1 = 2?

Friday, June 21, 2013

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Maybe you are wondering why or how can 1 be equal to 2. Let us look at the following proof:

First, let us have two real numbers a and b where

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Using the addition property of equality, let us add b on both sides.

If we simplify both sides by combining like terms, it will become

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Now let us subtract 2a on both sides

Combining like terms, we will arrive at

The right side has a common factor, which is 2. Using distributive property, it can be rewritten as

If you notice, both sides has a common factor, which is b - a. To simplify the equation, let us divide both sides by the common factor.

The process will arrive at

Are you convinced? No no no...

Seems like the proof is valid but there is something wrong with it. Look over the proof once again. Can you identify which of the process is not valid?

There is nothing wrong with the given. Real numbers can be equal. There is nothing wrong also with adding b and subtracting 2a on both sides. Likewise, there is nothing wrong with combining like terms on both sides. Then, where is the mistake?

There is nothing wrong with dividing both sides by any number but in this case it becomes invalid. The reason is that b - a = 0 since a = b. Subtracting equal numbers will yield 0. Since b - a = 0, then the result will be UNDEFINED. We cannot also cancel out b - a on both sides because of that.

Therefore, 1 is not equal to 2.

Here is a copy of the proof in pdf form. You may download and print it for educational purposes. You may share it to your friends. Your comments and suggestions are also welcome here.