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In the radix-ten, or 'denary', system and its decimal notation the numbers
between 0 and 1 all begin with 0. and are followed by one or more
of the figures 0 to 9 for integers.
(The use of integers for what have always been called "fractions"
perfectly betrays traditional arithmetic's pro-integer bias towards
what less denigratingly should be called "portions".)
If the 0. number is a rational number, there is always a repetend,
which is 0 if the number is terminating as well in the denary system.
I myself will use vertical bars for such a repetend, since the use of an
overline (preferred by others) should be reserved for cases in which over-
and underlines together can very well be applied to express some kind of
opposition (between numbers which are each other's inverse, for instance).
We have been made to believe that a 'fraction' such as ½=0.5 (or, for
example, that ½=0.8 in the hexadecimal system), but the equal sign
is misleading here, for, conversely, 0.5≠½.
In the decimal notation 0.5 does not refer to a particular number;
it refers to a range of numbers, a range such that
0.45≤x<0.55.
Strictly speaking, ½≠0.5, because ½=0.5|0| and 0.5|0|=½.
When we treat the numbers between 0 and 1 with zero repetends this
carefully, they stop having any necessary connection with numerical
approximation.
As a matter of fact, the exactitude of binary 'portions' (as i* shall
continue to call these numbers in their own right) is no different from
the exactitude of integers.
Thus, any negative-exponential power of two can and must be
calculated and given until the last digit, just like any
positive-exponential power of two can and must be.
(Note that as a rule i will use positive- and
negative-exponential here, and 'positive' and
'negative' only informally, because it is not really the powers
which are positive or negative, but the exponents.)
If we do not go to the very end, then we must make clear that we are
giving an approximation only, whether it be of the power with a positive
or with a negative integral exponent.
How can we be sure, however, that we have reached the last digit of a
negative-exponential power of two before the zero repetend when our
calculating machine or computing program is (almost) at the end of its
tether?
It may come as a surprise, but the behavior of these powers of two in
the denary system can help us tremendously in this task of being exact
instead of approximating, for as the absolute value of the exponent in
the power becomes larger and larger the final sequence of digits
repeated becomes longer and longer.
Without any feeling for regularities in general, and for the regularities
among numbers in particular, no one will notice that the
negative-exponential powers of two display such an impressive pattern in
their decimal notation.
Yet, even with the slightest ability to recognize patterns, and, perhaps,
a nonexistent affinity with numbers, one is bound to discover at least
one regularity in the decimal notation of the denary representation of
these powers of two (usually confused with the numerical
representation itself).
Just calculate 2^-1=½=0.5, 2^-2=¼=0.25 and 2^-3=⅛=0.125.
(Forget about the zero repetend for a moment.)
Those who see that these three numbers end in 5 may know 'the
reason' too, for each number, from ½ on, is half the previous number.
That previous number ends in 5, which, in final position, may
be replaced with 50.
When we divide 50 by 2, the outcome 25 ends in 5 again in decimal
notation.
Not only will every negative-exponential power of two end in 5, it
will also be seen to end in 25 from 2^-2=¼ on for precisely
the same reason.
Those who go a little bit further and also calculate, or are confronted
with, 2^-4= 1/16= 0.625, 2^-5= 1/32= 0.03125 and 2^-6= 1/64= 0.015625
will, first of all, find a confirmation of both the final-5 and
the final-25 rules.
Moreover, they may discover in an empirical way that from the exponent -3
onward the three final digits 'seem' to alternate between 125 and
625.
But also this phenomenon can be explained in simple terms, because after
a permitted addition of a final 0 ½*1260= 625 and
½*6250= 3125 (and ½*31250= 15625 and ½*15625= 78125, and
so on).
Altho** these are not elegant mathematical proofs, it will work so far.
From here on we may also discover all other regularities by a combination
of intuition for numbers and an affection for the powers of two.
However, we will not be able to 'prove' them as easily anymore.
But let us first have a look at what these regularities are for the first
thirteen negative-exponential powers of two.
(The number thirteen is not an entirely arbitrary one: it allows me to
show a complete five-digit cycle of final digits with the first term
repeated in the second round, where the exponent is -13.)
