A Curious Problem (3)

Algebra Level 2

Let x=1.

Step 1:

Multiply by x to get x = x 2 x=x^{2} Step 2:

Subtract 1 from each side to get x − 1 = x 2 − 1 x-1=x^{2}-1 Step 3:

Divide both sides by x-1. x − 1 x − 1 = x 2 − 1 x − 1 \dfrac{x-1}{x-1}=\dfrac{x^{2}-1}{x-1} = ( x + 1 ) ( x − 1 ) x − 1 =\dfrac{(x+1)(x-1)}{x-1} Which simplifies to 1 = x + 1 1=x+1 Therefore, 1 = 1 + 1 1=1+1 , and 1 = 0 1=0 . However, this is wrong. Can you spot the fallacy?

The fallacy was in step 1. The fallacy was in step 2. The fallacy was in step 3.

Hollow Knight by Hollow Knight , 21, Hong Kong

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1 solution

Hollow Knight
Oct 25, 2020

Step 3 shows division by ( x − 1 ) (x-1) . Unfortunately, we have set x x to 1 1 here, therefore we are dividing by x − 1 x-1 which is 1 − 1 1-1 which is 0 0 here. Dividing by 0 0 is meaningless, hence this is not valid.

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