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Im quite confused over the use of the radical. I can understand its definition in $\mathbb{R}$, with $\sqrt[n]{x}$ being defined for $x>0$ if $n$ an even integer greater than one and $x$ any real number if $n$ an odd integer greater than one. I believe its referred to as the principal $n$th root for even $n$ as it outputs the positive root rather than the negative root. In $\mathbb{R}$ there are no square roots of negative numbers, so on this domain it's undefined. I also know it defines the same function of $x$ as $x^{1/n}$ for $n$ integer greater than $1$.

In $\mathbb{C}$, a principal $n$th root function can be defined using the same radical notation which for me is confusing. The fact there are so many ways to define the principal $n$th root function in $\mathbb{C}$ makes it worse, and is $x^{1/2}= \sqrt{x}$ in $\mathbb{C}$ or do you have to define it that way after you have defined exactly what you mean by $\sqrt{x}$. Should this principal $n$th root function in $\mathbb{C}$ be viewed as a different function altogether from the one in $\mathbb{R}$, or just an extension of the domain?

Thanks

Karn Watcharasupat
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Carlos Bacca
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    Giving things names and making up definitions don't in any way make things more "real" or valid. A principal root is whatever one person makes it up to be and if someone else comes up with another then that's no reason for you to be confused. Let them fist-fight it out. Whether we define the "principal root" to be the smallest angle, or to closest to a real number or the one which looks most like an elephant when you hold it to the light won't make any difference in any math you actually do so long as once the dumb definition is made you apply it consistently. – fleablood Jul 29 '18 at 16:21
  • You can use the notation $\sqrt z$ for something, but that isn't a definition. You have to make arbitrary choices to define complex roots, and in particular you have to choose where you want your root function to be discontinuous. Some of the possible root functions are extensions of their real counterparts, others are not. For example, for the popular choice $q_n(re^{i\theta})=r^{1/n}e^{i\theta/n}$, where $0\le\theta<2\pi$, you have $q_3(1)=1$ but $q_3(-1) \neq -1$. – Kusma Jul 29 '18 at 16:23
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    There's a conception, a strange one if you think about, that somehow if you define something you force reality to obey the definition rather than you are using a definition to do the human best to describe reality. For example a naif may try to define $\frac 10=\infty$. Okay, that's fine whatever... If you divide something into smaller and smaller packages you need more packages. As packages of size zero can't have anything you can never have enough so the number needed is infinite. That makes a certain amount of sense. ... to be continued.... – fleablood Jul 29 '18 at 17:37
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    ... continued... we are talking about $\frac10=\infty$... but once we say that we somehow think we have changed reality and $\infty$ is now a number and so $\frac 100 = 0\infty$ so $1=0$ and OMG! I'M FREAKING OUT!!! ... except... nothing's changed giving an an extreme situation a name so that it looks like a normal situation doesn't somehow magically make it a normal situation. We couldn't do that before we defined $\frac10=\infty$ because dividing by zero can't give a finite quantitative value and defining it to have an infinite value doesn't change that... to be cocluded... – fleablood Jul 29 '18 at 17:45
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    .. concluded... anyway, ... in a way defining principal roots are little like that. We can define the principal sixth root and principal tenth root to be one way be the suddenly the half, fifth, and third are inconsistent. Well... so what? The definition was only a convenience. It didn't in anyway change reality. It was only a convenient way to temporarily describe a rather localized incident of reality in terms that were consistant and a good idea at the time but have no fundamental "truth" about them. – fleablood Jul 29 '18 at 17:52

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