CYCLES OF FINAL DIGITS
in the first thirteen
'negative' powers of two
|
Final x-digit cycles for 1 ≤ x ≤ 5 |
| m |
Power of two |
1 |
2 |
3 |
4 |
5 |
| 0 |
1 |
— |
— |
— |
— |
— |
| 1 |
0.5 |
5 |
— |
— |
— |
— |
| 2 |
0.25 |
5 |
25 |
— |
— |
— |
| 3 |
0.125 |
5 |
25 |
125 |
— |
— |
| 4 |
0.0625 |
5 |
25 |
625 |
0625 |
— |
| 5 |
0.03125 |
5 |
25 |
125 |
3125 |
03125 |
| 6 |
0.015625 |
5 |
25 |
625 |
5625 |
15625 |
| 7 |
0.0078125 |
5 |
25 |
125 |
8125 |
78125 |
| 8 |
0.00390625 |
5 |
25 |
625 |
0625 |
90625 |
| 9 |
0.001953125 |
5 |
25 |
125 |
3125 |
53125 |
| 10 |
0.0009765625 |
5 |
25 |
625 |
5625 |
65625 |
| 11 |
0.00048828125 |
5 |
25 |
125 |
8125 |
28125 |
| 12 |
0.000244140625 |
5 |
25 |
625 |
0625 |
40625 |
| 13 |
0.0001220703125 |
5 |
25 |
125 |
3125 |
03125 |
|
Final x-digit cycles for 6 ≤ x ≤ 10 |
| 6 |
7 |
8 |
9 |
10 |
m |
| — |
— |
— |
— |
— |
0 |
| — |
— |
— |
— |
— |
1 |
| — |
— |
— |
— |
— |
2 |
| — |
— |
— |
— |
— |
3 |
| — |
— |
— |
— |
— |
4 |
| — |
— |
— |
— |
— |
5 |
| 015625 |
— |
— |
— |
— |
6 |
| 078125 |
0078125 |
— |
— |
— |
7 |
| 390625 |
0390625 |
00390625 |
— |
— |
8 |
| 953125 |
1953125 |
01953125 |
001953125 |
— |
9 |
| 765625 |
9765625 |
09765625 |
009765625 |
0009765625 |
10 |
| 828125 |
8828125 |
48828125 |
048828125 |
0048828125 |
11 |
| 140625 |
4140625 |
44140625 |
244140625 |
0244140625 |
12 |
| 703125 |
0703125 |
20703125 |
220703125 |
1220703125 |
13 |
|
Final x-digit cycles for 11 ≤ x ≤ 13 |
| 11 |
12 |
13 |
m |
n |
| — |
— |
— |
0 |
— |
| — |
— |
— |
1 |
2 |
| — |
— |
— |
2 |
3 |
| — |
— |
— |
3 |
5 |
| — |
— |
— |
4 |
8 |
| — |
— |
— |
5 |
13 |
| — |
— |
— |
6 |
22 |
| — |
— |
— |
7 |
39 |
| — |
— |
— |
8 |
72 |
| — |
— |
— |
9 |
137 |
| — |
— |
— |
10 |
266 |
| 00048828125 |
— |
— |
11 |
523 |
| 00244140625 |
000244140625 |
— |
12 |
1036 |
| 01220703125 |
001220703125 |
0001220703125 |
13 |
2061 |
The value m is the absolute value of the exponent of the power of
two given in the second column of the first part of this three-part
table.
The value n in the last column of the third part is the absolute
value of the exponent of the power of two in which the same
m-digit sequence of final digits restarts for the first time.
For example, for 2^-5, m=5, and the final five-digit cycle
begins with 03125, a sequence which restarts in the power 2^-13
(n=5+8), a power shown in this same table on purpose.
Similarly, for m=13, the final thirteen-digit cycle is
0001220703125, which restarts in the power 2^-2061
(n=13+2048), a power extremely far away from the thirteen ones
shown in this table.
The sets of one, two, four or more figures between horizontal lines and
with their digits in red denote the second occurrence of a cycle
as the first occurrence of the same set does not yet prove anything by
itself.
The sets of ten and more figures in blue are assumptions for which i
have not been able to find any verification yet due to powers of two
with exponents of which the absolute value was 266 or larger.
(My calculator would give an approximation, not even the exact number
without its zero repetend.)
The first thirteen negative-exponential powers of two show already a
clear picture of the structure underlying the presence and distribution
of sets of final digits which are being repeated.
We can distinguish nine 1- to 9-digit cycles (periodic sequences of
digits in the decimal expansions of the powers) of which the first
five restart at or before the thirteenth exponent.
