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Nothing is relevant or irrelevant in any absolute sense.
The use of an arithmetic mean, for instance, may be relevant in a
one-dimensional space, whereas there is no reason to assume that it will be
relevant in a two- or more-dimensional space, that is, where it concerns
data which are fundamentally of a two- or more-dimensional nature, such as
surface areas (where you would expect the geometric mean to be relevant).
Similarly, base-10 or 'denary' arithmetic may be relevant in
five-dimensional space
—i* will not venture to prove this
here—, it may be relevant for humans still using their ten fingers
in counting —which needs no proof, i would say—, it is
irrelevant in (almost?) all other contexts.
It might be argued that the only nonarbitrary base to choose is 2 and that
we should stick to the binary representation of numbers in all contexts,
just like computers do in secret.
Whatever one may think of this argument, it only confirms that 'tenness'
is not a universal given whatsoever, and that too great a reliance on it
is naïve.
How naïve (if not faulty) i will show by means of the approximation of
pi, the ratio of the circumference of a circle to its diameter.
The apex of irrational naïvety is reached by those who try to find a
repetend (an infinitely recurring sequence of digits) in pi itself;
the apex of rational naïvety is reached by those who try to find a
repetend in a decimal approximation of pi right after the radix character,
that is, without considering an ever-increasing nonrepeating fractional
part.
In the short story
Miss Hecksenbasen, a maths teacher, defends the use of the senary system by
her community of warlocks against a 'bloody pack of tenfold calculating
wolves' who try to impose the denary system on them.
The reader is confronted with a great number of examples to illustrate why
the use of base 'six' (spelled in this way on purpose) is superior to the
use of base 'ten' in three-dimensional space.
Hecksenbasen pours scorn on the 'deca buffs with their silly pi of 3.14'.
In her eyes nothing is more logical than the use of base 2n in
n-dimensional space, even for the calculation of pi.
But let us not slip up with the
here, for while the base-2n part is my conjecture too, pi is not at all a
number characteristic of three-dimensional space; it is in the first place
a number characteristic of two-dimensional space, as squares and
circles are two-dimensional objects.
If base numbers do matter, it is therefore not base-10, nor base-6,
arithmetic which counts with respect to pi, but base-4 arithmetic.
(It is said that North and Central American natives too used base 4 to
represent the four cardinal directions.
This idea of four cardinal directions is a two-dimensional notion:
it would be two directions in one-dimensonal and six
directions in three-dimensional space.)
It is not entirely self-evident and certain that there is nothing
fundamental at stake in this kind of discussion.
The possibility of turning to Excel or some such program, or some fancy
officially-sanctioned formula, to convert base-10 numbers into base-4
numbers and vice versa, may only give the appearance of a complete
interchangeability of these systems.
Sure, such a conversion works for three: 3[D(enary)] =
3[Q(uaternary)].
(Numbers in italics are quaternary numbers in this text.)
And sure, it works for 4.5 = 10.2, which has a
terminating fractional part in both systems.
Unfortunately, it does not work for 5.1 in the sense of a terminating
5.1000..., which equals 11.02 but not exactly, or 11.012 but
not exactly, or 11.0122 but not exactly, and so on, because
0.1[D] =0.012121212...[Q] with an infinitely long
numeral expansion in the quaternary system which does not even converge to
a terminating number.
(But even the fact that 0.0999... converges to 0.1 in the denary system and
0.0333... to 0.1, in the sense of 0.1000..., in the
quarternary system can be taken both as a proof that they produce the
same result and as a proof that they never produce the same result.)
In practice and in general, all you get in the base-4 system is an
approximation of the denary number, in the base-10 system an
approximation of the quaternary number.
These approximations conceal at least one question of infinity: the
infinity at the lower end.
After all, base 4 is not base 5!
There are several ways to calculate, sorry, approximate pi.
In my own approximations i will start out from the Leibniz series of
pi, which is in denary notation
(4/1)-(4/3)+(4/5)-(4/7)+(4/9)-(4/11)+... and in
quaternary notation
(10/1)-(10/3)+(10/11)-(10/13)+(10/21)-(10/23)+... .
Especially in the latter notation, this series, while immediately showing
the special role of 4[D] = 10[Q], does not show any
base-10 bias and can be fully trusted for not containing one or more hidden
decimal initial or intermediate approximations.
It is simple, perhaps even simplistic in that it takes a long time before a
pi emerges close to 3.14[D], but in principle it can be
tackled without any recourse to calculators or calculating programs with
too narrow a denary view of the world.
Unless you are absolutely sure of what you are doing, all calculations in
the quaternary system must be done the quaternary way, as if there were
no other system of numeration around.
Did you ever see, let alone do, a long base-4 division on paper before?
