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September 14, 2009 | by  | in Opinion |
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1 equals 0


If I told you that one is equal to zero you’d think I was stupid. After all, something isn’t nothing. But read through the proof below of just that, recounted in Charles Sefie’s fascinating and entertaining book Zero: The Biography of a Dangerous Idea.

Let a = 1, and let b = 1.

Obviously, a2 = a2 (equation 1).

Because a is equal to b, b2 = ab (equation 2).

Subtract equation 2 from equation 1 to get a2 – b2 = a2 – ab (equation 3).

If you haven’t studied any maths for a while, you’ll have to cast your mind back to fifth form for this next step. Factor both sides of equation 3 to yield equation 4:
(a + b)(a – b) = a(a – b)

You should convince yourself that equations 3 and 4 really are the same thing.

Now, divide both sides of equation 4 by (a – b) to get a + b = a (equation 5).

Subtract a from both sides of equation 5 to get b = 0 (equation 6).

Take a look at the first line of the proof: we set b to equal 1, so 1 = 0

So, there you have it: one equals zero! And in fact, this proof works for 1, 5, 12345, and any number you care to try.

Okay, so obviously one does not equal zero, and nor does five, and nor does 12345—if this were true maths as we know it wouldn’t work, and consequently, neither would engineering or science. Bridges would collapse, computers would stop working, and we’d probably all explode. Well, maybe not that last one, but it’d be a pretty dire thing if thousands of years of mathematics and science was based on illusory foundations.

But before you jump out of the window in a ‘the end is nigh’ panic, don’t! The ‘proof’ wasn’t really a ‘proof’ but merely a mathematical sleight of hand. The trick was in between equations 4 and 5 where we divided by a – b. Because a = b = 1, a – b equals 1 – 1, which equals 0. And like your high school maths teachers and your calculator tell you, anything divided by zero is undefined; disallowed; meaningless.

One doesn’t equal zero, but if we were to allow division by zero it would lead to all sorts of craziness.

Take a look at Seife’s book to see why the above ‘proof’ that 1 = 0 is equivalent to a ‘proof’ that Winston Churchill is equal to a carrot. And even if this proof is old news to you, The Biography of a Dangerous Idea is full of lots of other cool things—like how to make a wormhole time machine in four easy steps. Check it out.

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Comments (3)

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  1. Electrum Stardust says:

    Division by zero: Feedback loops, Chaos theory, or shameless “pseudo-science”?

  2. Electrum Stardust says:

    Last time I checked, The Annotated Hobbit is selling at an unbelievable price at Unity Books. Well worth a look.

    P.S. And Many Happy Returns to Bilbo and Frodo!

  3. ZHAO says:



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