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It is a contingent matter, but since we have the power to procreate our
species sexually, and since there have always been exactly two, not three
or more, sexes involved in human procreation, the impact of the powers of
two in our lives has been tremendous.
Until today, all of us still have two biological parents, four biological
grandparents, eight biological great-grandparents, sixteen
great-great-grandparents, and so on and so forth.
Of course, these are maximum numbers, because not all these human beings
need involve different persons.
Conversely, we all are the biological children of parents, but our parents
need not have two (surviving) children: they could have (had) one child
only or three or more children.
And we all are the biological grandchildren of grandparents, but our
grandparents need not have four (surviving) grandchildren: they could have
(had) one, two or three or more than four grandchildren.
However, for a natural human population equilibrium the number of
biological children we (will) have on average in the end should be the same
as the number of biological parents we have; the average number of
biological grandchildren we (will) have in the end the same as the number
of biological grandparents; the average number of biological
great-grandchildren the same, in the end, as the number of biological
great-grandparents; and so on and so forth.
(For those interested, see
The Powers of Two — dedicated to the
future of nature and the nature of people's future.)
The powers of two are the same for everyone, for parents and children, for
grandparents and grandchildren, for great-grandparents and
great-grandchildren, ad infinitum.
Are they really?
(Apart from the dubious suggestion that there are accidental things which
go on forever.)
Are the powers of two the same for laypeople as well as for theorists of
numbers and computer experts, for instance?
First of all, there is the radix or 'base' of the numeral system used or
to be used.
Only for the most naive or orthodox that radix cannot be anything else
than ten (10 in denary or 'decimal', that is, radix-10, notation).
But in the context of the powers of 2, a numeral system with a radix which
is itself a power of 2, such as 2, 4, 8 or 16, may be much more
appropriate.
(It is for computers using the binary system with radix 2.)
Even without such a context, the more prime factors a radix has the more
suitable its general use might be.
Radix 10, with the prime factors 2 and 5, may be a better choice than 2,
but as far as prime factors are concerned there is no reason why it would
be better than radix 6 or radix 12, with the prime factors 2 and 3.
Radix 30 tho* would beat the radixes 2 to 16 altogether in this respect,
because it is the smallest natural number with three prime factors: 2, 3
and 5.
As numbers the powers of two may be the same regardless of the radix used,
their notations definitely do not look alike in different numeral systems.
The following table is an illustration of this.
THE POWERS IN DIFFERENT NOTATIONS
| n |
RADIX
2 |
RADIX
4 |
RADIX
6 |
RADIX
8 |
RADIX
10 |
RADIX
12 |
RADIX
16 |
RADIX
30 |
| 0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
| 1 |
10 |
2 |
2 |
2 |
2 |
2 |
2 |
2 |
| 2 |
100 |
10 |
4 |
4 |
4 |
4 |
4 |
4 |
| 3 |
1000 |
20 |
12 |
10 |
8 |
8 |
8 |
8 |
| 4 |
10000 |
100 |
24 |
20 |
16 |
14 |
10 |
G |
| 5 |
100000 |
200 |
52 |
40 |
32 |
28 |
20 |
12 |
| 6 |
1000000 |
1000 |
144 |
100 |
64 |
54 |
40 |
24 |
| 7 |
10000000 |
2000 |
332 |
200 |
128 |
A8 |
80 |
48 |
| 8 |
100000000 |
10000 |
1104 |
400 |
256 |
194 |
100 |
8G |
| 9 |
1000000000 |
20000 |
2212 |
1000 |
512 |
368 |
200 |
H2 |
| 10 |
10000000000 |
100000 |
4424 |
2000 |
1024 |
714 |
400 |
144 |
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Note: the numbers in this table are not segmentized by means of spaces or
commas.
The radix-10 number 1024 could also be written "1 024" or "1,024".
If the spoken language uses basic terms such as thousand for 1,000,
million for 1,000,000 and billion for 1,000,000,000 the
division of the number (if any) should be into 3-digit segments; in
languages such as Chinese a division of a radix-10 number in 4-digit
segments will be more convenient.
If the radix is a power of 2, no other division will make sense than one
into segments of 4 digits or of multiples of 4 (16 in particular).
