# 1 + 1 = 3 can someone prove this?

Discussion in 'After Hours Lounge (Off Topic)' started by Mark Giles, Jun 17, 2005.

1. ### Mark Giles Second Unit

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There is a way. This is not a riddle. Purely mathematical. Wont give any hints yet, but curious who can prove this first. Good luck!

2. ### Chris Bardon Cinematographer

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It works, but only for extremely large values of 1.

3. ### Rob Gardiner Cinematographer

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If I remember correctly, the solution to this puzzle requires dividing by zero, which is a big no-no.

4. ### ThomasC Lead Actor

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I saw this on a professor's door. The logic is flawed, though. "Proof" that 1 = 2. Let a = 1 and b = 1.

a = b

a^2 = ab

a^2 - b^2 = ab - b^2

(a + b)(a - b) = b (a - b)

a + b = b

Unfortunately, (a - b) = 0, negating (a + b) and b.

5. ### Mark Giles Second Unit

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Nope, that's not the answer. sorry

6. ### Bill Williams Screenwriter

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Here's one way of looking at it...

Husband - 1
Wife - 1
Husband and wife have baby
Husband, wife, and baby makes 3!

It takes two to make three!

7. ### Mark Giles Second Unit

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Very good ThomasC

I'll think of something harder next time :b

8. ### Haggai Producer

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Rob, you remembered right, as ThomasC's post explains in the spoiler.

Here's a fallacy that illustrates a different concept. Can you spot the error in this "proof" that 1 = 0? The ... in each line here means to repeat the pattern on into infinity.

1 = 1 + 0 + 0 + 0 + ...
= 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ...
= (1 - 1) + (1 - 1) + (1 - 1) + ...
= 0 + 0 + 0 + ...
= 0

9. ### Jason Harbaugh Cinematographer

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10. ### Danny Tse Producer

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I had a math professor who did this on the board one day and it was quite an eye-opener.

Now that I am a government analyst, I do see this type of "new math" used regularly.

11. ### RobertR Lead Actor

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You left out the first 1 in line 3, Haggai

Here's the "proof" that girls are evil:

girls = time * money

There's an old saying that time is money, so

girls = money * money

Another saying is that money is the root of all evil:

money = sqrt(evil)

substituting, we get

girls = sqrt(evil) * sqrt(evil), or

girls = evil

12. ### ThomasC Lead Actor

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Well, I had it handed to me as I mentioned in my first post.

13. ### ThomasC Lead Actor

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And that knowledge is evil as well:

Knowledge is power
Power corrupts
Corruption is evil
Thus, knowledge is evil

14. ### andrew markworthy Producer

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One plus another one is either eleven or two - it depends whether you're using arabic or roman numerals.

15. ### Haggai Producer

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Nope, that wasn't the intention. The formatting here isn't ideal, but look at the difference between lines 2 and 3, the intent there is that I've just re-arranged the parentheses from line 2, all of them being shifted one spot to the left. I'll change it to make the # of 1's match up, now there are 7 of them in both line 2 and in line 3:

1 = 1 + 0 + 0 + 0 + ...
= 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ...
= (1 - 1) + (1 - 1) + (1 - 1) + (1 - ...
= 0 + 0 + 0 + ...
= 0

Of course, that step between lines 2 and 3 is where the fallacy/error is...any takers?

16. ### RobertR Lead Actor

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But that was what I was getting at. In line 2, it's of the form 1 + n(1-1), where n can be infinite. In line 3, it's just n(1-1), which isn't the same thing.

17. ### Kenneth Supporting Actor

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So where does that put knowledge about girls?

Cheers,

Kenneth

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Yeah well, the question is how a seemingly correct operation using the associative property of addition produced such a result.

I can only say that the associative property doesn't apply to infinite sums/series?

--
H

19. ### Haggai Producer

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You're getting close...that property does apply to certain kinds of infinite sums, but not to others. Full explanation soon, unless someone tries to kill me first.

20. ### Haggai Producer

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OK, I'll go ahead and fill in all the details.

Using the associative property (i.e., moving parentheses around without changing the value of anything) for infinite sums only works if the series, when written without any parentheses, converges to a finite number (and in a specific way, to boot, convergence alone isn't always enough, but I'll leave that alone for now). For instance, the infinite sum:

1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...

...equals 1 when summed to infinity. You can re-arrange the terms any way you like, group them in any way with any pattern of parentheses, and the infinite sum will still converge to the value of 1. But let's see what the infinite sum from my previous posts looks like without any parentheses:

1 - 1 + 1 - 1 + 1 - 1 + 1 - ....

And so forth. Does this infinite sum have a finite value? No, it diverges. Certain sums are obviously divergent because they keep getting bigger in an unstoppable way, i.e. 1 + 2 + 3 + 4 + ..., or 1 + 1 + 1 + 1 + ..., for instance. The series in question here doesn't do that, as each successive 1 is cancelled out by the next -1, but it diverges because it never approaches one particular value in an inexorable way. If you stop after finitely many steps, you get a sum of either 1 or 0, depending on where you stop.

So you can't re-arrange any parentheses in this case because the infinite sum does not actually represent a finite value in and of itself.