Mysterious number 6174
plus.maths.org
plus.maths.org
If you're looking for something more humanly profound, how about this: "Arbitrary operations on integers can yield arbitrary results." That's actually profound (in the Zen-like way a tautology can still be profound) in the context of "human understanding" and how people argue with numbers, but it would take a long time to explain why, if you don't already know.
Feels like a more mysterious version of Collatz conjecture :) which is - start with a positive integer, iterate as such:
x = (x & 1) ? (3*x + 1) : (x/2);
What real mathematical operation involves rearranging digits in an integer?
Yes I know that there is a whole derivation to the simple derivative example I've given, but who is to say that the 6174 transformation might have preceded an actual use?
Who's to say that this odd mathematical transformation actually leads to something useful one day?
Now you have some new number who's only relationship to the old number is that it's one of (432)-1 possible permutations of it. Other than that, what's the significance?
Now you have some new number who's only relationship to the old number is that it's one of (432)-1 possible permutations of it. Other than that, what's the significance?
In base 10, 7641 - 1467 = 6174.
In base 11, 7641 - 1467 = 4808.
Finding tricks that work in base 10 is interesting, but it's a much more fun to look for patterns that work across several number systems. A simple example is for any base > 3, (base - 1) times x where base > x > 1; you get a 2 digit number [x - 1],[base - x]
You are mistaking the map for the territory, sort of. The positional aspects of the number system get encoded the equations and then the mere "numbers" don't matter anymore.
In base 11 using 6174 you cycle:
4808, 87A3, 7094, 9272, 3098, 9452, 7094 (cycle)
I was thinking of a http://en.wikipedia.org/wiki/Logistic_map for this equation abcd - dcba > might be interesting across different bases.
Everything in mathematics can be seen from a "meaningless" perspective. In some sense everything from algebra to complex analysis can be stripped of meaning by thinking of them as just "a rearrangement of symbols according to a set of arbitrary rules".
And, somewhere, that's really all they are.
It's only once we apply an interpretation to these operations that they become "math". Same with numerical rearrangement. Just because it seems like it's just a algorithmic manipulation doesn't mean there isn't an interesting interpretation of the meaning somewhere.
To be pathological, take the phrase X2YZ to mean "X + Y = Z if X < Y and Y - X = Z if Y > X" and both 1234 and 3241 are true. Now that manipulation suddenly has a whole lot of (silly) meaning.
Always be suspicious of any statement about numbers that relies on the base. This property of 6174 is really a property of the number's representation in base 10. In another base, it's the same number, but doesn't have the same property. That's why it seems suspiciously meaningless.
My reply was more of a meta argument than really a defense of the meaning of 6174. There's no reason to say that there isn't some property of its representation in base N that isn't useful, but it's pretty unlikely to be general in any interpretation we're used to dealing with (thus the re-representation of "2").
for an example see:
http://en.wikipedia.org/wiki/International_Bank_Account_Numb...
or google for 'eleven check'.
A snippet from a yahoo answers page:
"a number is divisible by 11 if and only if when you add the digits that are an odd number of places from the right, and then you add the digits which are an even number of places from the right, the difference of these is divisible by 11."
On another note the author uses the word kernel when I think he wants "fixed point". Kernel doesn't really have a meaning in this context.
Please add some information to your profile so I can place your comment in some kind of reference. I love that you write clearly and to my untrained, therefore naive, eyes your arguments seem valid. But without an understanding of where you are coming from, I do not want to necessarily accept what you say on face value.
I mean no insult, I just want more information.
Cordially,
Shamiq
I personally found this submission to be very exciting and interesting. I guess even with calculus, linear algebra and computability theory under my belt I still find number representation interesting. I don't think I'm alone. I'll use this idea later when tutoring kids in an attempt to share mathematical and programming excitement (would make for a fun script).
This submission may not be significant in advanced mathematical theory, but getting distracted for 20 minutes doing arithmetic or programming is no less important, albeit at quite a personal level.
If you want to know if something is true, do research, don't just assume that someone's credentials make everything they say true.
I find this attitude an incredible strength in the hacker community, but also occasionally a real weakness. It leads to a complete dismissal of a source of information, and that, surely, can't be right - namely, the mindset and knowledge of the person making the statements. When making decisions you should use all the information available.
Yes, some information should be tagged as "hearsay" (you can't check the person's credentials) and some should be marked as "suspicious," but simply throwing it out is wrong.
When you teach or train it is critical that you know people's background so you can pitch the explanations and demonstrations at the right level. When you listen to someone, you should assess what level of confidence to place in what they say.
Work since Shannon has shown that the most efficient communication of information (as opposed to data) is achieved when the sender has an effective model of the receiver, and the receiver has an effective model of the sender.
Certainly many advances in breaking ciphers are achieved by knowing more about the source. Cryptographers know not to throw away information, even when it is suspect.
You can dismiss this, or go and research it. Does it sound interesting? Should you ignore it? Modern information theory says you shouldn't. You don't know my credentials, so you have to make that decision in a void. If I told you my credentials, would your opinion remain completely unchanged? Modern information theory says it shouldn't.
Yes, be skeptical, but use what information you have.