This is what 10/31 (the equivalent of 4/13) looks like,
the first term in the Leibniz series yielding a repetend in which all
four quaternary digits appear:
31/100 \0.103230...
31
300
213
210
122
220
213
100
..
|
In the formula (4/1)-(4/3)+(4/5)-(4/7)+(4/9)-(4/11)+..., pi is an
alternating series.
But it is also possible, and has been proposed, to look at pi as a
nonalternating series in which each term consists of two terms of the
alternating series: 4/(4n+1)-4/(4n+3).
The first term (for n=0) is then 4-4/3=2.67, which is also the first
approximation of pi; the second term (for n=1) is then 4/5-4/7=0.23 with
2.67+0.23=2.90 as the second approximation of pi; the third term is then
(for n=2) 4/9-4/11=0.08, with 2.90+0.08=2.98 as the third approximation of
pi; et cetera.
In my own proposal and calculations i will treat the Leibniz series as a series of pairs
of which the first element, after adding its positive value to the sum of
all previous pairs, gives the upper limit and the second element, after
adding its negative value to the sum plus the value of the first element,
gives the lower limit of the approximation.
In this interpretation 'Leibniz' does not really start by saying that pi
equals 4/1=4, 'Leibniz' starts by saying that pi is a number between 4 and
(4/1)-(4/3)=2.666... or, if you prefer, a number between 2.666... and 4.
This is, properly speaking, 'the first approximation of pi'.
The upper limit of 'the second approximation of pi' is then
(10/1)-(10/3)+(10/11) and its lower limit
(10/1)-(10/3)+(10/11)-(10/13);
the upper limit of 'the third approximation of pi'
(10/1)-(10/3)+(10/11)-(10/13)+(10/21) and its lower limit
(10/1)-(10/3)+(10/11)-(10/13)+(10/21)-(10/23);
and so on and so forth.
By itself the value of pi in the quaternary system is as little
interesting as its value in the denary system.
All we need for pi is something like 39 decimal places for an accuracy
which is physically sufficient for use on a universal scale.
Yet, what has been intriguing people for ages is the possible occurrence of
interesting series of digits in pi or, if they knew what they were doing
and seeing, in its approximation.
The greatest discovery so far has been the Feynman point, a sequence of six
9s which begins at the 762nd decimal place of pi's denary representation.
The difficult question of a much longer series of digits recurring not only
once but an infinite number of times has remained a mystery.
This is the riddle of pi's repetend, in its approximation to be precise
— and to be honest.
With our present-day knowledge we may be sure that the value of π=3.1 or
3.14 or 3.142 or 3.1416 or 3.14159 or 3.141593 or 3.1415927 or
3.14159265 in denary or 'decimal' notation.
The calculated conversions will be in quaternary notation: π=3.1
or 3.02 or 3.021 or 3.0210 or 3.02101 or
3.021010 or 3.0210100 or 3.02100333.
(A converter
may show 2-, 10-, 50-digit and longer repeating
fractional parts, but these repetitions do not mean anything as 3.1 is read
as "3.1000...", which it is not; 3.14 as "3.14000...", which it is not, et
cetera.)
Altho** in the end we are
basically interested in real quaternary values, these conversions are meant
to help 'denarians' find their bearings again when base-4 figures leave
them in limbo.
(A good converter can be found at
http://www.knowledgedoor.com/[in the folder]2/calculators/[on the
page]convert_a_number_with_a_mixed_fractional_part.html.)
The following two tables show the first six lower- and upper-limit, denary
and quaternary approximations of pi.
(Unless noted otherwise, the numbers will be rounded off.)
| LIMITS OF THE FIRST SIX BASE-10
APPROXIMATIONS OF PI |
| APPR. |
LOWER LIMIT (L) |
UPPER LIMIT (U) |
PI |
| 1 |
2.66666667 |
4.00000000 |
2.66<π≤4.00 |
| 2 |
2.89523809 |
3.46666667 |
2.89<π<3.47 |
| 3 |
2.97604618 |
3.33968254 |
2.97<π<3.34 |
| 4 |
3.01707182 |
3.28373848 |
3.01<π<3.29 |
| 5 |
3.04183962 |
3.25236593 |
3.04<π<3.26 |
| 6 |
3.05840277 |
3.23231581 |
3.05<π<3.24 |
| LIMITS OF THE FIRST SIX BASE-4
APPROXIMATIONS OF PI |
| APPR. |
LOWER LIMIT (L) |
UPPER LIMIT (U) |
PI |
| 1 |
2.22222223 |
10.00000000 |
2.22<π≤10.00 |
| 2 |
2.32110232 |
3.13131313 |
2.32<π<3.20 |
| 3 |
2.33213132 |
3.11123311 |
2.33<π<3.12 |
| 4 |
3.00101133 |
3.10202203 |
3.00<π<3.11 |
| 5 |
3.00222312 |
3.10002123 |
3.00<π<3.11 |
| 6 |
3.00323303 |
3.03231321 |
3.00<π<3.10 |
As may be expected, the lower limit increases and the upper limit decreases
at every step, but, given the rate at which the changes in the numbers
slow down, we are still a far way off from even 3.14 or 3.02.
