I saw this T-shirt:
x = y
x[sup]2[/sup] = xy
x[sup]2[/sup] - y[sup]2[/sup] = xy -y[sup]2[/sup]
(x + y)(x - y) = y(x - y)
((x + y)(x - y) / (x - y)) = (y(x - y) / (x - y))
[therefore] (x + y) = y
If x = 1 and y = 1,
(1 + 1) = 1
The first four steps work. The fifth step contains division by zero, and so will not work.
Question: Is this anything more than a math joke? Or is it used in teaching math to illustrate that ‘figures don’t lie, but liars figure’?
I think in the t-shirt context it’s definitely a joke, although you could pose it as a kind of riddle in the classroom. Challenging your students to find the fallacy.
nevermind
The fallacious step is:
On line 5, when you divide each side by (x - y).
Since x=y, x-y=0.
You can’t divide by zero.
When you see anything along these lines, simply say to yourself “Right - division by zero.”
You then look for - and soon find - where this has happened.
Not always the case. I forget the details but I came across a similar one where the overlooked detail was that log(z) has more than one value in the domain of complex numbers.
Another good one is a nice geometric proof that all triangles are isosceles: which relies on a slightly distorted diagram that implies a certain point is contained within the triangle; whereas the actual construction means that it is always exterior. The point to get from that one is that good geometric reasoning can stand without needing a diagram (and is bloody difficult to do.) I can spot the error with the diagram, but if it was presented to me as text only I would have a very hard time of it.
I think these kind of conundrums have value in that they emphasis the need for precision and rigour in communication and they force a person to reflect properly on what they think they know.
Well if you take one drop of water, add in another drop of water, you still get ONE drop of water (albeit twice as large). So when it gets down to it, (1+1) can equal 1 under the right circumstances.
Or if you round to the nearest whole number:
0.51 + 0.51 = 1.02, so
1 + 1 = 1
Raymond Smullyan, in What Is The Name Of This Book? discusses the logical theorem that a false proposition implies any proposition. (That is, if something false is true, then anything is true.) He uses this to prove that, if 2 = 1, then I (that is, Smullyan) am the Pope.
He starts with the classic proof that 2 = 1 (as shown in the OP), then continues:
The proof shown in the OP is a classic example of a fallacy, quite commonly shown to beginning algebra students.
Beginning students, asked to find the fallacy, will often argue that the very first line,
Let x = y
is the error. The thinking is always something along the lines of: You can’t just assume that two variables have the same value, absent any evidence of that. (Or any variation of substantially that reasoning.)
Similarly, I once saw a “proof” (the details of which I forget) that 2 = 1 that somehow involved the square root of -1, with some incorrect rules for working with square roots of negative numbers.
ETA: Here’s an example of such a proof that I just found. As best I recall, the proof I saw was much simpler than this one, though.
Not with that attitude, you can’t.
1 + 1 = 1 for small values of 1.
It can indeed be used in teaching math. What lesson this example teaches depends on how specific you want to get.
At the lowest level, it’s “Be careful when dividing: watch out for division by zero.” I’ve seen students “solve” equations like
x[sup]2[/sup] = 6x
by dividing both sides by x and getting the solution x = 6 (thereby missing the other solution x = 0).
More generally, it teaches, as j_sum1 said, “the need for precision and rigour,” and that a false assumption or invalid step in an otherwise sound argument (mathematical or otherwise) can lead to a ludicrous conclusion.
On a T-shirt, it also teaches the lesson: “Beware. The wearer of this shirt is a raging math geek.”
(Not that there’s anything wrong with that.)
It’s like 7x13=28
Cite Enjoy!
The second, Pope-related argument also displays the problem of equivocation, in which “one” is taken to mean both the number one as well as a term meaning “the same person”.
That type of thinking is also encountered in formal software requirements specifications (e.g. SRS’s). There is a critical difference between the following clauses:
Those proof seem to always center around a number having two swuare roots, a positive and a negative one. Ultimately they boil down to:
1 = 1
sqrt(1) = sqrt(1)
therefore -1 = 1
When I was learning Algebra I was shown a similar proof, although it’s end result IIRC was that 2+2=3. In hinged on the same division by zero snag.
It was used to illustrate that one can get lost in algebra.
The entire proof works, and it seems pointless to go back and redo each step using hte numbers instead of the variables, but that’s the only way you’ll notice that, if x or y is 1 (same thing, per step 1), then line 5 includes division by zero.
So what you’ve really proven is that x can’t = 1.