Subject: Re: What is binary?? (or: how do you explain it to an idiot?)
From: edew@netcom.com (Eric Dew)
Date: Sat, 11 Jan 1997 19:22:15 GMT
In article <32D48276.E0A@rmii.com> sockeye@rmii.com writes:
>Mark wrote:
>>
>> For some reason, I just can't seem to get the meaning of binary!
>>
>> I first noticed binary when I first got into computers. I noticed how
>> everything was based on "two's", like 2, 4, 8, 16, 32, 64
>> (i.e. 16 bit, 32 bit software etc.)
>>
>> So I went and got a book at the library and it said, binary is not
>> based on 10, but on two!
>>
>> What really freaked me out, was that instead of "1" and "2", it had a
>> big string of numbers like:
>> (pardon me)
>> 001010110000011101001101001010101010101111
>>
[overly explicit description snipped]
Here's a simple answer: binary is what we would use if humans were born
with one finger on each hand. In that case, we would never have
created the symbols like ``2'', ``3'', etc. The only two symbols we'd
use would be something to denote zero (say, 0) and something to denote
one (say 1). Then, we'd count and add and do arithmetic in binary.
Actually people in the computer industry do more in hex than in binaries.
EDEW
Subject: Re: Infinitude of Primes in P-adics
From: Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium)
Date: 11 Jan 1997 20:31:19 GMT
In article
dik@cwi.nl (Dik T. Winter) writes:
> In article <5b0mi4$3k7$1@dartvax.dartmouth.edu> Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium) writes:
> > But you Dik assume that 3 is prime in the first place, do you not, in
> > order to construct the 3-adics.
>
> Yup, a prime in the (finite) natural numbers. 3 is not a prime in the
> rationals or in the reals. There are no primes in the rationals or in
> the reals because they form a field.
>
> By starting with (finite) natural numbers.
>
Wrong. Since you, Dik and virtually all other mathematicians have
assumed that you start with Peano axioms of Finite Integers. I would
think that you would have by now have incorporated that idea as a axiom
also. That mathematics requires a starting point.
I say that is a falsehood, a *hidden assumption*, to believe you
cannot have numbers , unless you have such primitive numbers from which
all other numbers are built.
> You have got this backwards. I never proof that the 3-adics form a field,
> I assert they do *not* form a field. And in the *not-field* (but a ring)
> 3 is the (only) prime.
>
> Again: if there is a field there are no primes, if there is a prime there
> is no field. The 3-adics do not form a field because 3 has no inverse.
> Moreover, 3 generates a prime ideal in the 3-adics, and so they are also
> not a field for this reason.
I concur, if there is a field, there are no primes. But that in no way
stops the truth of the fact that there exist an infinitude of primes in
p-adics, namely each prime of a prime-adic. And thus, the Infinitude of
Primes still exists in Naturals = P-adics.
The source of our disagreement is not that, whether 3-adics is a
field. We both accept the fact that 3-adics is a field and that no
primes exist in 3-adics. The source of our disagreement is that you Dik
wants to believe that Number theory must be constructed from one class
of numbers that are accepted as given. And that whatever you want to
say about these primitive numbers is true in whereever else you *think*
you see them, such as the 10-adics which look similar to many Reals.
I on the other hand say that all Numbers exist already and exist as a
point of a geometry. This is an idea that you Dik just do not seem to
want to consider. That the Reals exist already and that it is ludicrous
and a sham to try to construct these Reals from some primitive _first
step_ axiom system.
The points of Riemannian geometry exist already without some
hornswaggling first step primitive axiom system to build them up. Same
also with Loba geometry.
Dik, I have never read where you, or any other mathematician for
that matter, read where you tell me what the intrinsic numbers of Riem
or Loba geometry are? Is it Dik, and I suspect this is the case, is it
that you believe along with all the other math people that the Reals
and imaginary i and j ( you can throw in k if that fancies you). Is it
that you Dik , still believes that the Reals + i + j cover Riem geom
and Loba geometry. I suspect you do, because, like I say, you have
never given an alternative.
I on the other hand say that the P-adics, more accurately All-Adics
are the points of Riemannian Geometry and that Doubly Infinites are the
intrinsic points of Loba geometry.
How in the world, Dik, since you have been in mathematics for longer
than I have, how in the world can you at your prime age assume that
Reals are the points of , excuse me, the intrinsic points of Riem and
Loba geometry.
And if you still believe that Dik, then you still believe that the
Peano Axioms not only beget Reals and Euclidean Geometry but that those
Peano Axioms also beget Riemannian geometry, in fact also the
trigonometry and pi and e itself.
