The point is that most of the time when digits are mentioned, it’s only a coincidental fact about one inelegant representation of the number — and often people are insufficiently aware of that.

It's only a coincidence if you ignore the fact that "digit" first and foremost refers to the things sticking out of your hands, and so was repurposed to talk about numbers because we have 10 digits on our hands.

That's the word's origin, not its current mathematical meaning. Also, number systems that are not base IIIIIIIIII have been used historically. That humans can only compute in a base that matches the number of fingers we have is a relatively recent myth.

Base-10 isn’t inelegant, is it? I mean there are good arguments for 12 being better but it isn’t like 10 is prime or anything.

Happened across a neat comment yesterday that presents a defense of ten. Not 100% convinced but it is interesting to see pushback.

https://news.ycombinator.com/item?id=39000882

Rational fractions will terminate only if the denominator’s prime factors are the base’s factors.

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+2+3+4=10

And you can swear by that, if you know what I mean.

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In some cases, the fact in base-10 has analogous facts in other bases. A trivial example that adding N-1 to any base-N number yields a value with the same digit sum. That makes it a bit more interesting. But I can't think of an example that doesn't pivot on the representation rather than something more fundamental.

1+9=10

Right, the digit sum of 10 is 1...perhaps I should have said "final digit sum." Same for 10000, or 1 with any number of zeroes after it.

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

i'm not sure what you are demonstrating?

1 sums digitwise to 1

1 + (10-1) = 10 which also sums to 1 in the same way

1 = 1+0