The periods of these five cycles are 1, 1, 2, 4 and 8 (their lengths in
terms).
As far as what is shown in the table is concerned, the length of the
6-digit cycle is at least 8, of the 7-digit cycle at least 7, of the
8-digit cycle at least 6, etc. until the 13-digit cycle with a length of
at least 14-13=1 term.
However, given that 1, 2, 4 and 8 are the first four nonnegative powers
of two, we have reason to assume that the lengths of the last four cycles
will probably be 16, 32, 64 and 128 terms.
The 6-digit cycle starts at 2^-6, which means that it should restart at
2^(-6-16)=2^-22.
The power 2^-22= 0.0000002...1015625 (without repetend) and the last
six digits are indeed 015625 again.
Until the 9-digit cycle there is no further problem: the 9-digit cycle
itself starts at 2^-9, and 2^(-9-128)=2^-137=
5.7397185098...7001953125E-42, of which the last nine digits are
001953125, the same final digits as in the decimal notation for
2^-9.
The 10-, 11-, 12- and 13-digit cycles should have a length of 256, 512,
1024 and 2048 terms respectively.
The 10-digit cycle should therefore restart at 2^(-10-256)=2^-266.
Unfortunately, beyond 2^-201 the exact power values are not available
to me, and at this moment i must forgo any type of verification of these
last four cycles.
There is one more point i would like to make: the number of decimals
increases in steps of 1 from 2^-1 to 2^-13 in the above table.
In itself this does not prove that all powers smaller than 2^-13 will
also have the same number of decimals as the absolute value of their
exponent.
Yet, this is precisely the case, as Rick Regan proves in Number of
Decimal Digits In a Binary Fraction.
(See https://www.exploringbinary.com/[ ]number-of-decimal-digits-in-a-binary-fraction/.)
In spite of the fact that not all data in my table up to the thirteenth
power have been verified, i feel pretty confident of the following
conjecture:
- at each power 2^-m, with m>1, a new cycle of
m final digits will start with period
2^(m-2)
- the notation of the pth term (p≥1) of an
m-digit cycle starting at 2^-m will be no different
from the notation for 2^-(m+p-1), inclusive of
so-called 'leading zeros', apart from the first p-1 digits
after the radix point
(that is, for the first term it will be no different from
the notation for 2^-m; for the second term no different from
2^-(m+1) apart from the first digit after the radix point;
for the third term no different from 2^-(m+2) apart from the
first two digits after the radix point; and so on and so
forth)
When we want to have a mathematical explanation for the pattern(s) we
find in the radix-ten notations of the negative-exponential powers of
two, we should humbly start at the beginning: 10=2*5.
In other words, the factors of 10 are 1, 2, 5 and 10; and its prime
factors 1, 2 and 5, of which only 2 and 5 are of interest in the present
context.
Realizing this, we need not at all be surprised by the title of Rick
Regan's article entitled "Seeing Powers of Five in Powers of Two and Vice
Versa" (dated "November 16th, 2009" and accessed at
https://www.exploringbinary.com/[
]seeing-powers-of-five-in-powers-of-two-and-vice-versa/).
In this article Regan points out that 'a negative power of two in decimal
form looks like a positive power of five, except that it has a decimal
point and possibly some leading zeros'.
In the above table the powers of two are 0.5, 0.25, 0.125 and 0.0625 for
the exponents -1 to -4 respectively.
And the powers of five?
They are 5, 25, 125 and 625 for the exponents +1 to +4 respectively!
This would still not solve our problem, if it were not for another
article on the same website by Regan entitled "Patterns in the Last
Digits of the Positive Powers of Five" (dated "December 11th, 2009" and
accessed at https://www.exploringbinary.com/[
]patterns-in-the-last-digits-of-the-positive-powers-of-five/).
In this article the author argues that 'the positive powers of five ...
have a compact, repeating pattern in their ending m digits,
in the powers of five from 5m on'.
Also Regan sees cycles, and also these cycles come in lengths of powers
of two.
Under a table with the caption Cycle Length for Number of Ending
Digits (1 to 10) Regan writes that this table 'implies that the
ending m digits of the positive powers of five cycle with period 2m-2,
m ≥ 2, starting at 5m'.
This conclusion corresponds entirely with the first part of my
conjecture, except that instead of positive powers of five one
will have to read "negative(-exponential) powers of two", and instead of
5^m it will be 2^-m.
