# 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:** 20\ **Page:** 1
**Author:** ![mandala](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mandala/32/95_2.png) [@mandala](https://boards.straightdope.com/u/mandala)\ **Post date:** [September 14, 2013, 8:59pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/1 "2013-09-14T20:59:30Z")
Came across an equation that seeks to prove 3 = 2, and turn our world upside down. Can someone explain, at a high-school algebra level, why this might be incorrect? Here are the equations: -6 = -6 So, 9-15 = 4-10 adding 25/4 to both sides: 9-15+25/4 = 4-10+25/4 Changing the order: 9+25/4-15 = 4+25/4-10 This is of the form a square + b square - 2.a.b = (a-b) square. So this equation can be written as: (3-5/2)(3-5/2) = (2-5/2)(2-5/2) Taking positive square root on both sides: 3 - 5/2 = 2 - 5/2 So, 3 = 2. Thoughts? ---
**Author:** ![Carmady](https://avatars.discourse-cdn.com/v4/letter/c/f04885/32.png) [@Carmady](https://boards.straightdope.com/u/Carmady)\ **Post date:** [September 14, 2013, 9:11pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/2 "2013-09-14T21:11:49Z")
> [@mandala](#): > > Taking **positive** square root on both sides: > 3 - 5/2 = 2 - 5/2 (bolding mine) ---
**Author:** ![Sunspace](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/sunspace/32/1250_2.png) [@Sunspace](https://boards.straightdope.com/u/Sunspace)\ **Post date:** [September 14, 2013, 9:21pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/3 "2013-09-14T21:21:09Z")
Somebody divided by zero! I’m telling! ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [September 14, 2013, 9:21pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/4 "2013-09-14T21:21:32Z")
The trick is that you can’t take that square root without evaluating the ^2 first. Normally you can say sqrt(n^2) = 5 where “n” is some expression because they’re functionally equivalent. The trick here is that “n” is a negative number in one case. 4+25/4-10 = -1/2 9+25/4-15 = 1/2 So sqrt(1/2^2) = sqrt((-1/2)^2) =\> sqrt(1/4) = sqrt(1/4) =\> 1/2 = 1/2 ---
**Author:** ![John\_DiFool](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_difool/32/19543_2.png) [@John\_DiFool](https://boards.straightdope.com/u/John_DiFool)\ **Post date:** [September 14, 2013, 9:28pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/5 "2013-09-14T21:28:04Z")
Thanks-I knew it was an order of operations error of some sort, that would do it. ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [September 14, 2013, 9:34pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/6 "2013-09-14T21:34:55Z")
> [@Jragon](#): > > Normally you can say sqrt(n^2) = 5 where “n” is some expression because they’re functionally equivalent. The trick here is that “n” is a negative number in one case. Uh… sqrt(n^2)=n Actually, nevermind. Every square root is equal to five, screw it, it makes everything easier. ---
**Author:** ![leahcim](https://avatars.discourse-cdn.com/v4/letter/l/b4bc9f/32.png) [@leahcim](https://boards.straightdope.com/u/leahcim)\ **Post date:** [September 14, 2013, 10:07pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/7 "2013-09-14T22:07:22Z")
> [@Jragon](#): > > Uh… sqrt(n^2)=n No, it’s not. At least if you take “sqrt” to mean the positive square root. sqrt(n^2) = n for n \>= 0, and sqrt(n^2) = -n for n \< 0. 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. ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [September 14, 2013, 10:17pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/8 "2013-09-14T22:17:19Z")
> [@leahcim](#): > > No, it’s not. At least if you take “sqrt” to mean the positive square root. > > sqrt(n^2) = n for n \>= 0, and sqrt(n^2) = -n for n \< 0. > > 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. Which is exactly what I explained in my first post, I was correcting the phrase “we can usually say sqrt(n^2)=5” where I meant “sqrt(n^2)=n”. I then went on the explain exactly what you explained – that in this case it doesn’t work because “n” is negative, so we can’t use the same sqrt(n^2)=n identity that we use when n \>=0. ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [September 14, 2013, 10:21pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/9 "2013-09-14T22:21:57Z")
2 and 3 turn would turn out to be the same number all along if arithmetic had been axiomised by M Night Shyamalan. ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [September 14, 2013, 10:24pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/10 "2013-09-14T22:24:49Z")
> [@Sunspace](#): > > Somebody divided by zero! I’m telling! I’m actually surprised it wasn’t division by zero. These kinds of proofs **always** involve dividing by zero. The mathtrolls are getting smarter. ---
**Author:** ![leahcim](https://avatars.discourse-cdn.com/v4/letter/l/b4bc9f/32.png) [@leahcim](https://boards.straightdope.com/u/leahcim)\ **Post date:** [September 14, 2013, 10:25pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/11 "2013-09-14T22:25:08Z")
> [@Jragon](#): > > Which is exactly what I explained in my first post, I was correcting the phrase “we can usually say sqrt(n^2)=5” where I meant “sqrt(n^2)=n”. I then went on the explain exactly what you explained – that in this case it doesn’t work because “n” is negative, so we can’t use the same sqrt(n^2)=n identity that we use when n \>=0. Sorry, I mis-parsed and thought that you were actually arguing that sqrt(n^2) = n in all cases in the post I was replying to, rather than making a correction to your first post. ---
