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nth root

Defines: 
index
Keywords: 
nth root, square root, cube root, root, complex geometry, complex plane, complex, imaginary
Synonym: 
complex root, principal root
Type of Math Object: 
Definition
Major Section: 
Reference

Mathematics Subject Classification

30-00 no label found12D99 no label found

Comments

I should not have taken the square root of (sqrt[4](2), pi/4). I should have taken the cube root.

As a result, the entire rest of that example is wrong. I'll fix it I promise, but I don't have time right now..

Phew! I thought I remembered trigonometry. Then I did the last example!

Here, in PlanetMath, the real n'th root $\sqrt[n]{x}$ is defined only for non-negative real $x$ and for natural number $n$ (what does mean $\sqrt[0]{x}$?). I have always teached in school so that
the n'th root of a real number x is such a real number y that
(1) $y^n = x$
and
(2) $sign(y) = sign(x)$
(n = 1, 2, 3, ...).
Then the root is _uniquely_ _determined_ in the cases it exists.
Is this right?

The definition in PlanetMath allows two values in the example case
$\sqrt[4]{81}$, although here is given only the value 3.

The odd n'th root (cube root etc.) of a real number b can not be identified with the "fractional power" a^{1/n}, although so has been done in the entries "n'th root" and "cube root". Viz., the fractional power with a negative base is not uniquely determined -- it depends not only on the value of the exponent but also on the form of the exponent; e.g.,

(-1)^{1/3} = the 3'rd root of -1, i.e. = -1

(-1)^(2/6) = the 6'th root of (-1)^2, i.e. = 1

What should we do concerning the fractional powers?

Old discussion, I know, but this entry turned up on the sidebar,
and I think the content of the discussion merits some emphasis
(and an entry on PlanetMath)

The problem is, of course, that there is no algebraic way
to distinguish between the different nth roots of a number.
(Note: positivity does not count because positivity is not part
of the axioms of a field.)
This is already well-known in the case of complex numbers
(the nth root is not even a continuous function then)

Of course it is useful to have a notion of a principal square root
for real numbers. But we must be aware that some of the "distributive properties"
of exponents will then fail to hold, unless we restrict the bases
to be positive too --- naturally, since we imposed a positivity condition
to define real roots to begin with.

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