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My professor showed the following false proof, which showed that complex numbers do not exist. We were told to find the point where an incorrect step was taken, but I could not find it. Here is the proof: (Complex numbers are of the form $\rho e^{i\theta}$, so the proof begins there) $$\large\rho e^{i\theta} = \rho e^{\frac{2 \pi i \theta}{2\pi}} = \rho (e^{2\pi i})^{\frac{\theta}{2\pi}} = \rho (1)^{\frac{\theta}{2\pi}} = \rho$$ $$Note: e^{i\pi} = -1, e^{2\pi i} = (-1)^2 = 1$$ Since we started with the general form of a complex number and simplified it to a real number (namely, $\rho$), the proof can claim that only real numbers exist and complex numbers do not. My suspicion is that the error occurs in step $4$ to $5$ , but I am not sure if that really is the case.

Joe
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Madhav Nakar
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  • $e^{2\pi i+2\pi k i}=1$, where $k\in\mathbb{Z}$ for starters. Also, what is $\left(e^{z}\right)^w$ with $z,w\in\mathbb{C}$? – pshmath0 Dec 27 '17 at 17:19
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    The special case $\theta=\pi$ is $-1=(-1)^{2/2}=((-1)^2)^{1/2}=1^{1/2}=1$, so negative numbers do not exist either. – Carsten S Dec 27 '17 at 19:41
  • you first find the total exponent then perform potentiation. –  Dec 28 '17 at 03:31

2 Answers2

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The error lies in assuming that $(\forall a,b\in\mathbb{C}):e^{ab}=(e^a)^b$.

It's worse than wrong; it doesn't make sense. The reason why it doesn't make sense is because $e^a$ can be an arbitrary complex number (except that it can't be $0$). And what is $z^w$, where $z,w\in\mathbb C$? A reasonable definition is that it means $e^{w\log z}$, where $\log z$ is a logarithm of $z$. Problem: every non-zero complex number has infinitely many logarithms: if a number $\omega$ is a logarithm, then every number of the form $\omega+2k\pi i$ ($k\in\mathbb Z$) is also a logarithm.

José Carlos Santos
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    I think it’s worth explaining to OP why it makes no sense. When your background is real, it’s a completely natural thing to expect or assume must be okay. – Randall Dec 27 '17 at 17:21
  • I agree with Randall. Would you be able to explain why this is the case? So far most of what I have dealt with is in the set of real numbers. – Madhav Nakar Dec 27 '17 at 17:23
  • @Randall You are right. I have edited my answer. – José Carlos Santos Dec 27 '17 at 17:25
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    @MadhavNakar What do you think now? – José Carlos Santos Dec 27 '17 at 18:26
  • I understand the reasoning now. Thanks a lot for the help. – Madhav Nakar Dec 27 '17 at 19:04
  • @BCLC He is saying the reverse. Your first statement is wrong, your second is right. – eranreches Dec 28 '17 at 02:16
  • @eranreches Thanks. Pressed enter while in rush. Edit: $$\large \rho e^{\frac{i\theta*2\pi}{2\pi}} = \rho (e^{2\pi i})^{\frac{\theta}{2\pi}} \tag{1}$$ $$\large \rho (e^{2\pi i})^{\frac{\theta}{2\pi}} = \rho (1)^{\frac{\theta}{2\pi}} \tag{2}$$ I guess (1) is wrong but (2) is right. If so why? If not what then? I think (2) is wrong because of what Antinous pointed out – BCLC Dec 28 '17 at 03:53
  • Strictly speaking he is only assuming $b \in \mathbb R$, and $a^b$ is defined when $a \in \mathbb C$ and $b \in \mathbb R$. So this answer isn't correct is it? – DanielV Dec 28 '17 at 04:31
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    While the opening sentence of this answer is correct, the OP only involves a case of $(e^a)^b$ where $e^a$ is (positive) real (more precisely, it is $1$), so the second paragraph going on about the (very real, no pun intended) difficulties of defining $z^w$ when $z$ is not real (or I would add, when it is real but negative) do not really pertain to the original question. – Marc van Leeuwen Dec 28 '17 at 07:24
  • @MarcvanLeeuwen The second part of my answer was ment to explain why the equality $(e^z)^w=e^{zw}$ makes no sense in general. – José Carlos Santos Dec 28 '17 at 08:39
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Because for complex numbers, $e^{zc}\ne(e^z)^c$ for $z,c \in \mathbb{C}$.

ArsenBerk
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  • Does this imply that only one-to-one functions have this property? – Madhav Nakar Dec 27 '17 at 17:21
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    $f(z)=1$ is also multiplicative. – Kevin Dec 27 '17 at 18:39
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    Sorry, failure of this identity and the exponential function not being injective are entirely different matters. Maybe exponential function not being injective is a reason that defining $(e^z)^c$ is problematic and cannot be done so as to have this identity, but "in other words" is just not right. – Marc van Leeuwen Dec 28 '17 at 07:15