Originally Posted by
whiteflags
This is not a repeated sum. As proof of the idea that multiplication is repeated sums, I would accept that you added up 1/2 a total of 3/4 times. I'm genuinely interested in you demonstrating this idea as you stated it -- that multiplication is repeated addition, for reals other than integers.
But let's do it your way anyway!
If you want to multiply a number like 0.9... You could imagine multiplying each term in a equivalent series:
9/10 * 10 + 9/100 * 10 + 9/1000 * 10 and continuing. I don't know where you get the idea that there would be a last 8, because all you do when you multiply a number by its base is shift places. It's plenty easy to think of doing this for every 9 in 0.9...
Well, I had in mind that that is the way you define multiplication, but the way you use it makes sense (of course) as well.
The point is there that you have one less decimal digit. So if you do that with finite number of digits , lets say N, you get N-1 9s. So you are implying that when N is infinite then N is the same as N-1. In a way having one more 9 is still infinite 9s. Which is true if you say that N->oo but then the number itself doesn't have a specific quantity, it has a quantity always growing to some limit (that is 1). So is it really a number or just some process? I mean, would you define an ever growing quantity as a number?
Putting what I said earlier more clearly maybe
1) A number is a specific quantity, something static. A process changes.
2) Infinity is either the biggest possible number or a process of a quantity going bigger and bigger.
Using infinity in the sense that it is the biggest possible number would result that
Code:
0.99... (N=inf digits)* 10 = 9.999....(X digits) where X = N-1 < inf
if you use infinity as a process going bigger and bigger then 0.99... itself always grows, so I wouldn't consider it a number.
The way we use infinity is normally as something growing bigger without an upper bound, thus from the beginning of this topic I see 0.999... as a process, not really a number.
Of course I use my own definition of number. So my conclusion is that we can see the argument as having two types of quantity
a) Static quantities with a specific valued
b) Growing quantities. They either grow without a limit, which then we call infinity and negative infinity, or they go towards a limit.
Lets get away with the lexical difference and just use both. In this case how would you define the equality between a numberA and a numberB??
The way it is usually defined is that they are equal if there distance is as small as possible. Then how do you define the order? To differentiate you say that one is bigger than the other if there difference is bigger than the smallest possible quantity NOT if its bigger than zero.
Again, not to get subtracted, my thought is that when you use a number as both a) and b) I describe above, that we usually do, then you imply that a quantity very very small is the same as zero. In other words, when you measure a quantity you can just measure its limit, since it might change. So the limit and the quantity itself cannot be distinguished thus we use them the same way. More or less that is the background of all these proofs in my opinion. But this also implies that the equality sign the meaning of two quantities being equal, means something a little bit different, as I phrase above.
