There’s no such thing as a “four-digit number”, only a four-digit base-10 numeral. And facts about base-10 numerals aren’t facts about numbers.
("Binary digit" and "hexadecimal digit" are weird terms that abuse the language a bit.)
I see what you did here
Lots of people seem to think that, hence the -2 rating of my comment, but that's not the modern definition: https://en.wikipedia.org/wiki/Numerical_digit
Similar numbers (I presume) exist for other number bases, and it's an interesting question of whether they constitute some sort of strange attractor. istm quite a few mathematical discoveries have emerged from just farting around with inconsequential-seeming numerical oddities.
I do feel your frustration though. I'm into electronic music and math, but I regularly run into people who insist that tuning to 432hz instead of 440hz (the common default for western tonality) is better because 432 is numerologically interesting. I've wasted a lot of time trying to persuade people that yes, 432 is a very cool number, but the interval of a second (from which we derive tuning frequencies) is fundamentally arbitrary. I suppose it's true that if you tune everything slightly flat people will subconsciously feel like time is expanding, man.
You might be able to satisfy yourself by replacing "the digits of" with "the decimal digits in the base-10 representation of".
Happened across a neat comment yesterday that presents a defense of ten. Not 100% convinced but it is interesting to see pushback.
And you can swear by that, if you know what I mean.
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* * * *So for example, 1/2 = 0.5 and 1/5 = 0.2, but 1/3 = 0.333… and 1/7 = 0.142857….
1/4 = 0.25 works because the prime factors of 4 are 2 and 2… but 1/6 fails because 6 is infected by 3.
Now, base 12 has 2 prime factors (2 and 3) so it much any better than 10 really. But may I introduce base 30 (235)? Or perhaps base 210 will strike your fancy?
1 sums digitwise to 1
1 + (10-1) = 10 which also sums to 1 in the same way
The point of this trickery is that N-1 added to any number is really adding N (which adds 1 to the second position, by definition) and adding -1 (which subtracts 1 from the first position).
In base 10, this is the adding 9 trick. It can be extended by using any multiple of 9. That applies to the N-1 version, so that adding M*(N-1) to a base N number yields the same digit sum.
1+9 = 10 = 1
1 + 27 = 28 = 10 = 1
In hex:
1 + F = 10 = 1
1 + 2D = 2E = 10 = 1
For example, per another's link in these comments, this 'trick' works for 3 digits, but hits 1 of 3 possible loops for 5 digits. From this, interesting but likely useless questions can arise, such as finding an easy way to test for these loops, seeing if there is a way to calculate the loop without brute forcing it, and understanding the problem enough to know how much of this holds true when swapping to a new base.
In general, most of this is just for fun and doesn't lead to anything serious. But sometimes a fun problem can be hard to solve, possibly leading to discovering something new, which ends up being applicable to more serious mathematics. Other times it can become a trap that just seems to waste time without ever leading anywhere, like the 3n+1 problem.
I don't think this should be considered numerology, though I do think sometimes people treat tricks as if they have some more serious meaning that they don't deserve, at least not based on how they are presented. 3 Blue 1 Brown goes into the spiral pattern of the primes as something that appears to be deep, but ends up being an unique way to present an otherwise boring tidbit about prime numbers.
Being further pedantic - aren't all digits base ten? I thought that was part of the definition of digit.
Other bases would have different words for their numbers - bit in binary, for example (which, yeah, I know, it a combination of the words "binary" and "digit").
Do we have another example? I don't think there are special terms for "octal digits" or "hexadecimal digits".
We call computer circuits "digital" even though they work in base 2.
Regardless of the word's origin, digits are simply the symbols in a positional number system: https://en.wikipedia.org/wiki/Numerical_digit
An argument against being overly pedantic in this case is that this is a neat and accessible example of something quirky about numbers, and so even people who don't know much about numbering systems can approach it. If you instead emphasize that it's base 10 or that there is "no such thing as a 4 digit number", the main thing you'll probably do is cause disinterest in anyone who is sometimes overwhelmed by math. :)
Randomly, one of my sons told me about 6174 just a week ago, and it turned into an interesting conversation following by a little programming to find more of these numbers. After we went down that rabbit hole for awhile, then the conversation shifted to how these numbers might look in e.g. hexadecimal, and that seemed about the right time for that topic to come up.
The point of the parent comment is that this is not a property of numbers in general. It's just a coincidence that only works in base-10.
For example, a prime number is prime in every base. An irrational number is irrational in every base. Collatz conjecture is valid in every base. This one is not.
What? Not at all. In fact, trying it in other bases, as well as with other numbers of digits (in both base 10 and other bases), is a useful way to get some insights into why it happens.
So, yes, the described "special" thing about 6174 is actually a special thing about the string 6174 (representing a number in base-10). And I'd say the fact so many people in this very thread don't understand it is exactly the proof that the GPs comment actually has some merit. People kinda mix up properties of numbers and properties of some other mathematical objects — like their representations in base-10. Most of numerological games are concerned with the latter. Which is why it's especially interesting, when something like that happens to hold in other bases, which sadly just isn't the case with Kaprekar's constant.
Here are some additional examples:
dec, 3 digits: 495
hex, 3 digits: 7F8
hex, 30 digits: EECCAA88664421FFDDBB9977553312
dec, 30 digits: 988766544332209987766554332111
hex, 100 digits: FFFFEEEEDDCCCCBBAAAA9998888776666554444333222210FFFFEEDDDDCCCBBBBAA999988777766655554433332211110001
dec, 100 digits: 9999998888888877666666665544444444332222222210999999998877777777665555555544333333332211111111000001
Finally someone noticed the irony, thank you.
(On the other hand, at least some people also criticize my tone rather than the point I'm making, which I guess is fair as well.)
Comparing this to numerology is just combative and doesn't help get your point across (as you can see by the downvotes).
Besides, going to a place where people are discussing something fun and explaining to them why it isn't really fun is just not a good way to get points across to people, no matter how valid you think they are.
A much better way to approach this IMO - don't say this is wrong, give something analogous that would work for all bases, which by the way would teach people this concept. E.g. extending "do all digits appear infinitely and evenly in the decimal representation of pi" to talking about "normal numbers".
But it's not like it's somehow less worthy than other mathematical games. After all, there could have been some meaningful property hidden in there. Doesn't appear so in this case, but you'd never know beforehand.
Numerology is far stupider than this admittedly useless arithmetic game.
no that's highly opinionated compressionn in the domain of crazy