0 = -1

Back when I was a freshman in high school, and had an inadequate grasp of higher mathematics, I came up with an algebraic “proof” that I thought violated, well, something in math. I had “proved” that 0 = -1 using infinity. It was pretty basic. I don’t know why I remembered this today, but I thought it would be amusing to post.

It’s like this:

The symbol ∞ represents, well, infinity. So, you whittle infinity down to a simple variable and start with:

∞ = ∞

Nothing earth-shaking. But infinity being infinity, you could also say that infinity minus one (∞ – 1) is also infinity, since it still goes on forever. Then you’d have:

∞ = ∞ – 1

Then, following the rules, drop out the variable ∞ from the equation by subtracting it from both sides of the equation:

∞ – ∞ = ∞ – ∞ – 1

Which of course leaves you with:

0 = -1

Proof! :)

Then, of course, you could further apply various equality rules and come up with all sorts of non-zero results equalling zero.

I remember being pretty disappointed when it turned out to be appallingly wrong. Fortunately, I still went on to the Advanced Math and then Calculus courses…

2 Replies to “0 = -1”

  1. It’s similar to the proof that 1 = 2:
    Let a and b each equal 1.
    Then a^2 = ab (because a=b).
    And a^2-b^2 = ab-b^2.
    Factor, and you get
    (a+b)(a-b) = b(a-b).
    Then cancel the (a-b) factors, and you get
    a + b = b
    Which is the same as 1 + 1 = 1, or 2 = 1.

  2. Yeah Jon’s idea fails thanks to various infinity related rules. Charlie Nafziger, My Calculus teacher at COCC would have said "No. Wrong. Infinity minus one equals the end of this conversation." and carried on corrupting any questions about infinity with interuptions focusing on paranoid nonsense. It was really quite amusing 😉

    Isaac’s statement fails because he is dividing both sides by zero a=b -> (a-b) = 0.

    ‘Proving’ a falsehood, or any two contradictory statements, in any propositional system can be used to prove any proposable statement inevitably. I read that in GEB. 🙂

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