-1 = 1

Algebra Level 2

I have found a proof that − 1 = 1 -1=1 . Or have I? If you think one of these steps is wrong, input your answer as the step number.

Step 1: − 1 -1
Step 2: = i 2 ={ i }^{ 2 }
Step 3: = i × i =i\times i
Step 4: = − 1 × − 1 =\sqrt { -1 } \times \sqrt { -1 }
Step 5: = − 1 × − 1 =\sqrt { -1\times -1 }
Step 6: = 1 =\sqrt { 1 }
Step 7: = 1 =1

5 5 7 7 3 3 4 4 6 6 None of these choices , − 1 = 1 is correct \text{None of these choices}, -1=1 \text{ is correct} 1 1 2 2

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Michael Fuller by Michael Fuller , 22, United Kingdom

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2 solutions

Michael Fuller
May 9, 2015

The calculation rule a × b = a b \sqrt { a } \times \sqrt { b } =\sqrt { ab } is only valid for real, non-negative values of a a and b b . Therefore step 5 is invalid.

Source: Wikipedia: Imaginary Unit
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Moderator note:

Yes correct. You might want to check out a relevant post recently shared that "abuses" the complex numbers.

Akiva Weinberger
May 13, 2015

− 1 × − 1 = i × i = − 1 \sqrt{-1}\times\sqrt{-1}=i\times i=-1 , while − 1 × − 1 = 1 = 1 \sqrt{-1\times-1}=\sqrt1=1 . So they clearly are not equal.

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