In radix-2 notation the number 10000000000 (denary 1024) may also be
written "100 0000 0000" or "100,0000,0000".
The same number is 400 in radix-16 notation.
There is a direct many-to-one correspondence between the two 'macrobinary'
notations: each 4-digit segment of the radix-2 notation corresponds with
one digit in the radix-16 notation, such as 0100 or 100
with 4 and 0000 with 0.
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Are the powers of two all the same once we take the differences between
numeral systems into account?
What to think of number theorists who are definitely using radix 10, but
who claim that the powers of 2 are 2, 4, 8 —so far, so good—
and then 7, 5 and 1?
Is this just anecdotal evidence of a great-great-grandparent that has seven
instead of sixteen great-great-grandchildren, and a
great-great-great-grandparent that has five instead of thirty-two
great-great-great-grandchildren, and a great-great-great-great-grandparent
that has one instead of sixty-four great-great-great-great-grandchildren?
(On a large scale, that could leave us, not with a problem of human
overpopulation, but with one of underpopulation, even the threat of an
extinction.)
No, it is nothing to do with over- or underpopulation; it is part of an
entirely different way of looking at numbers, devoid of any concern about
environmental problems or natural prospects, for that matter.
Number theorists like to play and juggle with numbers.
Numbers or number notations can be addictive and, if making use of
radixes other than ten, they may rouse you from your 'decimal' slumber.
(Read, for example, my
Final Digit Information in Different Numeral
Systems.)
The following table demonstrates how the powers of two can become an
object of cyclical mathematical interest:
THE POWERS IN CYCLES
| WITH RADIX 10 |
WITH RADIX 16 |
| n |
POWER
OF TWO |
FINAL-
DIGIT
CYCLE |
DIGIT
SUM
CYCLE |
POWER
OF TWO |
INITIAL-
DIGIT
CYCLE |
DIGIT
SUM
CYCLE |
| 0 |
1 |
— |
1 |
1 |
1 |
1 |
| 1 |
2 |
2 |
2 |
2 |
2 |
2 |
| 2 |
4 |
4 |
4 |
4 |
4 |
4 |
| 3 |
8 |
8 |
8 |
8 |
8 |
8 |
| 4 |
16 |
6 |
7 |
10 |
1 |
1 |
| 5 |
32 |
2 |
5 |
20 |
2 |
2 |
| 6 |
64 |
4 |
1 |
40 |
4
| 4 |
| 7 |
128 |
8 |
2 |
80 |
8 |
8 |
| 8 |
256 |
6 |
4 |
100 |
1 |
1 |
| 9 |
512 |
2 |
8 |
200 |
2 |
2 |
| 10 |
1,024 |
4 |
7 |
400 |
4 |
4 |
| 11 |
2,048 |
8 |
5 |
800 |
8 |
8 |
| 12 |
4,096 |
6 |
1 |
1000 |
1 |
1 |
| 13 |
8,192 |
2 |
2 |
2000 |
2 |
2 |
| 14 |
16,384 |
4 |
4 |
4000 |
4 |
4 |
| 15 |
32,768 |
8 |
8 |
8000 |
8 |
8 |
| 16 |
65,536 |
6 |
7 |
1,0000 |
1 |
1 |
| 17 |
131,072 |
2 |
5 |
2,0000 |
2 |
2 |
| 18 |
262,144 |
4 |
1 |
4,0000 |
4 |
4 |
| 19 |
524,288 |
8 |
2 |
8,0000 |
8 |
8 |
| 20 |
1,048,576 |
6 |
4 |
10,0000 |
1 |
1 |
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Note: the sets of four or six numbers 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.
There seems to be an initial-digit cycle in the radix-10 notation as well.
It spans ten numbers, starting each time with 1, 2, 4, 8, 1, 3, 6, 1, 2
and 5, shown in red from 1,024 to 524,288.
However, the illusion holds only until the 46th power of 2
(70,368,744,177,664), which does not start with 6 'as required'.
The next spoiler is 2^53=9,007,199,254,740,992.