**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 14, 2013, 10:35pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/12 "2013-09-14T22:35:12Z")
> [@Asympotically\_fat](#): > > 2 and 3 turn would turn out to be the same number all along if arithmetic had been axiomised by M Night Shyamalan. It’s not only tricks involving division by zero and tricks involving sqare roots that have been used to equate all numbers and destroy arithmetic. _Basic differential calculus can be used to “prove” 1=2 and thus all numbers are equal to each other._ The “proof” has few steps, is easy to read, and is very straight-forward. Or so it seems. Will I hear crickets chirping or will someone here beg me to show it? One math enthusiast I showed it to described it as “delicious” and only half of his instuctors spooted the mis-step. ---
**Author:** ![Leo\_Bloom](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/leo_bloom/32/10377_2.png) [@Leo\_Bloom](https://boards.straightdope.com/u/Leo_Bloom)\ **Post date:** [September 14, 2013, 10:58pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/13 "2013-09-14T22:58:25Z")
tl;dr Just caught this thread. OP’s wrong, right? Otherwise I gotta start packing because something’s gonna blow… ---
**Author:** ![Roland\_Orzabal](https://avatars.discourse-cdn.com/v4/letter/r/c2a13f/32.png) [@Roland\_Orzabal](https://boards.straightdope.com/u/Roland_Orzabal)\ **Post date:** [September 14, 2013, 11:01pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/14 "2013-09-14T23:01:51Z")
> [@Trans Fat Og](#): > > It’s not only tricks involving division by zero and tricks involving sqare roots that have been used to equate all numbers and destroy arithmetic. _Basic differential calculus can be used to “prove” 1=2 and thus all numbers are equal to each other._ The “proof” has few steps, is easy to read, and is very straight-forward. Or so it seems. > > Will I hear crickets chirping or will someone here beg me to show it? One math enthusiast I showed it to described it as “delicious” and only half of his instuctors spooted the mis-step. Never seen a calculus version, so sure, post away. If nothing else I can use it to screw with the Calc 1 kids I tutor. ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [September 14, 2013, 11:06pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/15 "2013-09-14T23:06:04Z")
> [@Trans Fat Og](#): > > It’s not only tricks involving division by zero and tricks involving sqare roots that have been used to equate all numbers and destroy arithmetic. _Basic differential calculus can be used to “prove” 1=2 and thus all numbers are equal to each other._ The “proof” has few steps, is easy to read, and is very straight-forward. Or so it seems. > > Will I hear crickets chirping or will someone here beg me to show it? One math enthusiast I showed it to described it as “delicious” and only half of his instuctors spooted the mis-step. Yep, there’s a lot of 2=1 ‘proofs’, I guess the one you may be thinking of is (IIRC): x[sup]2[/sup] = x\*x x\*n = x+…+x with the number of x’s on the RHS being n Therefore x[sup]2[/sup] = x+…+x with x number x’s Setting f(x) = x[sup]2[/sup] and g(x) = x+…+x (with x number of x’s) Then f(x)=g(x) and therefore f’(x)=g’(x) From basic calculus: f’(x)=2x and g’(x) = 1+…+1 (with x number of 1’s) = x Therefore: 2x=x Divide by x: 2=1 QED Of course it’s fairly obvious where that one goes wrong. ---
**Author:** ![Roland\_Orzabal](https://avatars.discourse-cdn.com/v4/letter/r/c2a13f/32.png) [@Roland\_Orzabal](https://boards.straightdope.com/u/Roland_Orzabal)\ **Post date:** [September 14, 2013, 11:15pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/16 "2013-09-14T23:15:14Z")
Oops, misread the setup. Ignore this post. Will follow up later. :smack: ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [September 14, 2013, 11:29pm UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/17 "2013-09-14T23:29:08Z")
> [@Roland\_Orzabal](#): > > Oops, misread the setup. Ignore this post. Will follow up later. :smack: 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. ---
**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, 12:15am UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/18 "2013-09-15T00:15:42Z")
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 feel dumb. ---
**Author:** ![SCAdian](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/scadian/32/3292_2.png) [@SCAdian](https://boards.straightdope.com/u/SCAdian)\ **Post date:** [September 15, 2013, 1:20am UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/19 "2013-09-15T01:20:37Z")
> [@Jragon](#): > > The trick is that you can’t take that square root without evaluating the ^2 first. > > Normally you can say sqrt(n^2) = 5 where “n” is some expression because they’re functionally equivalent. The trick here is that “n” is a negative number in one case. > > 4+25/4-10 = -1/2 > > 9+25/4-15 = 1/2 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 ---
**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, 1:37am UTC](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866/20 "2013-09-15T01:37:44Z")
> [@Jragon](#): > > I’m actually surprised it wasn’t division by zero. These kinds of proofs **always** involve dividing by zero. The mathtrolls are getting smarter. You can always convert a wrong proof that assumes a^2 = b^2 -\> a = b to one that wrongly assumes that x_a = x_b -\> a = b, even if x is 0, via the process: a^2 = b^2 -\> a^2 - b^2 = 0 -\> (a + b)(a - b) = 0 -\> a - b = 0 -\> a = b I wonder if there is a way to make the reverse transformation to have a full “equivalence of wrong proofs” theorem. [Next page](https://boards.straightdope.com/t/mathematical-proof-for-2-3/668866.md?page=2)