# Mathematical proof for 2 = 3 **URL:** \ **Category:** Factual Questions\ **Created:** [September 14, 2013, 8:59pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866 "2013-09-14T20:59:30Z")\ **Posts on this page:** 15\ **Page:** 2
**Author:** ![leahcim](https://avatars.discourse-cdn.com/v4/letter/l/b4bc9f/32.png) [@leahcim](https://boards.straightdope.com/u/leahcim)\ **Post date:** [September 15, 2013, 1:47am UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/21 "2013-09-15T01:47:27Z")
> [@Jragon](#): > > I… uh… I can’t find the error. The best I can say is that the setup of g only really works if x is integral, but that’s kind of a weak refutation since it shouldn’t even work if x **is** integral. The other issue is that x might be 0, but that doesn’t feel like a magic bullet either. I want to say that, assuming there was a way to make the notation make sense for non-integral x, the derivative of (x + x + … + x) (x times) is being taken wrong. The contribution from each x is 1, but there is an additional contribution from the fact that the stack of x’s is growing with the size of x that is being ignored. The notation makes it awkward, but that is kind of the point. The derivative actually being taken is that of n_x, with n set to be equal to x afterward, which is not the same thing as the derivative of x_x. ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [September 15, 2013, 2:44am UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/22 "2013-09-15T02:44:23Z")
> [@SCAdian](#): > > Huh? > > 25/4 = 6.25 > > 4+25/4-10 = 4+6.25-10 = 10.25-10 = .25 > > 9+25/4-15 = 9+6.25-15 = 15.25-15 = .25 Derp, I meant that the factored terms come out to .5 and -.5 (before squaring). ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [September 15, 2013, 2:47am UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/23 "2013-09-15T02:47:47Z")
> [@Asympotically\_fat](#): > > Well you were right though, it isn’t particularly sophisticated. I think I may’ve seen a much better one involving calculus, but if I did I can’t for the life of me remember what it was. Here’s one possible “proof” of 2=1 using calculus. Two students were trying to evaluate the indefinite integral of 1/(2x). Using S to stand for the integral sign, that’s S(1/(2x))dx. One of them decided to use substitution. Let u = 2x. Then du = 2dx, so dx = (1/2)du. So S(1/(2x))dx = S(1/u)(1/2)du = (1/2)ln|u| + C = (1/2)ln|2x| + C The other student factored 1/(2x) into (1/2)(1/x). So S(1/(2x))dx = S(1/2)(1/x)dx = (1/2)ln|x| + C Neither student made any errors, and both answers, when differentiated, yield the original integrand of 1/(2x). So (1/2)ln|2x| + C = (1/2)ln|x| + C, from which we can derive that 2 = 1. I leave it as an exercise for the interested reader to find the flaw in the proof. ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [September 15, 2013, 2:53am UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/24 "2013-09-15T02:53:03Z")
I assume the trick is that spoilerln|2x| + C = (1/2)ln|x| + (1/2)ln|2| + C = (1/2)ln|x|+C Since C is arbitrary, we can roll the constant addition into it. Or, alternatively, since the two "C"s aren’t the same, we can set the C1 (the factoring term) to (C2(the u-substitution term) - (1/2)ln|2|).[/spoiler] ? ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [September 15, 2013, 3:01am UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/25 "2013-09-15T03:01:52Z")
> [@Jragon](#): > > I assume the trick is ? Yep ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [September 15, 2013, 3:06am UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/26 "2013-09-15T03:06:31Z")
I’m surprised so many people miss that one, it’s actually a surprisingly useful property of integrals, I’ve used it quite a few times, though I don’t recall the situation. ---
**Author:** ![Isilder](https://avatars.discourse-cdn.com/v4/letter/i/cc9497/32.png) [@Isilder](https://boards.straightdope.com/u/Isilder)\ **Post date:** [September 15, 2013, 5:37am UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/27 "2013-09-15T05:37:12Z")
> [@leahcim](#): > > The OP’s proof fails because when it gets to (3-5/2)(3-5/2) = (2-5/2)(2-5/2), he is saying: > > (1/2) ^2 = (-1/2) ^2 > > Which is entirely true, but then he concludes, through careless application of a positive square root that: > > 1/2 = -1/2 > > Which is false. The simplification is an example of the wrongness. The wrongness is in trying to use square roots in the proof of LHS=RHS. Actually after the square root, the mathematician knows to check for if the LHS is absurd compares to the RHS… Where the result is absurd, he blames the square root process for introducing an error. The idea is to check for the +ve and -ve of each square root to see if specific choices of +ve and -ve leads to a non-absurd result. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [September 15, 2013, 7:32pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/28 "2013-09-15T19:32:49Z")
Every such “proof” either involves division by 0 or taking the wrong square root. Every incorrect geometric proof, on the other hand, involves ignoring “betweenness”. For example, I know a simple proof that every angle is a right angle. ---
