ( a l s o remember not to confuse a
1 with a -1 (1 ' 1 11 -11 # 111bll -$I) and not t o make t h e mistake of "expanding" a - 1 i n t o two + 1's. ) After a l l - 1% have been removed by c a n c e l l a t i o n , proceed as you would with an addition example.
Mu l t ip 1 i c a t ion
M u l t i p l i c a t i o n involves nothing new. We simply p l a c e t h e p a r t i a l prod u c t s as we do i n t h e decimal system, and add. For instance:
MATHEMATICS MAGAZINE
10101 o r
S e t t i n g up t h e p a r t i a l p r o d u c t s , we g e t
10 10 1
and (from now on. i t i s simple a d d i t i o n ) expanding one of t h e 1 ' s
i n t h e u n i t s column:
We now s i m p l i f y t h e p a i r i n t h e u n i t s and
7-1
column
and t h e p a i r we t h u s produce:
and a g a i n t h e p a i r t h i s s i m p l i f i c a t i o n produces, g e t t i n g :
Next we expand one of t h e 1's arr. t h e
4,'s i n t h e r -4 column:
P - ~ column,
and one o f t h e
F i n a l l y , we s i m p l i f y t h e p a i r i n t h e r - 2 and r - 3 columns, and t h e n
t h e one in the r -4 and B -5 columns, giving our final answer:
or
lOOOOO.1O4OO1
Division
E i v i s i o n i s quite d i f f e r e n t i n t h i s system, and i s , i n f a c t , r a t h e r
odd, The 0nl.y things it has i n common with ordinary division are t h e
basic prira.ciples behind it, t h e way the example looks, and t h e mov.ema.t
106
NUMBER SYSTEM
of the "decimal point" to eliminate any figures to the right of it in the
divisor. It is best explained by an example: 12 divided by 2, or
which, after moving the "decimal point", i s
I I
Now, since we are dividing by 11
11 , if there are anywhere two 1's
with two spaces between them (the spaces can be empty or full), they can
be crossed out and a 1 placed in the quotient above the rightmost of the
two. This crossing out, in no way signifies that the 1's should 110e be
there, but i s merely equivalent, to, in long division (decimal) the subtraction of the product of the number placed in the quotient and the divisor
from the dividend, Since the number placed in the quotient can only be 1
(placed in any column, of course), we merely subtract the dividend (placed
in that same column), i.e. cross i t out. It i s obvious that once the whole
dividend has been crossed out, the group of 1's in the quotient, after
being changed to the simplest form, will be the complete quotient. Getting
back to ouroriginal problem, we s e e that we do have just such a s e t of
1's in the 7-I and 7-* column, and s o we cross it off and place a 1 in the
quotient, getting
But now, you may say, there are no more pairs of 1's spaced in that way;
what shall we do? The answer i s our old pair of friends, expansion and
simplification. Since they do not change the value of a number, if either
of those processes yields a s e t of 1's spaced correctly, that s e t can be
crossed off and a I placed in the quotient just a s though that s e t were
part of the original number. Since in our problem it i s s o far impossible
to simplify, we shall expand. Expanding the 1 in the r 7 column, we get
No such s e t yet. However, when we expand the 1 we've just placed in
the '7 column, giving
MATHEMATICS M A G A Z I N E
we have not one but two such s e t s (remember that all w e need i s two
with that certain separation, regardless of intervening and crossed
1 9 s ) , one made of the 1's in the a' and r4 columns, and the other of
1's in the r 3 and r 6 columns. Crossing them both out and placing the
in the correct places in the quotient, we get
107
1's
out
the
1's
I
and since there are no more 19sin the dividend, our number in the quotient
is the complete quotient, and s o 1010.0001, or 6 , i s our answer. This
time the general rule is: Always take that course of action that will place
your next 1 (i.e., a 1 in the quotient-set in the dividend) farthest to the
left. By a course of action, I mean a series of expansions and simpiifications and the exchange of a s e t for a 1 in the quotient that fo29ows; or
simply that exchange, if the s e t i s already there, (1 did not obey this rule
in my demonstration so that I could show the proceqs in a simpler way.)
This is s o that the answer be in i t s simplest form.
By the way, the processes of addition, subtraction, aid often ~nult'aplication, can be performed together by writing the addends, the subtrahends,
and the partial products in one s e t of columns; for instance: 2 x 3 + 4 + 3 - 3:
and working it out:
1
1
partial products
addends
1-1
1-1
subtrahend
Now that we know these four processes, we have a much better way of
finding a number in this system than merely repeatedly adding 1's until
we reach it. For instance, to find thirty-seven, we can multiply 6 x 6 and
add 1; to check the arithmetic, we multiply 7 x 5 and add 2:
1010.0001
x 1060.001 + 1
(to check)
1000.1001
x 10080.000%+ 110.61
108
N U M B E R SYSTEM
What's more, we are not only able to find integers, but, since we can
divide, we ought to be able to find fractions also. L e t us try. First, we
shall attempt to find 1/,. We begin by setting up our division:
Our first set, a s can be seen, will have i t s leftmost I in the '7
We therefore expand the 1 in the r 2 column:
column.