Such magnificent power does your Finite numbers beget. Dik, you sure
you are not a Sunday school preacher?
Subject: Re: Palindromic Numbers
From: ksbrown@seanet.com (Kevin Brown)
Date: Sat, 11 Jan 1997 21:04:53 GMT
On Fri, 10 Jan 1997 02:05:41 +0100, Patrick De Geest
wrote:
>I'm always looking for all sorts of information about palindromic
>numbers...
>
>Record_nr(?) for palindromic tetrahedrals; Function = (b*(b+1)*(b+2))/6
>
> Function( 336 ) =
> 6.378.736 [length 7]
>
>If you happen to know of similar basenumbers that are larger than the
>ones above, please let me know your source.
If you don't restrict yourself to decimal representations you can
find many such numbers. For example, the base-9 representation of
the tetrahedral number k(k+1)(k+2)/6 is palindromic for the following
values of k (shown in decimal):
k k(k+1)(k+2)/6 (in base 9)
------ -----------------------------
1 1
2 4
3 11
4 22
12 444
13 555
120 488884
588 71066017
1093 505555505
88573 505055555550505
Of course the "tetrahedral" numbers are just the binomial coefficients
C(k,3). The base-3 representations of C(k,3) are palindromic for the
following values of k
k C(k,3) (in base 3)
------ -----------------------------
1 1
2 11
3 101
4 202
6 2002
12 111111
14 202202
39 112121211
120 112222222211
392 201000222000102
496 1102111111112011
Similarly we have
C(1105,3) = 1534114351 in base 8
C(521,3) = 1122123212211 in base 4
C(1166,3) = 6364334636 in base 7
C(82332,3) = 12 5 9 13 18 18 13 9 5 12 in base 27
There is also a tetrahedral number C(k,3) whose base-B representation
is palindromic where both k and B are perfect squares (k=441, B=16).
Of course, every positive integer has a palindromic representation in
SOME base. If we let f(n) denote the smallest base relative to which
n is palindromic, then clearly f(n) is no greater than n-1, because
every number n has the palindromic form '11' in the base (n-1).
Interestingly, for almost all integers n the value of f(n) is actually
quite small, as can be gathered from the table below:
n f(n) n f(n) n f(n) n f(n) n f(n)
--- ---- --- ---- --- ---- --- ---- --- ----
1 2 21 2 41 5 61 6 81 8
2 3 22 10 42 4 62 5 82 3
3 2 23 3 43 6 63 2 83 5
4 3 24 5 44 10 64 7 84 11
5 2 25 4 45 2 65 2 85 2
6 5 26 3 46 4 66 10 86 6
7 2 27 2 47 46 67 5 87 28
8 3 28 3 48 7 68 3 88 5
9 2 29 4 49 6 69 22 89 8
10 3 30 9 50 7 70 9 90 14
11 10 31 2 51 2 71 7 91 3
12 5 32 7 52 3 72 5 92 6
13 3 33 2 53 52 73 2 93 2
14 6 34 4 54 8 74 6 94 46
15 2 35 6 55 4 75 14 95 18
16 3 36 5 56 3 76 18 96 11
17 2 37 6 57 5 77 10 97 8
18 5 38 4 58 28 78 5 98 5
19 18 39 12 59 4 79 78 99 2
20 3 40 3 60 9 80 3 100 3
The numbers for which f(n) = n-1 are
3, 4, 6, 11, 19, 47, 53, 103, 137, 139, 149, 163,
167, 179, 223, 263, 269, 283, 293, 311, 317, 347, 359, 367,
389, 439, 491, 563, 569, 593, 607, 659, 739, 827, 853, 877
977, 983,...
These might be called "the non-palindromic numbers" because they
are the only numbers n that are NOT palindromic in ANY base from
2 to n-2.
Proposition: If f(n) = n-1 for n>6 then n is a prime.
Proof: If n=ab with a<(b-1) and b>2 we have n=a(b-1)+a, so n has
the palindromic form 'aa' when expressed in the base (b-1). This
covers all composites greater than 6 except for squared primes,
but for squares we have a^2 = (a-1)^2 + 2(a-1) + 1, so every square
n>4 has the palindromic form 121 in the base sqrt(n)-1. Done.
The ratio of non-palindromic primes to all primes drops as the
range increases. For example, of the 1229 primes less than 10000
we find that 218 are non-palindromic, which is about 1 out of
every 5.6. Of the 9295 primes less than 100000 we find only 1199
are non-palindromic, which is only about 1 out of every 7.75.
