Proof that 1=-1 using algebra

Recall that i is defined to be the square root of -1.
We write this as i = ( -1 ) ^{1 / 2}.

Proof that -1 = 1

Step 1: -1 / 1 = 1 / -1     Looks harmless
Step 2: ( -1 / 1 ) ^{1 / 2} = ( 1 / -1 ) ^{1 / 2}     After taking the square root
Step 3: ( -1 ) ^{1 / 2} / ( 1 ) ^{1 / 2} = ( 1 ) ^{1 / 2} / ( -1 ) ^{1 / 2}     After splitting terms
Step 4: i / 1 = 1 / i     After simplifying, using (-1) ^{1 / 2} = i
Step 5: i = 1 / i
Step 6: -1 = 1           On multiplying by i

Explanation

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