Re: Help with confirmation on a small bit of assembly?

[dempson@actrix.gen.nz (David Empson)]

John B. Matthews <nospam@nospam.com> wrote:

> In article <pvgcr2ljj2nilnh3em53656c16eiihj3to@4ax.com>,
>  "Jeff Blakeney" <jeff.blakeney@a2central.com.remove-xme-this> wrote:
> 
> >   To: mdj
> > On Tue, 23 Jan 2007 05:32:11 -0800, "mdj" <mdj.mdj@gmail.com> wrote:
> > 
> > >David Empson wrote:
> > >
> > >> I'm sure someone can come up with a proof that 16 = 15, based on the 1 =
> > >> 0 concept. :-)
> > >
> > >Damn you Empson! I'd never seen that before, and now that I've Googled,
> > >I'll be awake for days trying to decide if it's funny or not
> > 
> > A college classmate showed me one of those on a chalk board one day
> > many years ago but the fact that he used a divide by zero in the
> > process made the result meaningless.  :-)
> 
> It is well known that 16 = 15 for sufficiently large values of 15:-)

Tee hee.

> 1. -240 = -240 ; identity
> 2. 256 - 496 = 225 - 465 ; still equal
> 3. 16^2 - 16(31) = 15^2 - 15(31) ; factor each side
> 4. 16^2 - 16(31) + (31/2)^2 = 225 - 15(31) + (31/2)^2 ; complete square
> 5. (16 - 31/2)^2 = (15 - 31/2)^2 ; factor each side again
> 6. 16 - 31/2 = 15 - 31/2 ; take square root & add 31/2, each side
> 7. 16 = 15 ; qed?
> 
> No division by 0; can you spot the bogus step?

Without looking at the answer, it must be step 6 (square root).

If A^2 = B^2, it does not follow that A = B. There are two possible
solutions: either A = B, or A = -B.

In this case, 16 - 31/2 is 0.5, and 15 = 31/2 is -0.5. Squaring both of
these produces the same value (+0.25) and the square roots are therefore
also equal (+0.5), but the square root of (15 - 31/2)^2 is not (15 -
31/2), it is (16 - 31/2).

So there. :-)

Enough sillyness for one night.
-- 
David Empson
dempson@actrix.gen.nz