Mathematical puzzle

[altivo]

This thing popped up as a fortune cookie on Linux this morning. It took me a while to figure out where the error was. Let's see who can find it first.

Proof that 2 equals 1:

x = y	[Given]
x2 = xy	[Multiply both sides by x]
x2 - y2 = xy - y2	[Subtract y2 from both sides]
(x + y)(x - y) = y(x - y)	[Factor]
(x + y) = y	[Cancel out (x - y) term]
2y = y	[Substitute x for y, by original equation]
2 = 1	[Divide both sides by y]

I'll supply the answer after folks take a crack at it. It isn't as hard as it looks, and someone may get it right away, so replies are screened for now. It was credited to Omni Magazine.

Answer:
I declare quaxo the first to put his paw right on the heart of the error, though you were all on the right track. If X = Y, then (X - Y) = 0. And the fourth step ("Cancel out (X - Y) term") is actually a divide by zero error. Up to that point, everything was perfectly kosher, even though some of it was nonsensical. After that, nothing is right.

Comments

[daikitsune.livejournal.com]

I don't know if it's fair for me to do this, being an engineering major. I'll just say "0=0" and let someone else figure out the rest. ;)

[vakkotaur.livejournal.com]

The "Factor" step is the key. If X=Y then X-Y=0 and anything multiplied by 0 is 0.

[scruff.livejournal.com]

Isn't (X - Y) zero by the first definition?

Can't divide by 0...

[kakoukorakos.livejournal.com]

If x=y, then x-y=0. You cannot divide by zero.

[kakoukorakos.livejournal.com]

Oh, that said, I miss Omni. It was an awesome magazine. I have an old issue around that I nabbed out of the magazine bin at Eco-cycle. It had a nice mix of science, philosophy, imagination, and generalized intellectual pursuits.

[gldm.livejournal.com]

X = Y ok
X^2 = XY ok
X^2 - Y^2 = XY - Y^2 <----- If X = Y both sides of this must be 0
(X+Y)(X-Y) = Y(X-Y) <--- if X = Y, X-Y = 0, both sides multiplied by 0, 0 = 0.
(X+Y) = Y <---- false, factor does not apply as you can factor any term out of an equation with a 0 term on both sides as 0 * N = 0 where N is a real number.
2Y = Y <--- false unless Y = 0
2 = 1 <---- always false

[darkhorseman.livejournal.com]

.
..
...
DEVIDE BY 0 ERROR
64K OK

Crap now I know why I slept through that class

[kistaro]

The error is on the third-to-last line. There's no such thing as "cancel terms"- that's a euphemism for either subtraction or division, depending upon context. This time, obviously, it's division. So we're dividing by (X-Y). Now there's this nifty little rule in mathematics that you can't divide by zero becuase your results are total nonsense otherwise. Therefore, (X-Y) != 0. So let's add Y to both sides, and we get X != Y.

Unfortunately, it was established as the given on line 1 that X == Y. Therefore, as soon as the division step occurs, the expression becomes a contradiction with an empty range of validity; the entire expression becomes the Contradiction, F.

Amusingly, there's another wrong-er move available that instead concludes that all numbers are zero. 2Y=Y, if it appears in a real expression, is not an invitation to divide by Y. It is, instead, a conclusion that Y = 0, because it's the only number that works for, but it's more obvious when instead of trying to divide (which is actually the second divide-by-zero in the same formula), subtract Y from each side (which is always a legal move as long as the expression isn't yet F): you get Y = 0. Of course, since the only given was X = Y, then you could use this to instead conclude that all numbers are zero, because any number can be set equal to itself since there are no constraints on X.

This is why we don't divide by zero.

[cats-haven.livejournal.com]

AAAAAAHHHHHH!!!!!!! *runs and hides from the scary math equations*

Actually, it makes some sense when I do it on paper like my teachers taught me, but I still don't get it. Go ahead and laugh, I know I suck at numbers and algebraic equations. :)

Ode To Omni

[pioneer11.livejournal.com]

Omni.

My first slick mag love.

Today, Wired holds that slot, but Omni did something that Wired
doesn't. Omni had some amazing Science Fiction, introducing
us to Cyberpunk among other things.

It was published by a pornographer.

http://en.wikipedia.org/wiki/Bob_Guccione

Top shelf girls was what he got, along with things Playboy
didn't allow; namely pussy and erections.

So any mag Bob was gonna spring for was gonna have
the BEST. And really, Omni did have the very best
fiction you could ask for.

Along with great articles, science intervies, and
things that even today (looking at my collection) are
still forward looking as of this late date.

Bob didn't end well, at least in terms of his family.
Its hard to imagine him taking the kids out for
ice cream without some naked dark haired woman coming
along, naked, on his arm. Thats why he never did
the family thing, he was too busy destroying it.

And it showed in his personal life, his kids turned
against him, the same way most kids of powerful,
corrupt, intelligent, men do.

But as for the question.

This is why I don't work for NASA.

XD

[leopanthera.livejournal.com]

You can't cancel out X-Y, because X=Y and therefore X-Y=0. You can't divide by zero.

[brunbera.livejournal.com]

Ah, the classic "hide the zero" conundrum. Well played, sir!

X=Y; X-Y=0; (X+Y)(X-Y)=Y(X-Y); ((X+Y)(X-Y))/(X-Y)=(Y(X-Y))/(X-Y)
((X+Y)(0))/0)=(Y(0))/(0) - DIV BY ZERO ERR.

^.^

[duraji-synth.livejournal.com]

I remember seeing this once before. When I had a far sharper mathematical mind in high school, I figured it out in a couple minutes. I remember it having to do with dividing by zero or something.

