Another interesting thing is that these seem to work:

base 16: 123456789ABCDEF~16 * (16-2) + 16 - 1 = FEDCBA987654321~16

base 10: 123456789~10 * (10-2) + 10 - 1 = 987654321~10

base 9: 12345678~9 * (9-2) + 9 - 1 = 87654321~9

base 8: 1234567~8 * (8-2) + 8 - 1 = 7654321~8

base 7: 123456~7 * (7-2) + 7 - 1 = 654321~7

base 6: 12345~6 * (6-2) + 6 - 1 = 54321~6

and so on..

or more generally:

base n: sequence * (n - 2) + n - 1

This is in the original post, in the form

  num(b)/denom(b) = b - 2 + (b-1)/denom(b)

so you just need to clear the denominator.

> the last 2 digits are swapped

They are also +9 away from being in order.

And then 12345678 * 8 is 98765424 which is +9 away from also being in order.

Oops - the second one was supposed to say +8

I also went about looking at the difference rather than the order. In the hexadecimal case, the difference is 15 (0xEF vs 0x12). I thought, then, that for any base B with ascending digits A and descending digits D, (D-(B-1))/A=B-2.

For binary, it looks like (1-(b-1))/1=b-10 or (1-(2-1))/1=2-2=0 in decimal.

For trinary, it looks like (21-(b-1))/12=b-2 or (7-(3-1))/5=5/5=1 in decimal.

For quaternary, it looks like (321-(b-1))/123=b-2 or (57-(4-1))/27=54/27=2 in decimal.

Essentially and perhaps unsurprisingly, the size of the slices in the number pie get smaller the bigger the pie gets. In binary, the slice is the pie, which is why the division comes out to zero there.

On an 8 digit calculator the common variant of this was

    12345679 * 8 = 98765432