At the opposite extreme, there are primes that are already
palindromic in the base 2, although these are even more scarce
than non-palindromic primes. The first few are
3, 5, 7, 17, 31, 73, 107, 127, 257, 313, 443, 1193, 1453, ...
Returning to the function f(n) for general values of n (not restricted
to primes), it's clear that f(n), n=1,2,3,... fluctuates between 2 and
n-1, but I think the average value of f(n) for all n from 1 to
infinity is polynomial in n. Specifically it appears that
f(n)_average = n^(1/c)
where c is a constant approximately equal to 2.68. In other words, I
conjecture that
1 n ln(k)
lim --- SUM ------- = c
n->inf n k=1 ln(f(k))
but I don't know how to prove that this limit actually exists.
Anyway, it's also interesting to observe the values of f(n) when n
is a prime power. For example, powers of 2 have the following values
of f(n):
n f(n) representation of n in the base f(n)
--- ---- ------------------------------------
2 3 2
4 3 1 1
8 3 2 2
16 3 1 2 1
32 7 4 4
64 7 1 2 1
128 7 2 4 2
256 15 1 2 1
512 7 1 3 3 1
1024 7 2 6 6 2
2048 31 2 4 2
4096 7 1 4 6 4 1
8192 15 2 6 6 2
16384 15 4 12 12 4
In general it appears that the min-base palindromic representation
of 2^m is a multiple of a binomial expansion, i.e.,
2^m = 2^r [(2^s -1) + 1]^t
Obviously we have m = st + r. The values of r,s,t for the first
several m are tabulated below:
m r s t m r s t m r s t
- - - - - - - - - - - -
1 1 2 0 11 1 5 2 21 1 5 4
2 0 2 1 12 0 3 4 22 2 5 4
3 1 2 1 13 1 4 3 23 2 7 3
4 0 2 2 14 2 4 3 24 0 6 4
5 2 3 1 15 0 5 3 25 0 5 5
6 0 3 2 16 0 4 4 26 1 5 5
7 1 3 2 17 1 4 4 27 3 6 4
8 0 4 2 18 3 5 3 28 0 7 4
9 0 3 3 19 1 6 3 29 1 7 4
10 1 3 3 20 0 4 5 30 0 5 6
In each case the triple (r,s,t) is such that s is minimized under
the constraint that
/ (2^r) * C(t,t/2) if t is even
2^s - 1 > (
\ (2^r) * C(t-1,(t-1)/2) if t is odd
The min-base palindromic representation of other prime powers also
tend to be multiples of binomial expansions, but not always. For
example, the min-base representation of 3^7 is [3 19 3] in the
base 24. Also, the min-base representation of 7^6 is [1 2 3 2 1]
in the base 18. This raises the question of whether the min-base
representation for powers of 2 is always of the binomial form.
Can anyone supply a proof or counter-example?
____________________________________________________________
| /*\ |
| MathPages / \ http://www.seanet.com/~ksbrown/ |
|____________/_____\_________________________________________|
Subject: Math Problems
From: landmark@vcn.bc.ca (Scott Phung)
Date: 11 Jan 1997 13:56:03 -0800
Hello, i have two questions:
The first question is:
Find the exact value of x,
1/x = e^x
I couldn't find the exact value, but i've found an
approximate.
ln(x^-1) = ln(e^x)
-ln(x) = x
x + ln(x) = 0.
I then applied newton's method and came up with the answer
of 0.5671432904097838, which is approximate but not exact
and as far as my computer will take me.
But how do i go about calculating the exact value?
------------------------------------------------------------------------
My second question is:
Given the equation x^2 + y^2 + z^2 = n,
with x, y, z being integers and n a positive integer,
how many different solutions (x,y,z) are there?
(ie (0,1,2), (0,-1,2),(-1,2,0), and (-2,0,1) are
all different solutions and are counted seperately).
(This is just like Jacobi's Four-Square Theorem but with
three squares).
Also, can someone please point me to web site(s) containing
problems / interesting math stuff that are like the above
questions.
Thanks,
S. Phung
Subject: Vietmath War: DEAR AMERICA: LETTERS HOME FROM VIETNAM
From: Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium)
Date: 11 Jan 1997 20:52:02 GMT
My favorite two Vietnam movies are the series VIETNAM WAR: A TV HISTORY
and recently I saw DEAR AMERICA: LETTERS HOME FROM VIETNAM
As I watch these movies I analogize with my own fight of moving the
Mathematics community. They still believe their old ways of Naturals =
Finite Integers. So, what better way to point out the errors of their
old habits than by making a satire of the math community.
