1=2: A Proof using Complex Numbers

This supposed proof uses complex numbers. If you're not familiar with them, there's a brief introduction to them given below.

The Fallacious Proof:

•Step 1: -1/1 = 1/-1

•Step 2: Taking the square root of both sides: sqrt(-1/1) = sqrt(1/-1) (where "sqrt" denotes the square-root symbol which cannot be displayed on text-only browsers.)

•Step 3: Simplifying: sqrt(-1) / sqrt(1) = sqrt(1) / sqrt(-1)

•Step 4: In other words, i/1 = 1/i.

•Step 5: Therefore, i / 2 = 1 / (2i),

•Step 6: i/2 + 3/(2i) = 1/(2i) + 3/(2i),

•Step 7: i (i/2 + 3/(2i) ) = i ( 1/(2i) + 3/(2i) ),

•Step 8: (i^2)/2 + (3i)/2i = i/(2i) + (3i)/(2i),

•Step 9: (-1)/2 + 3/2 = 1/2 + 3/2,

•Step 10: and this shows that 1=2.

See if you can figure out in which step the fallacy lies. When you think you've figured it out, click on that step and the computer will tell you whether you are correct or not, and will give an additional explanation of why that step is or isn't valid.

See how many tries it takes you to correctly identify the fallacious step!