And while there are some recurring digits and pairs of digits in the first
two approximations no such pattern is visible in the third to sixth
approximations.
In the above two tables the base-4 system appears to be as little
interesting as, or no more interesting than, the base-10 system.
Before proceeding with other tables some conceptual clarification should be
worth our while.
Let us start with the word repetend.
In its simplest form a number with a repetend looks like this:
0.123123123... .
In this number the repetend repS='123' (S stands for string) and its
length lenRepN=3 (N stands for natural number).
A fraction like 6/5 is terminating in the denary system and may be written
as "1.2". (It is 1.030303... in the quaternary system!)
But conversely 1.2 need not be the equivalent of 6/5, it may also be short
for 1.19 or 1.21, for instance.
Only if we interpret it as "1.20000000..." is 1.2=6/5, and
interpreted in this way it has a repetend too: repS='0' with lenRepN=1.
(There is nothing sacred about 'being terminating' anyhow, as it is
base-dependent.
It is the relevance of the base selected which could turn this
feature into something special in that particular system.)
The nonrepeating fractional part between the radix point and the beginning
of the repetend may be called "the lead".
In 1.00000000... the repetend starts immediately after the point, and hence
leadS='' (empty) with lenLeadN=0.
In 1.20000000... the repetend starts after one digit after the point, and
hence for this number leadS='2' with lenLeadN=1.
With the above approach and terminology in mind it can now be easily proven
that, from the second approximation on, (the Leibniz approximation of) pi
in the base-10 system always has a lead.
The first approximation starts without leads, because the upper limit is
4=4.000... and the lower limit is 4-(4/3)=8/3=2+(2/3)=2.666... .
But in the second approximation the culprit emerges: 4/5=8/10=0.8000...=0.8.
The upper limit is now (8/3)+(4/5)=(40/15)+(12/15)=52/15=3.4666..., the
same as 2.666...+0.8000... .
The lower limit on this level is
(52/15)-(4/7)=(364/105)-(60/105)=304/105=2.895238095238095238095238..., the
same as 3.466666666666666666...-0.571428571428571428... .
The new term in the calculation of this limit is
4/7=0.571428571428571428... .
This term has no lead and repS='571428' with lenRepN=6.
As repS='6' in the initial term, with lenRepN=1, the new lenRepN should be
1x6=6.
And indeed, it is the length of repS='095238'.
In this lower-limit approximation the initial leadS='4', with lenLeadN=1,
but the new lead will have the same length as the new repetend:
leadS='895238' with lenLeadN=6.
Yet, we see that only the first digit (the onset) differs, and keeps on
differing, from the first digit of the repetend; the second to last digits
of the lead are the same as the second to last digits of the repetend!
This is what the total picture looks like for the first twelve alternating
Leibniz terms or the first six upper- and lower-limit approximations in
the denary and quaternary systems:
LEADS AND REPETENDS
IN THE FIRST SIX BASE-10 APPROXIMATIONS OF PI
|
| APPR. |
ONSET
OF leadS |
ADD.
lRN |
(NEW)
lRN |
FIRST 32 DIGITS OF repS (TRUNCATED) |
| 1U |
'' |
- |
1 |
'0' |
| 1L |
'' |
1 |
1 |
'6' |
| 2U |
'4' |
1 |
1 |
'6' |
| 2L |
'8' |
6 |
6 |
'095238' |
| 3U |
'3' |
1 |
6 |
'539682' |
| 3L |
'9' |
2 |
6 |
'176046' |
| 4U |
'2' |
6 |
6 |
'483738' |
| 4L |
'0' |
1 |
6 |
'817071' |
| 5U |
'2' |
16 |
48 |
'05236593471887589534648358177769' |
| 5L |
'0' |
13 |
624 |
'24183961892945484271490463967243' |
| 6U |
'2' |
6 |
624 |
'43231580940564531890538083014862' |
| 6L |
'0' |
22 |
6864 |
'25840276592738444934016343884427' |
Note 1: in this and the following table lRN is short for lenRepN.
ADD(itional) lRN is the lenRepN of the last term added (U) or
subtracted (L).
Note 2: 13 is a prime number and hence for 5L the new lenRepN=13x48=624.
