3
$\begingroup$

This question already has an answer here:

Wolfram Alpha doesn't give $-2$ for $(-8)^{1/3}$, and it absolutely fails to draw $f(x)=x^{1/3}$ - does anyone know why? Am I missing something very 'deep' Wolfram Alpha is trying to teach me?

Here's what the graph should look like: enter image description here

And here's what Wolfram Alpha draws: enter image description here

$\endgroup$

marked as duplicate by Mark McClure, Kamil Jarosz, user228113, user223391, Eric Wofsey Mar 11 '16 at 21:13

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • 4
    $\begingroup$ Yes. You are missing the fact that cubic root multi-valued (3 values) and W|A gives you the principle value while the other program uses a simpler convention and spits out a non-complex value. Try plotting $x^{1/2}$ instead in both to understand the difference deeper. $\endgroup$ – A.S. Mar 11 '16 at 8:29
  • $\begingroup$ You can get the plot you expect by entering y=Abs(x)^(1/3)*Signum(x) into W/A. $\endgroup$ – LouisB Mar 11 '16 at 8:50
  • $\begingroup$ You can enter cbrt(x) for the cube root function in Wolfram|Alpha, which (as implemented in that particular piece of software) is different from $x^{1/3}$. $\endgroup$ – Mark McClure Mar 11 '16 at 12:56
3
$\begingroup$

What you are seeing is that every non-zero number has three distinct "cuberoots". In the first picture you've posted, the software very nicely graphed the real cuberoots. The problem is that Wolfram|Alpha is drawing a different branch of the cuberoot than you are used to, hence the labels of real and imaginary parts on the graph.

$\endgroup$
1
$\begingroup$

The principal value of the cube root is given by Mathematica or Wolfram Alpha. If you really just want the real value, input with Sign[x] Abs[x]^(1/3)

$\endgroup$
1
$\begingroup$

What you are probably thinking about is that for solving the equation $$z^k = a$$

we often have many solutions (for $z$) in the complex plane. More to the point, for a function one wants to assign one out-value for each in-value. Choosing a particular way to assign function values is called picking a branch for a function. For some functions there exist widely used conventions how to pick these branches and they get called principal branch. Integer roots and logarithms are famous examples of functions which have a principal branch.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.