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I'm a physics student and I seem to be having trouble accepting a glaring inconsistent with regards to $i^2$. From all the math sources I see, $i^2$ is defined as -1. While, on the other hand, physicists seem to think that $i^2 = 1$ so that $-i^2 = -1$.

So, which is right and why?

Ake
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    Which physicists think this way? Do you have examples? – abiessu Dec 06 '14 at 05:53
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    You are mistaken. The two square roots of $1$ are $1$ and $-1$. The two square roots of $-1$ are $i$ and $-i$. Unless $1$ and $-1$ are the same number (last I heard, they are not), $i^2$ cannot possibly be $1$. – MPW Dec 06 '14 at 05:57
  • Once upon a time a mathematician tried to solve the equation $x^2 + 1 = 0$. He realised that the solutions must be $x = \pm \sqrt{-1}$. From this, the notion of $i$ and imaginary numbers was brought to life. That is, $i = \sqrt{-1}$. That is the definition. As the others have stated, there is a mistake with your interpretation. – Gustavo Louis G. Montańo Dec 06 '14 at 05:59
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    @GustavoMontano I think historically $i$ came from solving cubics, not quadratics. – Cameron Williams Dec 06 '14 at 06:02
  • @Ake, See http://math.stackexchange.com/questions/264037/relationship-between-complex-number-and-vectors – lab bhattacharjee Dec 06 '14 at 10:19

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I've not once seen $i^2=1$ in any physics text I've read. (I am a former physicist-in-training.) $i$ is defined to be such that $i^2=-1$ (or something equivalent) in every text I've read.

Cameron Williams
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  • Really? So, why is the square of the momentum operator negative? It is defined as $-i\hbar\frac{d}{dx}$ (in the one dimensional position basis to be sure), so then wouldn't that imply $-(iħ\frac{d}{dx})^{2} = -(-{ħ}^2\frac{d^{2}}{dx^{2}}} = {ħ}^2\frac{d^{2}}{dx^{2}}$, but that is clearly not the squared momentum operator as defined in ANY QM textbook. – Ake Dec 06 '14 at 05:58
  • Well you made a slight algebraic mistake. $$\left(-i\hbar\frac{d}{dx}\right)^2 = (-1)^2i^2\hbar^2\frac{d^2}{dx^2} = 1\cdot(-1)\hbar^2\frac{d^2}{dx^2} = -\hbar^2\frac{d^2}{dx^2}.$$ – Cameron Williams Dec 06 '14 at 06:00
  • You forgot to square your negative sign. – Cameron Williams Dec 06 '14 at 06:01
  • Heh, thanks a lot. That was a very, very stupid mistake on my end. – Ake Dec 06 '14 at 06:06
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    It happens! We've all been there. – Cameron Williams Dec 06 '14 at 06:06
  • @CameronWilliams: I'll say "Amen" to that, brother! – Robert Lewis Dec 06 '14 at 07:35