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Does someone know what I'm doing wrong? I'm struggling with this for a while now and I don't see what I do wrong!

$$1+1=$$ $$1+\sqrt{1}=$$ $$1+\sqrt{-1*-1}=$$ $$1+\sqrt{-1}*\sqrt{-1}=$$ $$1+i*i=$$ $$1-1=0$$

Najib Idrissi
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Jan Eerland
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    The third "=" is wrong. – mvw Apr 08 '15 at 14:30
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    If you had to figure out where the mistake is, what would you try? Which step are you the least certain about? For example, you might try drawing vectors on a complex plane. – DanielV Apr 08 '15 at 14:30
  • Also $\sqrt{-1}=i$ is wrong. The square root of complex numbers that are not real and nonnegative is not single-valued. (In other words, why do you say that $\sqrt{-1}=i$ instead of $\sqrt{-1}=-i$? How can you choose one of the two?) – Giuseppe Negro Apr 08 '15 at 14:31
  • @GiuseppeNegro: You are wrong. The choice is built into the definition of $i$ and $\sqrt{\cdot}$. The choice does not matter because all field extensions of $\mathbb{R}$ that add a root of the polynomial $x \mapsto x^2+1$ are isomorphic. Also, the only field automorphism of $\mathbb{C}$ that fixes $\mathbb{R}$ is conjugation, which is simply a reflection about the real axis. – user21820 Apr 10 '15 at 02:24

3 Answers3

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$$\sqrt {ab}=\sqrt {a} .\sqrt b$$ For $a,b\ge 0$

Fermat
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$1+\sqrt{-1\cdot-1}\neq1+\sqrt{-1}\cdot\sqrt{-1}$. That is the issue with your "proof" that $1+1=0$.

Daniel W. Farlow
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You've run into a common problem of the square root. It can take on two value. usually we use the principal root, that is to say the positive square root of a number. In this case, you've split the root into two part, however, you can't do that because it could then have 4 combinations of signs, 2 visible combinations. Most of the time this isn't a problem, because splitting two positive roots results in 2 combinations of signs, but only one visible sign.

I should also mention that when you say $1=\sqrt n$, this equality has to hold true even after less than truthful manipulations. If you end up with $-1=1$ its your own fault, because you neglected the original equality.

Zach466920
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