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how does 1+1 = 3?

I've seen it around and I mean the obvious response is that its just wrong but people say stuff like there's just some special way to look at it or something. any ideas?

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closed as not a real question by Arturo Magidin, Jonas Meyer, Robin Chapman, KennyTM Dec 7 '10 at 12:02

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

They are wrong. Any proof that they present has a mistake, although it can be hard to spot. –  Qiaochu Yuan Dec 7 '10 at 3:48
Most commonly they involve dividing by zero, often disguised as cancellation (which is really division but people don't think of it that way) –  Ross Millikan Dec 7 '10 at 3:53
A recent example here used the principle that $x^2=y^2$ implies $x=y$: math.stackexchange.com/questions/13082/proved-2-2-5. As for your question, "How does 1+1=3?", I don't know what can be said except "It doesn't", unless you have a specific mathematical question. –  Jonas Meyer Dec 7 '10 at 3:57
Right now, this is not a real question. It is vague, there is no way to know what the poster is talking about (any argument that includes a false premise can be used to conclude that 1+1=3). If he has a specific "argument" he wants to talk about, fine, but I'm voting to close as it currently stands. –  Arturo Magidin Dec 7 '10 at 5:05
Letseatlunch: as you can see from the responses, there are false arguments with errors that are hard to spot, that end out with "1 + 1 = 3" or some such thing. The conclusions are false; the arguments are meant as entertainment and to challenge the reader to spot the error. –  Zarrax Dec 7 '10 at 5:17

4 Answers 4

up vote 2 down vote accepted

It follows directly from the elementary theorem 2=1.

Let $a=b$ be integers.

$a = b$

Multiply both sides by $a$: $a^2 = ab$

Subtract $b^2$ from both sides: $a^2 -b^2 = ab - b^2$

Factor: $(a-b)(a+b) = b(a-b)$

Cancel $(a-b)$ on both sides: $a+b = b$

Use the fact $a=b$: $2b = b$

Divide both sides by $b$: $2 = 1$

Now that we have $2=1$, we get $3 = 1 + 1$ as a corollary. And an elementary proof of the Riemann Hypothesis.

P.S. $a-b = 0$, so canceling is the same as division by $0$.

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I don't see how presenting another proof of a contradiction is supposed to help someone who is already confused by one. –  Qiaochu Yuan Dec 7 '10 at 4:09
By pointing out exactly where the trickery happens. Having a specific proof in front of you gives you the chance to really examine it and find the error. As opposed to trying to recall the proof from memory. Since he didn't give a specific argument in the question, it's a safe bet that he (she?) doesn't have it lying around. –  Mark Reitblatt Dec 7 '10 at 5:54
this was the kind of answer i was looking for thx –  Letseatlunch Dec 7 '10 at 22:40

I have no idea which people are saying $1+1 = 3$ and why they are saying it. Certainly, without any additional explanation this sounds like a simply false statement: in standard arithmetic, $1+1 = 2$, which does not equal $3$! (It is not even the case that $1+ 1 = 3$ modulo n for any $n > 1$, and considering numbers modulo $1$ is, literally, trivial: I believe there was another recent question about that.)

But there are real-world phenomena which are somewhat reasonably described as $1 +1 = 3$. For instance, suppose I am living on a desert island and I have fashioned a very crude scale out of coconuts, bamboo, crab exoskeleta, and so forth. My scale doesn't have decimals on it: it says that the weight of something is some whole number of pounds. Now it is entirely possible that I can have two rocks, each of which, according to the scale, weighs one pound, but when I put both rocks on the scale, it tells me that together they weigh three pounds.

Of course this is an instance of rounding error: e.g. maybe the true weight of each rock is something like $1.35$ pounds -- which gets rounded to $1$ -- so the true weight of the two rocks is something like $2.7$ pounds -- which gets rounded to $3$. I have explained this in the context of a somewhat unlikely scenario, but rounding errors are something that one has to take very seriously in applied mathematics and the sciences.

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It's actually much simpler than that. It's not a maths problem though.

OK, brace yourselves, I'm gonna tell you about the little bees and the pretty flowers...

More seriously, 1+1=3 is a colloquial expression for saying that when a man and a woman get together, they usually are soon 3 in the family.

No math there. More like a riddle, a word game.

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definitely the most creative answer –  Letseatlunch Dec 7 '10 at 22:39

More generally, if somewhere in our reasoning we make a mistake (by dividing by zero, assuming that $a^2 = b^2 \implies a = b$, assuming that $\lnot P \implies \lnot Q$ follows from $P \implies Q$), we are able to prove anything.

That is, $P \implies Q$ is always true if $P$ is false to begin with. Wikipedia calls this being "vacuously true", although I think the term is usually used when the hypothesis is not satisfied but the result is still true, as in the example given of all cell phones being turned off in a room simply because there are no cell phones.

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