[biancomago.livejournal.com]

for me... there seems to be something off with this..... *is boggled* I think there is a flaw somewhere....

then again... math was never my strong point *grins*

[songcoyote.livejournal.com]

Unfortunately I've seen this one before and know the answer (I was a math major originally and came across it pretty early).

The error is in the "cancel out (X-Y) term" step, and lies in the fact that it isn't actually just "cancelled out", but that you must divide both sides of the equation by that term, and since X=Y you are dividing by zero, thus invalidating the entire operation.

Whee! I like occasionally knowing answers :)

Light and laughter,
SongCoyote

[altivo.livejournal.com]

0=0 is perfectly OK, but dividing by it won't do. :)

[altivo.livejournal.com]

On the right track, but the factoring was OK and the equation is still valid. The next step, though, does a divide by zero.

[altivo.livejournal.com]

You got it, at least to my way of thinking. Some of the earlier steps were nonsensical given that X = Y but all were legal and the results still valid.

[duskwuff.livejournal.com]

You're more right than you think. The fallacy is that the fourth step divides by zero! :-)

[altivo.livejournal.com]

Second to get the answer. :)

[altivo.livejournal.com]

Yup, you got it.

[altivo.livejournal.com]

It IS a divide by zero, in fact. If X = Y then (X - Y) is zero.

[altivo.livejournal.com]

Or rather, we only do it if we want silly results. :)

Somewhere at home I have a poster that was released by Byte magazine, done by their frequent cover artist. It shows what goes on inside a computer, and is full of absurdities, but my favorite piece is way at the bottom. It's a dungeon cell with eyes peering out of the darkness behind the bars, and scary claws poking out under the door. The sign on the door says "To divide by zero, lift latch.)

[altivo.livejournal.com]

Read 'em, they're good for you. ;p

[altivo.livejournal.com]

Yup, you got it. I was actually composing the post to ask someone to point out the flaw to me, and I typed "Can someone show me where the divide by zero or whatever it is occurs" when I saw it. :)

[altivo.livejournal.com]

So if I divide a hug by zero do I get twice as many hugs? ;p

[altivo.livejournal.com]

Answers above. Yes, it's a disguised divide-by-zero error.

[altivo.livejournal.com]

It's a divide by zero. Explanations above.

[altivo.livejournal.com]

You were a math major? Hmm, I'd never have figured you for that geeky. :) Yes, I didn't see it at first and then felt all pleased when I figured out where it was. Obviously others are quicker at spotting such things than I am, though.

Re: Ode To Omni

[altivo.livejournal.com]

Answers now posted. :)

[scruff.livejournal.com]

Technically, you get infinite hugs :)

[kint.livejournal.com]

If X = Y then X - Y = 0, therefore you cannot divide both sides of the equation by (X - Y) to cancel them out

[kakoukorakos.livejournal.com]

I see others did a better job of explaining the concept though, I left out the part about "cancellation of terms" being by division. :D

It's really a pretty sneaky thing they did there, it *almost* had me going until I was working it through in my mind, saw all the zeros (very suspicious!), and then thought out the real method of dropping a term from an equation. Still, it was a fun exercise, I've always had a knack for mathematics even if I haven't had to deal with the more complex varieties much recently.

[altivo.livejournal.com]

*runs around looking for a zero to divide into the next hug he gets* XD

[altivo.livejournal.com]

You got it. :) Divide by zero and you can make anything happen.

[gldm.livejournal.com]

Yay! I can still do teh maths!

P.S. Never saw this one before, guess I should have subscribed to more old computer magazines.

[songcoyote.livejournal.com]

Yah, my original intention when going to college was to learn how to teach math in high school - mostly because one particular teach of mine was not a particularly good teacher; I learned more from my father than I did from him. However, my brain went a bit nuts when I went to college, a bunch of stuff happened, I met someone and moved to the SF Bay Area, and I got a Biotech career accidentally. So, here I am :)

Life is full of surprises! WHEE!

Light and laughter,
SongCoyote

[darkhorseman.livejournal.com]

Well son of a bitch.. I was just being a smartass

[daikitsune.livejournal.com]

That works. I just figure you're screwed anyway at the point you reach 0=0. Isn't really any way to make it 1=2 after that. ;)

[cabcat.livejournal.com]

You must be very bored at work ;)

Oh god I'm bored at work......really bored

[altivo.livejournal.com]

Sure there is. ;p

0 = 0
2 * 0 = 1 * 0 (Dumb but true)
2 = 1 (Divide by zero to get rid of the extra term)

*ducks and runs*

We need to get you another user pic that doesn't look so serious all the time. XD

[altivo.livejournal.com]

Teaching math to high school students? Wow, not only are you geeky, you're a masochist too. ;p

[altivo.livejournal.com]

Frequently bored, not in a good way if you take my meaning. ;p

[daikitsune.livejournal.com]

I suppose I just take it for granted that you can't divide by zero and have anything meaningful. I wouldn't even think it needs to be said.

This is valid:
5/0 = cheese

But it ain't math. ;D

[altivo.livejournal.com]

Well, that's a different look. :)

But you see, if you're an engineer, you're used to thinking at a higher level of math than most of us ordinary schmucks. I left all that behind when I finished my last astronomy elective course in about 1970. I was impressed that I could find the flaw at all.

[songcoyote.livejournal.com]

I'm more of a "sensation slut" than a masochist per se, but....

Oh, wait. Wrong context :D

Now that I think on it I was more interested in teaching junior high (algebra, geometry, and trig), for all the difference that makes. Besides, schools weren't quite so scary 18+ years ago when all this was bubbling in my brain.

To quote Tor the Calculate: 2+2=4!

Light and laughter,
SongCoyote