Wednesday, July 1, 2009

Mathematical Humor – Silly Proofs

Here is little quip for those of you who appreciate a bit of mathematical humor along with your tea and crumpets for breakfast. This genre is especially fun when observing the common mistakes of non-mathematicians in attempting to construct valid proofs. These kinds of jokes, however, reflect some insecurity that perhaps one’s own work may be based on false premises. There are many fallacious proofs of the type purporting to prove 1 = 2. Here is one example. Can you locate the problem with the proof?

Theorem. 3 = 4

Proof. Suppose
a + b = c.

This can also be written as:
4a – 3a + 4b – 3b = 4c – 3c.

After reorganizing:
4a + 4b – 4c = 3a + 3b – 3c.

Take the constants out of the brackets:
4(a + b – c) = 3(a + b – c)

Remove the same term left and right:
4 = 3.

[Note: courtesy of “Foolproof: A Sampling of Mathematical Folk Humor,” from Notices of the AMS, Volume 52, Number 1.]

7 comments:

  1. I don't see how the 2nd line follows from the 1st. It seems like an arbitrary statement. Gotta do something like:
    a+b=c
    4(a+b) = 4c (multiply both sides by four)
    4a + 4b = 4c
    etc.
    And no way does it correctly reach line 2 of above.

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  2. To reach line 2 from line 1, it helps to use parens: (4a – 3a) + (4b – 3b) = (4c – 3c) and evaluate each term in parens first. It yields a + b = c. Mathematicians frequently do seeminly simpleminded manipulations in order to get an expression in form that is useful later on.

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  3. Well, everything is ok except that you can't divide by zero, and a+b-c is zero.

    Apart from that this is beautiful. :-)

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  4. Excellent!! That's the fallacy in the proof. The supposition clearly states that a + b - c = 0, therefore if you try to divide by this expression, anything can happen. Good show!

    - Physics Groupie

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  5. Ahhhhhhh good trick... I like it! It had been making my brain hurt.

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  6. I am sorry to say, but you just proved zero equals zero. If a+b=c, then a+b-c=0. in the last line, you actually stated that 4(0)=3(0).

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  7. Oh, sorry, i did not see the other explanations.

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