About the principal nth root

Question

[$n_{th}$ root]

A difficulty with this choice is that, for a negative real number and an odd index, the principal nth root is not the real one. For example, ${\displaystyle -8}$ has three cube roots, ${\displaystyle -2}, {\displaystyle 1+i{\sqrt {3}}}$ and ${\displaystyle 1-i{\sqrt {3}}.}$ The real cube root is ${\displaystyle -2} $ and the principal cube root is ${\displaystyle 1+i{\sqrt {3}}.} $.

Cube root

In some contexts, particularly when the number whose cube root is to be taken is a real number, one of the cube roots (in this particular case the real one) is referred to as the principal cube root, denoted with the radical sign ${\displaystyle {\sqrt[{3}]{~^{~}}}.} $

I'm confused about these two descriptions of "principal cube root" which seems leading to different result, the second description denotes that the principal cube root of $-8$ is $-2$ while the first one is ${\displaystyle 1+i{\sqrt {3}}.}$. Or did I interpret this wrong?

There is a definition about principal $n_{th}$ root? . Is it ok with this definition? Thank you.

Answer 1

These are indeed two different definitions.

The first one describes how to select the principal $n$th root of a complex number. As described in the Wikipedia article you linked, for a nonzero complex number $z$, $z^{1/n}$ describes $n$ different values. In other words, $z\mapsto z^{1/n}$ is a "multi-valued function." Most of the time you will want to work with a single-valued function though. If you want a single-valued $f(z)$ which takes input $z$ and spits out one of the values of $z^{1/n}$, you need some kind of rule for choosing which one of the $n$ possible values is your output. The most common choice is the principal $n$th root, which is defined by the formula $f(z) = |z|^{1/n}e^{i{\rm Arg}(z)/n}$, where ${\rm Arg}(z)$ is the unique angle of the complex number which lies in the interval $(-\pi,\pi]$. Another way of writing this is $$f(z) = r^{1/n}(\cos(\theta/n)+i\sin(\theta/n)),$$ where $z=re^{i\theta}=r\cos\theta+ir\sin\theta$ is the polar form of the complex number $z$, and we again insist $\theta\in(-\pi,\pi]$.

Using different ranges of angle values describe different ways of defining an $n$th root function over the complex plane. Even mathematicians aren't in complete agreement over which range of angles is the "principal" one, but $(-\pi,\pi]$ seems to be the most popular.

In general, a single-valued version of a multi-valued function is known as a "branch" of the function. So you'll sometimes see the particular $f$ we defined above referred to as the "principal branch of $z^{1/n}$."

The other definition of principal root is much rarer, and I've only seen it in more elementary contexts (notice your link is to an college algebra course). If you're reading an advanced math book or talking to a mathematician, and they say "the principal $n$th root", they almost certainly mean the first definition I discussed.

Answer 2

The suggested links in the comments do not answer your question, which is a valid one.

In essence, you are asking why there seem to be two different definitions of "principal" for the cube root: one definition would say that $-8$ has the principal cube root $1 + i \sqrt{3}$, and the other has the principal cube root $-2$.

In fact, this apparent discrepancy in definitions would extend to any fifth, seventh, and any power of the form $$z^{1/(2m+1)} = \sqrt[2m+1]{z}$$ where $m$ is a positive integer.

One way to resolve the discrepancy is to use the first definition whenever we use the notation on the left-hand side of the equation above, and to use the second definition whenever using the notation on the right-hand side. So for example, the principal root of $$(-8)^{1/3} = 1 + i \sqrt{3},$$ but $$\sqrt[3]{-8} = -2.$$

This, however, is still unsatisfactory since it is not standard practice to do so. The truth is that "principal" simply does not unambiguously characterize what we mean in such situations. One way to address this is to use different terminology for each definition; e.g., to call the first definition the "principal root" and then to call the second the "principal real root" or "principal real-valued root."