Proof that 2=1… or not

Federico Pistono posted about the Wall Street Journal’s opinion piece on science proving the existence of God… or not, so I am going to post (in similar fashion) 2=1… or not.

Many involved with mathematics have already heard of this fallacious proof, however it is an eloquent one and I wanted to share it:

Objective: To prove that all numbers equal each other, namely, 2=1.

Proof

1. Let a and b be equal non-zero numbers.
a=b

2. Multiply both sides by a
a^2=ab

3. Add a^2-2ab to both sides
a^2+a^2-2ab=ab+a^2-2ab

4. Factor:
2(a^2-ab)=a^2-ab

5. Divide both sides by a^2-ab
2=1

Ah, brilliant, we have just proved step-by-step that 2 is equal to 1. Using these same steps we can prove that any number is equal to each other, right?

Well, obviously, there is something deeply wrong about this proof. Try to consider what might be wrong for a moment, it is a fun challenge. When you’ve done that, continue reading.

Lame_Math_Joke_by_ichiko_wind_griffin

The problem in this fallacious proof lies in step 5. In step 1 we stated that a=b, so dividing by a^2-ab would mean we would be dividing by 0. Strange things happen when you divide by 0, and this result is one of them. Therefore, this proof is invalid and 2 does not equal 1. Mathematics is still fundamentally sound and all can remain at peace.

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