Applicability of Maths to the Universe

[polymath257]

I'd actually rate it as fair. The problem with

.99999.... = 1

Here's something more formal:

Wikipedia -- 0.999..

I agree with the formal proof, which is what I think that the author of the Khan video (who, appears to be, at the time at least, a PhD finalist in mathematics) was trying to convey:

Yes. And the key should *always* be some way that shows the limit exists and is 1.

Just to give a *bad* example,

Suppose we let
x=1+2+2^2 +2^3 +2^4 + 2^5 + ...

Then,
2x=2+2^2 +2^3 +2^4 +2^5 +2^6 +...
=x-1

Which 'shows' that x= -1.

The problem is that the defining series for x doesn't converge, so the algebraic operations are not legitimate.

In the corresponding claim that
1=.9 + .09 + .009 +...

we would start with
x= .9 +.09 +.009 +...
and get
10x = 9 +.9 +.09 +...
=9+x,

leading to 9x=9, and then to x=1.

But this set of operations is ONLY legitimate if the series actually converges. And, yes, in this case, it does,
but that is a separate issue that needs to be addressed.

The Wiki proof is good because it effectively shows that convergence happens and what the convergence is to simultaneously.