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In the above table we discover 'two' initial-digit cycles, one (the
radix-10 one) spurious and one (the radix-16 one) genuine;
furthermore, one final digit cycle in the radix-10 notation, and two
digit sum cycles in both notations.
The digit sum of a number is the sum of the sum of the sum ... of the
values of the digits in the notation of that number which has a value
between 0 and, but not including, the radix of the numeral system.
The sum of the digit values of denary 64 is 6+4=10, but the notation
10 itself still has two digits with values whose sum is in turn
1+0=1.
Hence, the digit sum of 64 is 1.
Similarly, the sum of the digit values in 1,048,576 is 31, those of
the values in 31 is 4, which is also the digit sum of
1,048,576.
The digit sum of a power of 2 in radix-16 notation is the same as the value
of its initial digit, because all other digits, if any, are zeros.
With radix 16, the powers of 2 show a very simple sequence which keeps on
repeating itself: 1, 2, 4 and 8.
In this sequence 2=2x1, 4=2x2, 8=2x4, while 2x8=10 (denary 16), and
10 has the initial digit sum 1 again.
The digit sum cycle in radix-10 notation is longer and less simple, but not
at all complicated.
It starts like the cycle in radix-16 notation (1, 2, 4, 8) but now 2x8=16
and the digit sum of 16 is 7.
The next digit sum is the one of 32, obtained from 2x16, which is 5.
The multiplication 2x32=64 produces a digit sum of 1 for 10, and
there the cycle starts again.
Did i** mention this sequence before?
Yes and no.
I said that there are number theorists who claim that the powers of 2 are
2, 4, 8, 7, 5 and 1, a sequence which is, strictly speaking, not the same
as 1, 2, 4, 8, 7 and 5.
But, obviously, the former sequence begins at the first power of 2, which
is 2, whereas the latter one begins at the zeroth power of 2, which is 1.
Sometimes, this starting point makes a difference, as in the case of the
final digit cycles in radix-10 notation, where including the zeroth power
would immediately put an end to the possibility of recognizing such a
cycle, since the digit 1 occurs nowhere else at the end.
In general, however, it is very well possible to start with 1, the zeroth
power of 2.
Whether we consider the six-term sequence to be (1,2,4,8,7,5) or
(2,4,8,7,5,1), it is not a sequence of the powers of 2; it is a
sequence of the digit sums of the radix-10 notations of the powers
of 2; a sequence which incessantly repeats itself and is therefore called
"a cycle".
Why would number theorists be interested in this sequence?
Because they need the length of the sequence or the number of different
digit sums to get the order of 2, which is said to be 6.
In the radix-10 system, that is, because it is 4 in the radix-16
system.
In Order of an Element the authors Corn, Mishra, Liang and others
argue that 'there are six distinct congruence classes mod 9 of integers
that are relatively prime to 9, namely 1,2,4,5,7,8' and ask us to compute
their orders mod 9.
(See brilliant.org/[ ]wiki/order-of-an-element/.)
They explain that the powers of 1 are 1,1,1,…, and that the order of 1 is
1; that the powers of 2 are 2,4,8,7,5,1,…, and that the order of 2
is 6; and so on, until the powers of 8, which are 8,1,…, the reason why
the order of 8 is said to be 2.
If we accept a sloppy use of be (are in the text) and some
prosaic license it may all be interesting and correct, but arcane
knowledge ought, if necessary, to be couched in equally arcane terms which
do not negatively affect our common knowledge (and sense), whether
purposefully or inadvertently.
Calling something "a power of 2" that is not a power of 2, is like calling
something "a spade" which is actually not a spade at all.
We badly need to make people aware of the role of the (real) powers of 2
in the fight against human overpopulation as one of the two main causes of
the degradation of the environment, if not the destruction of nature.
Too many people still do not know that (starting with 2^1) they are
2, 4, 8, 16, 32, 64, et cetera.
It will serve nothing and no one, neither from a natural nor from a
cultural perspective, to confuse the human procreators on Earth even more
by offering them 2, 4, 8, 7, 5, 1 instead; or on top of it.
75.LSE
| * |
Where there is some existing orthographical
variation preference will be given to the (more) phonematic
variant |
| ** |
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. |
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