For 3L the new lenRepN=(1x2)x(2x3)/2=6;
for 4U lenRepN=(2x3)x(2x3)/2x3=6;
for 4L lenRepN=(1x2)x(2x15)/2=30;
for 5U lenRepN=(2x8)x(2x3)/2=48;
for 6U lenRepN=(1x6)x(6x104)/6=624;
for 6L lenRepN=(2x11)x(2x312)/2=6864.
|
REPETENDS
IN THE FIRST SIX BASE-4 APPROXIMATIONS OF PI
|
| APPR. |
LEAD
leadS |
ADD.
lRN |
(NEW)
lRN |
FIRST 32 DIGITS OF repS (TRUNCATED) |
| 1U |
'' |
- |
1 |
'0' |
| 1L |
'' |
1 |
1 |
'2' |
| 2U |
'' |
2 |
2 |
'13' |
| 2L |
'' |
3 |
6 |
'321102' |
| 3U |
'' |
3 |
6 |
'111233' |
| 3L |
'' |
5 |
30 |
'332131320221201120010102033330' |
| 4U |
'' |
3 |
30 |
'102022030111311010113332203220' |
| 4L |
'' |
2 |
30 |
'001011323101210000012322102211' |
| 5U |
'' |
4 |
60 |
'10002123003130300331331220122030' |
| 5L |
'' |
9 |
180 |
'00222312000011023322212003112310' |
| 6U |
'' |
3 |
180 |
'03231321003020032331221012121313' |
| 6L |
'' |
11 |
1980 |
'00323303132331012213120121320132' |
|
Note: 11 is a prime number and hence
for 6L the new lenRepN=11x180=1980.
For 3U the new lenRepN=(1x3)x(2x3)/3=6;
for 4U lenRepN=(1x3)x(3x10)/3=30;
for 4L lenRepN=(1x2)x(2x15)/2=30;
for 5U lenRepN=(2x2)x(2x15)/2=60;
for 5L lenRepN=(3x3)x(3x20)/3=180;
for 6U lenRepN=(1x3)x(3x60)/3=180.
|
Of course, what is most striking here is that the quaternary approximations
do not only have repetends but repetends which are not preceded by leads,
whereas the denary approximations have repetends too but preceded by leads
of the same length.
Judging by the last three alternating terms (5L, 6U and 6L) the repetends
in base 10 are much longer than those in base 4, but this could be a
coincidence as some intermediate terms (3L, 4U, 4L and 5U) are shorter.
It seems obvious that the repetends will get longer and longer in both
systems, their lengths tending to infinity.
Now, when looking at the approximations as estimates of pi, whether in
denary or in quaternary notation, the results may be considered at least a
little disappointing: they are still a far way off from 3.14 or
3.02.
However, if we stick to our interpretation of lower and upper limits, it
can be argued that any mean of the lower and upper limit is a legitimate
approximation as well, because a mean of numbers cannot be smaller than the
smallest number, nor larger than the largest number.
Not only are the means of the approximations legitimate approximations
themselves, they are much better estimates of pi, as the following table
shows:
PI AS A MEAN BETWEEN
LOWER AND UPPER LIMITS
|
| APPR |
ARITHMETIC |
GEOMETRIC |
HARMONIC |
| 1 |
3.333 |
3.266 |
3.200 |
| 2 |
3.181 |
3.168 |
3.155 |
| 3 |
3.158 |
3.153 |
3.147 |
| 4 |
3.150 |
3.148 |
3.145 |
| 5 |
3.147 |
3.145 |
3.144 |
| 6 |
3.145 |
3.144 |
3.143 |
Note 1: the arithmetic mean or 'average' of two numbers x
and y is (x+y)/2; their geometric mean is the square root of
their product, sqrt(x*y); and their harmonic mean is the
reciprocal of the arithmetic mean of the reciprocals, that is,
2/((1/x)+(1/y)).
Note 2: for the calculation of these means the quaternary lower and
upper limits were first converted to denary numbers.
Except for some minor differences in the second approximation, where the
quaternary-denary means are 3.180 (instead of 3.181), 3.167 (instead of
3,168) and 3.154 (instead of 3.155), the results are the same in the two
numeral systems.
|
Clearly, for the first six approximations the harmonic mean between the
lower and upper bounds gives the best estimate of pi.
In this way pi can be calculated correctly to two decimal places after only
10 alternating Leibniz terms, instead of the nearly 300 alternating terms
needed when treating each finite Leibniz series as an approximation of pi
itself rather than of its upper or lower limit.
Harmonic means do play a role in geometry, in triangles with inscribed
circles and in equilateral triangles with circumscribed circles, to be
precise.
But even in geometry itself there does not seem to be a direct connection
with the ratio of the circumference of a circle to its diameter.
The geometric mean, on the other hand, is the mean of
two-dimensional space, most evidently for surface areas.
While it starts out worse than the harmonic mean in the above table of
means, it soon catches up with the harmonic mean, at a moment that the
arithmetic mean still yields an inaccurate value for the first two decimal
places of pi.
| * |
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 |
67.ENW
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