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The principal square root of $x$, denoted $\sqrt{x}$, occurs strictly when $x\in \mathbb{R}^{\geq 0}$. Every mathematician knows that writing $\sqrt{25}=-5$ is clearly $\underline{\text{incorrect}}$ here as it is the positive solution we want and not the negative. Also, writing $\sqrt{x}$ for when $x<0$ would be contradictory to a set of solutions (see incorrect result) when dealing with different algebraic equations.

$\textbf{Question:}$ So, why is it acceptable to define $\pm i=\sqrt{-1}$ when I just said the same exact notation for the principal root is reserved for non-negative real numbers? How are we able to distinguish the notation for square roots over $\mathbb{C}$ instead of over $\mathbb{R}$ when dealing with algebraic equations? I mean there has got to be a notation out there somewhere that avoids conflating these concepts as to when it is and is not appropriate to venture into the complex world.

The problem is is that we want $\sqrt{-5}$ to not be defined so often; yet, it is perfectly acceptable to write a square root around a discriminant that is $<0$ when dealing with the quadratic formula.

  • Here is a link to a previous question I asked too.
W. G.
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    It is a classical, convenient and somewhat misleading convention. – copper.hat Mar 30 '19 at 18:51
  • Because for positive real numbers there is a distinguished square root - the one that is positive. For all other numbers (complex non real or negative reals) there is no unambiguous choice as we can shift argument ranges etc; intrinsically we can distinguish 1 from -1, but not i from -i – Conrad Mar 30 '19 at 18:53
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    I´m afraid of long discussions and many answers. – callculus42 Mar 30 '19 at 18:56
  • I'm for it! lol – W. G. Mar 30 '19 at 18:57
  • @W.G. I don´t think it is acceptable to write $-i=\sqrt{-1}$, since $i=\sqrt{-1}$ by definition. – callculus42 Mar 30 '19 at 19:00
  • That's pretty typical to define it that way; I personally define it slightly differently than you. I more so am concerned as to the notation of writing a square root of a negative number in general. – W. G. Mar 30 '19 at 19:05
  • @W.G. Maybe it is worth to mention that there is a difference between $x^2=-1$ and $\sqrt{-1}$. The first one is an equation and the second one is a number. – callculus42 Mar 30 '19 at 19:16
  • That is true.. I swear I am going to define $\sqrt[*]{x}$ to be the principal square root if no one calls dibs on it first. I hate ambiguity in math! – W. G. Mar 30 '19 at 19:27
  • Can you support the claim that it is acceptable to write $\pm i=\sqrt{-1}$? I know mathematicians can some times be very lax with notations, but I don't think this is something many people do. Also, this actually correct, but many people miss it. When you use $\pm$ with an equality you're actually abbreviating a disjunction, in this case it is $i=\sqrt{-1}\lor -i=\sqrt{-1}$, this is true. – Git Gud Mar 30 '19 at 19:50

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I personally find it acceptable when you are clear that you interpret all of $\sqrt{-1}$ as a single symbol, not as a square root with a $-1$ under it. In other words, the $-1$ and the square root carry no mathematical significance, but just serve as a remainder of what happens when you square it.

It's not so common to see with $\sqrt{-1}$, because that role is filled by the symbol $i$. But the same notation with other numbers, like $\sqrt{-5}$, I see from time to time.

Arthur
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