But, fortunately, Regan comes with a proof.
In the article entitled "Cycle Length of Powers of Five Mod Powers of
Ten" (dated "December 22nd, 2009" and accessed at
https://www.exploringbinary.com/[
]cycle-length-of-powers-of-five-mod-powers-of-ten/) Rick Regan presents a
proof consisting of two parts: (1) that, from 50 on, the powers
of five mod 2m start and restart in a cycle with period
2m-2 (m≥2); and (2) that, from 5m on, the powers of
five mod 10m start and restart in a cycle with the same period
as the powers of five mod 2m.
In the first part the author derives a formula to show that the period,
or order, of 5 mod 2m is 2m-2.
(For a single cycle, its order is equal to its period or length in terms.)
Regan claims that it is possible for the powers of five mod
2m to derive a formula for their order — "due to a
hidden, binary structure I've uncovered," the author writes (with
emphasis in the original).
This is indeed most interesting, but it does not only require some
algebra; it also requires sufficient knowledge and skill in the field of
modular arithmetic.
You may know that x mod[ulo] y is the remainder when
x is divided by y.
(E.g., 7 mod 2 = 1 and 7 mod 10 = 7.)
But modular arithmetic goes further than that: you can also express that
x and y have the same remainder when divided by z.
Two integers a and b are 'congruent modulo' n, if
n (the modulus, an integer larger than 1) is a divisor of their
difference.
The notation for this is a≡b(mod n).
Thus, 10 ≡ 6 (mod 4), with 10 and 6 having the
same remainder, which is 2, when divided by 4.
(Note that 10 ≠ 6 mod 4, because 6 mod 4 = 2!)
It is possible now to define a (multiplicative) order of a modulo
n.
Given a positive integer n and an integer a coprime to
n, the multiplicative order of a modulo n is the smallest
positive integer k such that a^k ≡ 1 (mod n).
(See https://en.wikipedia.org/[ ]wiki/Multiplicative_order.)
Two integers a and b are coprime, if the only positive
integer which is a divisor of both of them is 1.
I advise those familiar with modular arithmetic, or willing to
familiarize themselves with it, to read and study the articles by Regan
referred to here.
In my introduction to the subject of this article i have made it quite
plain that my interest in the final digits of the powers of two, both
the positive- and the negative-exponential ones, stems from my desire to
draw a sharp distinction between the exactitude of these numbers
themselves and their mere approximations.
That this interest is in the powers of two and not in the powers of ten,
or any other such number which could be the base of a power or the radix
of a numeral system, stems from my preference for an arithmetic which
is as methodical (as little arbitrary or skew) as possible from both a
linguistic and a mathematical perspective.
While i am but too aware of the mantra that 'any base will do for any
numeral system', i have taken the position that there is only one
'logical' choice for the base of a numeral supersystem:
two.
(This is not to argue by any manner of means that the radix of
the numeral subsystem most suitable for everyday use is also two.)
I have taken that position on universal theoretical and universal
practical grounds, albeit human practical grounds; i have definitely not
taken that position on the basis of what calculators can or cannot
manage, let alone on the basis of the architecture of contemporary
computers.
However, once you get to the details, there are clear signs of an
obscured relationship between humanity's denary thinking and the binary
calculating of its machines, even in that branch of human endeavor which
is so wishfully named "exact science".
Is 'one tenth' or 'ten percent' such a detail?
With a convincing illustration, Rick Regan demonstrates what everyone
but the most naive should already know: that, regardless of the number
of bits you store in a computer, you will never end up with the binary
equivalent of decimal 0.1.
And, still, new programmers will keep on wondering why 0.3+0.6=
0.89999999999999991 — instead of 0.90000000000000000.
(See https://www.exploringbinary.com/[ ]why-0-point-1-does-not-exist-in-floating-point/.)
True, this article about the negative-exponential powers of two is
totally useless for those raised or raising themselves with the binary
supersystem or any of its subsystems, for they need no conversion from
denary to binary, let alone from binary to denary.
Obviously, it has been written specially for those raised with the
denary system, a system of which it is impossible to know whether its
hegemony will last another century, another millennium, or another
myriad of years.
But last it will, for the time being.
77.ENW
| * |
The first-person singular pronoun is
spelled with a small i, as i do not consider myself a Supreme
Being or anything else of that Ilk |
| ** |
Where there is some existing orthographical
variation preference will be given to the (more) phonematic
variant |
|
