�PROVE� 2 = 0. CAN YOU FIND THE MISTAKE?

I received an email I wanted to share with you. Wolfgang came up with a false proof that 2 = 0. No one in his class, not even his teacher, could figure out the mistake. Can you?

I present the false proof and the mistake in a new video.

�Prove� 2 = 0. Can You Find The Mistake?

 

Here is the �proof� in text.

2 = 1 + 1
2 = 1 + v(1)
2 = 1 + v(-1 * -1)
2 = 1 + v(-1)v(-1)
2 = 1 + i(i)
2 = 1 + i2
2 = 1 + (-1)
2 = 0

Or keep reading for a text explanation.

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False Proof 2 = 0

2 = 1 + 1
2 = 1 + v(1)
2 = 1 + v(-1 * -1)
2 = 1 + v(-1)v(-1)
2 = 1 + i(i)
2 = 1 + i2
2 = 1 + (-1)
2 = 0

The mistake is between lines 3 and 4.

v(-1 * -1) ? v(-1)v(-1)

This is a misapplication of the product rule for square roots. The product rule is guaranteed to work only when both values are positive.

If x, y = 0, then
v(xy) = v(x)v(y)

When x = y = -1, the product rule may not apply, and as demonstrated, it is not a valid step because it leads to the conclusion that 2 = 0.

When you learn a property in math class, make sure to pay attention to the specific conditions when it applies. If you don�t you could end up with an absurd result like 2 = 0!

So how are we supposed to simplify a square root of a negative number? It is actually a mistake to use the product rule (which KhanAcademy teaches):

v(-52) = v(-1)v(52) = i v(52)

This is a mistake! You should not use the product rule unless both terms are positive�although in this case you do get the correct answer.

The correct way is that we define the square root of negative numbers as follows (see page 529 in here):

If b is a real number greater than 0 , then
v(-b) = i vb

So the correct way to find the answer is by definition:

v(-52) = i v(52)

You might think this is a nit-picking distinction as the KhanAcademy method gets to the correct answer. But remember that the process matters in math�it is not about getting the correct answer, it�s about getting the correct answer by the correct method.