The other one we need to complete the s e t i s a 1 in the r - 2 column; t h i s
we get by expanding the 1 in the units column:
After we exchange our s e t for a 1 in the quotient (7-2 column), we notice
that our remainder i s 1 (in the r - l column). Since 1 i s the number we
started with, the next figure in the quotient and the next remainder should
be the same a s these. However, the question i s , where in the quotient
shall we place it? Since our first 1 was in the r 2 column (because we
moved the decimal point) and our remainder i s three places to the right of
it, in the r - I column, our next 1 in the quotient should be three places
to the right of the first 1 there. Since the next remainder will bear the
same relationship to the first remainder as the first did to our original 1,
the following 1 in the quotient will be three places to the right of our
second 1. Since this can be carried on indefinitely, i t appears that '/,
expressed in the Tau System i s .01001001001........ (any 'Voubting
Thomases" may carry i t out a few places to see).
Before we go on to other fractions, i t would be wise to mention something about 1 in the Tau System. A s you can easily see, 1=.11=.1011=
.101011= .10101011 etc. It i s , therefore, equal to the endless "fraction"
.f0101010.... (just as in the decimal system 1=.9999999 ....). If we can
now take this fraction and expand the leftmost 1, and then expand the
1 in the r a 3 column, s o a s to prevent the occurrence of two 1's in the
same column, and then expand the 1 in the T - ~ column s o that there are
not two 1's in that column, etc., we will get . O l l l l l l f . . . , If, on the other
hand, we start by expanding 1's in other columns, we get: .1001111~1....,
MATIIEMATJGS M A G A Z I N E
109
.10100111111... .., 101010011111....., etc. Therefore, if you multiply
.OlOOlOOlOO1....
by 10.01 (2) and get .1001111111..,., this does not
mean that 2 x x = a fraction, but merely shows a different way of representing 1.
To get back to fractions, we can make a list of them just a s we did of
integers before:
l / 2 = .010010010010.. .........
1/3 = . ~ 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0......
.
1/4 = .001000001000001000001000.. ........
1/5 = .00010010101001001000001001010100100100. ........
(x)
1/10=.000010000100010100001010001010101000100101000001001000l00
00000001 ...
Of course, finding these fractions i s immeasurably harder than finding
'/2, and with 1/10 I had to work it out 5 or 10 times before I got the correct
answer, a s there i s much room for error.
By the way, no fraction can be terminating- in this system, since that
would mean that it could be expressed a s the sum of a group of integral
powers of Tau. Since all the powers of Tau can be expressed a s the sum
of an integer and an integral multiple of Tau; if the integral multiples
66
cancel" (e.g. 7 - I + r-'+ P - =~ l r - 1+ 2 r - 3 - 7 7 + 5 = I) the result will be
an integer, and if they don't (e.g. 27-3 - - 3 r + 5 = r + 2). it w i l l naturally be
irrational. However, when we have an endless serios, this paradox i s
detoured by admitting the fact that Eim A T + B with A and B always
integral can be a rational fraction if A and B -+
The Tau System has a good many other. ~ntersstingand unusual characteristics, and investigation by the readers of some, suclr a s tho frequency,
occurrence, and nature of numbers with a 1 in the u n i ~ scolumn (when in
simplest form) might prove interesting, d do not know of any useful applacation
for systems such as this, except a s a mental exercise and pascirne, though
it may be of some service in algebraic number theory. FOPinsta11ce, the
numbers expressible in the Tau System in terminating form consist of
all the algebrnlc integers in R ( f i ) , and some of the properties of numbers
in this andother systems might correspond to facts about associated fields,
-
W.
Definitions /%vented f o ~~ Y O in
T ~the Tau S y s t e m
Expand: alter three successive figures of a number by changing
...100.,.
to ...011... The result i s the same in value as the original, because
of rule (I), This does not mean change zeros to ones and ones to
zeros; just this specific change.
Simplify: the reverse of expand; alter the figures thus: chan e
011.
to
100... One speaks of simplifying the 19s in the r n-k and ,n-2
columns into the rn column. Also, one speaks sf expanding the 1 in
...
... ..
110
N U M B E R SYSTEM
the 7 column into the r "-l and 7 n-2 columns.
Simplest form: that form of a number which has been simplified until no
more simplification i s possible. Xt therefore has no two 1's in succession. It also has the fewest 1's and i s the easiest form to work with.
Columns: just a s in our decimal system we speak of a units column, a
ten's column, a hundred's column, a tenth's column, etc., in the Tau
system we speak of a units column, a 7' column, a 7 - I column, etc.
Pair: two 1's in succession.
Cancellation: a change of the form
Set: in division, two or more 1's arranged with the same spacing a s the
1's in the divisor (regardless of intervening 1's). A s e t can be "exchanged" for a 1 in the quotient.
'also true OF - I / ? ; there are other numbers which have similar properties,
e.g. there i s a number S between 1 and 2 for which s3 = s 2 + S + 1. E d .
2because 7-'+ r 0 = 7 .
3this i s the basic property defining the Fibonacci Series.
4 ~ h e r ei s also a more complex proof which involves multiplying the expressions like 2 r + 1 by 7, giving 2 r 2 + r , and expanding, r2 into r + 1.
5 ~ h i is
s a restatement of r n = r
+ 7 n-2.
6 ~ changing
y
the 1 in the 7 column to 1.1.
7 ~ you
f
come across words (like "expand")
used in an unfamiliar way,
look for them in the list of definitions at the end of this article. I have
put there all words which I have had to invent or alter for use in this
system, s o a s not to break up the text by explaining them.
Jr. High School 246
Brooklyn, N. Y.
(see page 91)