Shrike,
With respect, you're doing the naughty thing and saying 0 x 6 = 0. But that wasn't what I was saying! I said
10 x 0. (I'll use 10 rather than your 6)
10 comes first based on the logical left to right rule I was taught early on at school so we establish there is a 10. 10 exists. Then the 0 (zero) comes next in the sum. So we've got a real 10 and an abstract zero. If we multiply the real 10 by the abstract 'not real ' 0 the answer is always 10.
If you want to assert
0 x 10
I suppose you could say "well, if I start out with nothing and I multiply it by 10 then I must end up with nothing" but this is undermined by the fact 0 + 1 is 1 and not 0.
It seems maths wants to have its cake and eat it. If you have nothing and multiply it by any figure you always end up with nothing but if you add a number to nothing you get a real number!
For the rules to make true sense, I think it should be like this
10 x 0 = 10
When you start off with a real number the result is always a real number, never a zero. However, if you start off with a zero, a nothing, the results should be:
0 x 10 = 0
0 + 10 = 0
0 divided by 10 = 0
0 - 10 = 0
I think that makes more sense than the system we've got. Also, the idea of negative numbers sounds a bit silly. If you have a zero - that means nothing, right? - so how can you then go from 0 to minus 1. That makes no sense.
People will say "we have that with the temperature - we have zero degrees centigrade and minus degrees centigrade" but if you really think about it logically, if you establish the concept of NOTHING as a measurement, then nothing (pardon the pun!) can be below it. Once you have nothing there's no number below it. Nothing is the bottom. You can't go lower than that because nothing is absolute. That's the end of the line, folks. Nothing is the last station on the track. You can't carry on down the line to the next station because nothing is the end of the line.
The concept of minus numbers is bizarre because it negates the concept of zero actually representing an absolute 'nothing' numerical measurement.