All of it.
http://www.google.com/search?q=riderofgiraffes
"User Profile for: riderofgiraffes UserID: 383529 Name: Email: Registered: 1/11/07 Occupation: Mathematician Location: UK Total Posts: 350"
So much for that vague reputation, unless of course you've been fibbing ;) and / or someone else liked your moniker as much as you did, but I somehow doubt there are multiple riders of giraffes.
I fully agree with the above by the way, it feels just right.
I always tell my kid to distrust all sources of information, including his dad (there is a pardox in there somewhere) and to gather his own facts if the issue warrants the effort.
My point remains. With no credentials, with no idea of the author's background, you must perform independent verification. With some idea of the author's background, you have more information, and can decide whether the information is likely to be trustworthy, or at least worthy of pursuit or verification.
I am concerned that hacker culture teaches "Trust no one, verify everything." I think it is thereby unnecessarily impoverished.
Remember that math is only of use if it can be applied to real world problems. Other bases than 10 are only interesting for computing machines, but even there the result have to be human readable (again base 10), otherwise nobody would be able to understand.
I see the base 10 as the only really important base to us humans (machines may have different preferences...).
It probably just seems that way because we're used to it. At some point when designing a system of numbers, 10 was a convenient answer to the question of how many digits to use. This doesn't mean it has a neurological basis.
I searched the Web a little bit for research pointing to a neurological basis for decimal. I didn't find it, but I did find this article on the brain's handling of numbers: http://arstechnica.com/science/news/2007/01/6661.ars
I see the base 10 as the only really important base to us humans (machines may have different preferences...)
If communication is ever established with non-human intelligences, it will be important to move beyond speciesism. When aliens finally land, or AI awakens, we should be able to say "Welcome. All your base are okay with us."
First, the term kernel certainly does have meaning in this context. The kernel of a mapping is the set of arguments it sends to the identity element (zero here), so in this case the kernel is just {1111, 2222, 3333, ... , 9999}. Like you, I thought the author should have used the term "fixed point," but there are situations where his way would be better. Details if anyone asks.
Now, why could anyone possibly gain from studying permutations on the digits of integers represented in base ten?
EDIT: the short answer is "cryptography." I'm cutting this post in two. For the longer answer, see below.
First off, it's helpful to know that the integers (denoted {Z}) are the canonical example of what's called ring structure, which basically means + and * work normally, but you aren't guaranteed clean division. Since they're simple, mathematicians sometimes use {Z} as a stand-in for more persnickety structures in the same way they use n-dimensional Euclidean space as a proxy for thinking about infinite-dimensional vector spaces. Articles like this one usually get written when an algebraist or number theorist publishes an epic proof, and the only way to make it comprehensible to a lay audience is to project it down into toy examples in a simple space. That's kind of a "trust me" answer by itself, of course. So without further ado, the math:
There are two notable things about rings. You can make polynomials with + and * in them, and division generally leaves you with remainders. Remainders are okay, though, because they let you define new rings. {Z mod 12 Z}, for example, is shorthand for "integers mod 12," or more formally, "map the integers to a new space where all the multiples of 12 get sent to zero," so {1, 2, ..., 11 } act normal, 12 is actually zero, 13=12+1=0+1=1 and so on.
It turns out that this is all you need to represent numbers in place-value notation. A number N in base r has a representation of n digits:
N = (A_n)(r^n) + ... + (A_2)(r^2) + (A_1)r + A_0
which is just a polynomial whose coefficients are members of the ring {Z mod r*Z}. If you map that back to the integers, plug in the actual value of r, and crunch the numbers, you get your original value of N back.
So, "rearranging the digits of an integer" is actually "permuting the coefficients of polynomials with coefficients mod r."
Now, suppose you have an email message that's 12000 characters long. You can think of that message as the polynomial representation, modulo 256, of a really big number. Mapping that number into different modular spaces, or permuting the coefficients of its various representations so that you can get it back later ("encrypting the message") forms the basis of the field of cryptography, which in turn is closely related to information theory and the theory of ergodic processes, etc, etc, etc.
For more details, you might start with:
http://www.amazon.com/Applied-Cryptography-Protocols-Algorit...
However the article does specifically say that 6174 is the kernel. Now for that to be correct under your definition it would need to be true that: 7641 - 1467 = 0 Which we all know to be false. Your post seems to suggest that you think this use of kernel was a poor word choice instead of completely incorrect. Perhaps you could send me the details that your post promises.
"Number of digits in n" is shorthand for "floor(log(b)n) + 1" and the digits themselves can be expressed in terms of modulus (while n > 0: digits.append(n % b); n = n / b). Speaking in terms of digits just makes it easier to talk about what are really rather odd mathematical operations. It's true that their primary use is in our number representation system, but that doesn't make them any less interesting from a pure-mathematical viewpoint.
Secondly, if every time a mathematically oriented person stumbles upon something interesting instead of figuring out why they would respond like you just did mathematics would grind to a halt really quickly. After all, that's how math started - that curiosity about patterns - and mathematicians really can't help it but they have to figure out what causes this. There may be a trivial answer, or something profound may be found out, who is to know in advance what will come of it.
To everybody else the circumference of a circle may as well be three times its diameter.
Fail.