Lol I'm still making this :D
We'd have to come to this at some point. You would think that it's not
actually that bad for addition, just that you can't remove leading zeroes
from numbers anymore.
50+65+0=50650
0+39=039
But we run into a problem. Where would the number 039 be on the number line?
Well, we'd have to have another axis. It would be a number plane. Not the
complex plane that you're probably used to, but something like a normal
number line, with a kind of y-axis that would represent the number of leading
zeroes the number has. So the number 0038 would be 2 units up from the number
38 on the 'normal' number line.
This number plane would only have one quadrant. Quadrants III and IV don't really make sense because you can't have a negative number of leading zeroes. Quadrant II doesn't make sense becasue you can't really have a negative number with leading zeroes. 0+(-4) would be the number '0-4', which doesn't make sense. Also, the normal number axis would be continuous for all real numbers but the leading zero axis would only be available for whole numbers. You can't have 1.2 leading zeroes, but you can have a leading zero on 1.2. You have to put the number zero one unit up on the number line too, since it technically has one leading zero.
Very easy. 450-0=45. Note that you can also have leading zeroes on zero, so 4500-00=45.
Okay, what is 0*8? If you said 00000000, you'd be correct. But what is 8*0? Easy, you just write 8 zero times. So the answer is, of course, an empty void. Where would the empty void number be on our number plane? It would be 1 unit to the right of -1. The number 0 with 0 leading zeroes. The 'origin' of our plane. This is weird.
000000 ÷ 3 = 00, of course. But 5 ÷ 0 is still undefined...