Ugh - here's the complete version - sorry!

- Step 1: -1/1 = 1/-1
- Step 2: Taking the square root of both sides: sqrt(-1/1)=sqrt(1/-1)
- 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 / (2*i*), - Step 6:
*i*/2 + 3/(2*i*) = 1/(2*i*) + 3/(2*i*), - Step 7:
*i*(*i*/2 + 3/(2*i*) ) =*i*( 1/(2*i*) + 3/(2*i*) ), - Step 8: (i^2)/2+(3i)/2i=1/(2i)+(3i)/(2i)
- Step 9: (-1)/2 + 3/2 = 1/2 + 3/2,
- Step 10: and this shows that 1=2.

That's an old one - I don't want to give away the secret! Here'a another good "proof" that 2 = 1, but requires a little knowledge of imaginary numbers:

- Step 1: -1/1 = 1/-1
- Step 2: Taking the square root of both sides:
- Step 3: Simplifying:
- Step 4: In other words,
*i*/1 = 1/*i*. - Step 5: Therefore,
*i*/ 2 = 1 / (2*i*), - Step 6:
*i*/2 + 3/(2*i*) = 1/(2*i*) + 3/(2*i*), - Step 7:
*i*(*i*/2 + 3/(2*i*) ) =*i*( 1/(2*i*) + 3/(2*i*) ), - Step 8: ,
- 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.

I remember a mathematician friend of mine telling me that actually 2+2 could equal five and then she embarked upon a whole complicated reason to explain why this was so, but it was a more philosophical and scientific debate than simply using equations as you have done above. In addition, as #2 indicates, there appear to be some erroneous equations in what you have done above.

This seems rather random to me. Why are you multiplying both sides by x and then both sides by y squared? If x=y then x squared would equal xy. If x=y then x squared minus y squared would give you zero. Yes, your third equation is technically correct, but I'm not sure why you would multiply both sides by y squared. If x=y then we cannot say that x+y=y nor does x+x=x. There are definite faults here.

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