Concept of Quantity

Originally Posted by C_ntua

So to clarify one more time, 0.999... cannot be proven to be equal to 1
a) Without axioms about infinite sums
b) Without axioms about decimals with infinite digits
c) Without something else

AND if this is attempted the definition of infinity will save 90% of arguing, so people can give their definition of infinity at least in the sense of "infinite decimal digits"

b implies a, so you could shorten your list, if you want.

In general, aren't all decimal expansions infinite sums? Decimal representation - Wikipedia, the free encyclopedia

What makes equations that involve infinity wrong?

Any number multiplied by its inverse is 1 since N*(1/N)=1 because the N's cancel on each other. Now, since 0 is the inverse of infinity, is it safe so assume that 0*infinity=1? Methinks so.

0 isn't the reciprocal of infinity.

inf/inf = 1
1/inf = 0
0inf = 0

All of these are made rigorous by the limit concept.

Originally Posted by whiteflags

0 isn't the reciprocal of infinity.

inf/inf = 1
1/inf = 0
0inf = 0

All of these are made rigorous by the limit concept.

They are rigorous if you actually write them as limits. Written in the above way they are just nonsense.

Think of it this way:
Think infinity as the universe ( and ignore my signature ).
The universe is constantly expanding. Likewise, two infinities may not be the same...

I know... I think... I guess... who am I kidding?

Originally Posted by KAUFMANN

Any number multiplied by its inverse is 1 since N*(1/N)=1

Except 0. Otherwise...

0 * (1/0) = 1

because the N's cancel on each other. Now, since 0 is the inverse of infinity, is it safe so assume that 0*infinity=1? Methinks so.

Since the above shows you, I hope, that 0 can't be the inverse of infinity (infinity has no inverse), 0 * infinity must be constructed differently. The current agreement is that infinity multiplied by any number, including 0 is undefined.

Originally Posted by KAUFMANN

Any number multiplied by its inverse is 1 since N*(1/N)=1 because the N's cancel on each other. Now, since 0 is the inverse of infinity, is it safe so assume that 0*infinity=1? Methinks so.

It depends. Do you mean 0 multiplied by something that approaches infinity or something that approaches 0 multiplied by something that approaches infinity? If the former, no, it would be 0. If the latter, sometimes; It could approach anything, because it is an indeterminate form(Indeterminate form - Wikipedia, the free encyclopedia).

Originally Posted by whiteflags

inf/inf = 1
1/inf = 0
0inf = 0

lim(x->inf)x/lim(x->inf)x^2 = lim(x->inf)x/x^2 = lim(x->inf)1/x = 0
The second is correct. (I even used it above.)
lim(x->inf)1/x * lim(x->inf)x = lim(x->inf)x/x = lim(x->inf)1 = 1

The first and third are indeterminate, and can't be given a general rule.