(-1)^(2/3) evaluates to 1

[mezzarobba]

Likely a duplicate, but I couldn't find it elsewhere...

sage: (-1)^(2/3)
1

Of course, this is inconsistent with virtually all interpretations and conversions of symbolic expressions...

sage: CC(-1)^(2/3)
-0.500000000000000 + 0.866025403784439*I
sage: QQbar(-1)^(2/3)
-0.500000000000000? + 0.866025403784439?*I
sage: RR(-1)^(2/3)
-0.500000000000000 + 0.866025403784439*I

Also:

sage: (-1)^(-1/3)
-1
sage: 1 / ((-1)^(1/3))
-1

sage: (-1)^(1/3)*(-1)^(1/5)
1

but

sage: (-1)^(1/15)
(-1)^(1/15)

CC: @videlec

Component: symbolics

Author: Ralf Stephan

Branch: 69f75ec

Reviewer: Vincent Delecroix

Issue created by migration from https://trac.sagemath.org/ticket/15605

[mezzarobba]

Description changed:

--- 
+++ 
@@ -15,3 +15,10 @@
 sage: RR(-1)^(2/3)
 -0.500000000000000 + 0.866025403784439*I
 ```
+
+Also:
+
+```
+sage: (-1)^(-1/3)
+-1
+```

[mezzarobba]

Description changed:

--- 
+++ 
@@ -22,3 +22,9 @@
 sage: (-1)^(-1/3)
 -1
 ```
+
+```
+sage: (-1)^(1/3)*(-1)^(1/5)
+1
+```
+}}}

[fchapoton]

comment:4

Seems to be a problem about "even" powers:

sage: [(-1)^(i/77) for i in range(9)]
[1, (-1)^(1/77), 1, (-1)^(3/77), 1, (-1)^(5/77), 1, (-1)^(1/11), 1]

[fchapoton]

Description changed:

--- 
+++ 
@@ -27,4 +27,3 @@
 sage: (-1)^(1/3)*(-1)^(1/5)
 1
 ```
-}}}

[sagetrac-tmonteil]

Replying to @mezzarobba:

Likely a duplicate, but I couldn't find it elsewhere...

It is indeed a known issue since at least this ask answer, and i use this example in my presentations about Sage and the need to define objects within a reliable parent.

I did not report that on trac because this is somehow a feature of the symbolic ring : no semantics. I remember a discussion with you at sd49 (Orsay) about that precise example and the benefits (or not!) of having such kind of symbolic expressions that are able to make all kind of simplifications/derivatives/evaluations without context.

If one want to have something that both:

• applies rules such as a^(bc) = (a<sup>b)</sup>c systematically,
• takes principal branches of multi-valued complex function when evaluating,
  then we should accept that kind of behaviour.

It would however be awesome to have a well-designed object to work with holomorphic functions, along the same lines that what is done with polynomials, i guess it is a very hard task, and i do not know whether there exists libraries for that.