I am a revolutionary, and I believe Naturals = P-adics = Infinite
Integers. I believe that a concept of a number being purely finite is a
sham and that all numbers need a component of infinity. A Real Number
such as .500... is true because it has infinite digits, but a purely
finite number such as 5 which the math community wants you to accept ,
to me that finite number is a fake. There is no finite number of 5 but
that all numbers, in order for them to exist, must have infinite
digits. And so 5 is a fake, but ....00005 exists and is a p-adic.
I usually look for lines in these movies to incorporate into my
satire-harangue of telling the math people that they need to reexamine
their foundations. Upon seeing this movie of DEAR AMERICA, I was too
moved by its human compassion to want to incorporate passages and I did
not want to use other people's lived emotions for these are genuine
letters and not some fiction plot.
But I perhaps am vicariously experiencing some of the emotions that
these people went through as I battle the world mathematics community.
------------------------
Dear Red, ( or Dear Dad, southern accent cannot make it out)
Anyone over here who walks more than 50 feet through elephant grass
should automatically get a purple heart. Try to imagine grass
possessing razor sharp edges big, to 15 feet high, so thick as to cut
visibility to 1 yard. Then try to imagine walking through it while all
around you are men possessing the latest automatic weapons who
desperately want to kill you. You would be amazed how much a man can
age on but one patrol.
-----------------------------------
58,132 Americans were lost to the Vietnam War.... and nearly 3 million
came home.
---------------------------------------
Dear Folks,
.... All trickle into Supply and disbursed as needed... I've read
where officers were cordially saying "this is the only war we got,
don't knock it"
---------------------------------------
December 31, 1965
US Troops in Vietnam 184,300
Killed in action to date 1,363
Wounded in action to date 7,645
--------------------------------------
"We own the day, Charlie owns the night"
-- a G.I. saying
Dearest Beth,
Last night we had the VC all around us. Beth, don't ever tell them
this but there are times I feel I am never coming home. The VC are
getting much stronger. So I think this war is going to get worse before
it gets better. The days are fairly peaceful, but the nights are pure
hell. I look up at the stars and it is so hard to believe that the same
stars shine over you and in such a different world as you live in. All
my love, Al
-------------------------------------
December 31, 1965
US Troops in Vietnam 184,300
Killed in action to date 1,363
Wounded in action to date 7,645
--------------------------------------
December 31, 1966
US Troops in Vietnam 385,300
Killed in action to date 6,644
Wounded in action to date 37,738
------------------------------------
Dear Mom and Dad,
You know that joke about how hard it is to tell the good guys from
the bad guys over here ..... The enemy in our areas of operations is a
farmer by day and a VC by night. Every man we pick up says,
" Vietnamese number 1, VC number 10."
So, we have to let him go. By the way, number 1 means real good and
number 10 means real bad.
Other handy phrases are:
Titi -- very little
Boo Koo -- which means very much
Didi Mow -- get out of here
What more do you need to know?
Love always, Mike
---------------------------------
LBJ, Jan 12, 1967, State of the Union Address : I wish I could report
to you that the conflict is almost over. This I cannot do We face more
cost, more loss, and more agony, for the end is not yet. I cannot
promise you it will come this year or come next year. Our adversaries
still believe, I think tonight, that he can go on fighting longer than
we can. And longer than we and our Allies will be prepared to stand-up,
and resist.
----------------------------------
Jan 21, 1968 Khe Sanh
Almost 40,000 NVA regulars have surrounded 5,600 Marines. The siege
begins.
--------------------------------
To defend KheSanh, the US mounts the most intense bombing in the
history of war. The equivalent of 5 Hiroshima-size bombs are dropped
within a mile of Khe Sanh.
General Earle Wheeler, Chmn., Joint Chiefs of Staff : I do not think
that nuclear weapons will be required to defend Khe Sanh.
------------------------------------
Although the enemy's Tet Offensive has failed militarily, it succeeds
in changing public opinion in the US
The War has reached a turning point.
---------------------------------------
{Silent Night sung here }
December 31, 1968
US Troops in Vietnam 536,100
Killed in action to date 30,160
Wounded in action to date 192,850
--------------------------------------
" One thing worries me .... Will people want to hear about it? Or will
they want to forget the whole thing happened. " -- Lt.J.G. Richard
Strandberg
