When we subtract some number m from another number n, we are essentially creating a relationship between n and m such that whatever the difference is, can be treated as a 'local identity' (relative value of '1') for n, and the base then becomes '(base n/(n-m))%1' (the floating point component).

for example, take any number, say 512
697/(697-512)
3.7675675675675677

here, 697 is a partial multiple of our new value of '1' whose actual value is the difference (697-512) 185 in base 10. proper multiples on this example number line, based on natural numbers, would be
185*1,
185*2
185*3, etc

The translation factor between these number lines becomes

0.7675675675675677

multiplying any base 10 number by this, puts it on the 1:185 integer line.

Once on a number line other than 1:10, you must multiply by the multiplicative identity of the new number line (185 in the case of 1:185), to get integers on the 1:10 integer line back out.

185*0.7675675675675677 for example gives us

185*0.7675675675675677
142.000000000000

This value, pulled from our example, would be 'zero' on the line.

185 becomes the 'multiplicative' identity of the 1:185 line. And 142 becomes the additive identity.

Incidentally the proof of this is trivial to see just by example. if 185 is the multiplicative identity of 697-512, and and 142 is the additive identity of number line 1:185

then any number '1', or k=some integer, (185*(k+0.7675675675675677))%185
should equal 142.

because on the 1:10 number line, any number n%1 == 0

We can start to think of the difference of any two integers n, as the multiplicative identity of a new number line, and the floating point component of quotient of any number n to the difference of any number n-m, as the additive identity.

let n =697
let m = 185
n-m == '1' (for the 1:185 line)

(n-m) * ((n/(n-m))%1) == '0'

As we can see just like on the integer number line, n%1 == 0
or in the case of 1:185, it equals 142, our additive identity.

And now, the purpose of this long convoluted post: all so I could bait people into reading a rant on division by zero.