**Author:** ![Malacandra](https://avatars.discourse-cdn.com/v4/letter/m/45deac/32.png) [@Malacandra](https://boards.straightdope.com/u/Malacandra)\ **Post date:** [September 15, 2013, 7:43pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/29 "2013-09-15T19:43:35Z")
Also: $1 = 100¢ $0.1 = 10¢ ($0.1)[sup]2[/sup] = (10¢)[sup]2[/sup] $0.01 = 100¢ 1¢ = $1 ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [September 15, 2013, 8:17pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/30 "2013-09-15T20:17:57Z")
nm ---
**Author:** ![Find\_Friends](https://avatars.discourse-cdn.com/v4/letter/f/c37758/32.png) [@Find\_Friends](https://boards.straightdope.com/u/Find_Friends)\ **Post date:** [September 15, 2013, 8:46pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/31 "2013-09-15T20:46:28Z")
> [@Asympotically\_fat](#): > > Yep, there’s a lot of 2=1 ‘proofs’, I guess the one you may be thinking of is (IIRC): > > Of course it’s fairly obvious where that one goes wrong. Yep, you nailed it with your guess. It should be “obvious” where the flaw is, at least to the instructors, but it isn’t always. > [@Thudlow\_Boink](#): > > Here’s one possible “proof” of 2=1 using calculus. > > Two students were trying to evaluate the indefinite integral of 1/(2x). Using S to stand for the integral sign, that’s S(1/(2x))dx. > > One of them decided to use substitution. > Let u = 2x. > Then du = 2dx, so dx = (1/2)du. > So S(1/(2x))dx = S(1/u)(1/2)du = (1/2)ln|u| + C = (1/2)ln|2x| + C > > The other student factored 1/(2x) into (1/2)(1/x). > So S(1/(2x))dx = S(1/2)(1/x)dx = (1/2)ln|x| + C > > Neither student made any errors, and both answers, when differentiated, yield the original integrand of 1/(2x). > So (1/2)ln|2x| + C = (1/2)ln|x| + C, from which we can derive that 2 = 1. > > I leave it as an exercise for the interested reader to find the flaw in the proof. I had never heard of that one, or any at all involving _ **integral calculus** _. Thank you. ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [September 15, 2013, 8:49pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/32 "2013-09-15T20:49:32Z")
> [@leahcim](#): > > I want to say that, assuming there was a way to make the notation make sense for non-integral x, the derivative of (x + x + … + x) (x times) is being taken wrong. The contribution from each x is 1, but there is an additional contribution from the fact that the stack of x’s is growing with the size of x that is being ignored. The notation makes it awkward, but that is kind of the point. > > The derivative actually being taken is that of n_x, with n set to be equal to x afterward, which is not the same thing as the derivative of x_x. That seems right, you can’t ignore the “x” in the implicit summation. ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [September 15, 2013, 8:57pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/33 "2013-09-15T20:57:45Z")
> [@Malacandra](#): > > Also: > > $1 = 100¢ > $0.1 = 10¢ > ($0.1)[sup]2[/sup] = (10¢)[sup]2[/sup] > $0.01 = 100¢ > 1¢ = $1 Similarly: 1 yard = 36 inches 1/6 yard = 6 inches (1/6 yard)[sup]2[/sup] = (6 inches)[sup]2[/sup] 1/36 yard = 36 inches 1 inch = 1 yard The flaw in the second is more obvious because we know about square inches and square yards, but we don’t have a concept of square cents or square dollars. ---
**Author:** ![Carmady](https://avatars.discourse-cdn.com/v4/letter/c/f04885/32.png) [@Carmady](https://boards.straightdope.com/u/Carmady)\ **Post date:** [September 16, 2013, 12:01am UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/34 "2013-09-16T00:01:32Z")
> [@Trans Fat Og](#): > > Yep, you nailed it with your guess. It should be “obvious” where the flaw is, at least to the instructors, but it isn’t always. I’ll admit I don’t see what the “obvious” flaw is supposed to be, but maybe I missed something. It doesn’t make enough sense to me to have an obvious flaw. If you say you are going to take the positive square root, and then you take the negative square root, your reasoning is clear but you made a mistake. But if you say: x2 = x+…+x with x number x’s g(x) = x+…+x (with x number of x’s) g’(x) = 1+…+1 (with x number of 1’s) = x Then I am at a loss to understand your reasoning, so there are any number of possible flaws. 1. Writing a nonsensical statement (“x number of x’s for x a real number”) 2. Taking the derivative of something nonsensical (flaw: use of magic) 3. Taking the derivative of a function whose domain is the natural numbers 4. Assuming a property of natural numbers holds for all real numbers 5. Treating the first x in “x number of x’s” as a constant when differentiating I take it you mean flaw #5 to be the obvious one? But it doesn’t really work because the rest doesn’t make sense either. (again, I could be missing something) ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [September 16, 2013, 12:04am UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/35 "2013-09-16T00:04:03Z")
> [@Carmady](#): > > (flaw: use of magic) I swear to god if I ever have to grade a math-heavy class again and students just pull answers out of their asses, from now on I’m going to write “-\ Use of magic”. [Previous page](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866.md?page=1)