I get -1 as the cube root of -1,which is correct. But if I put -1 in parentheses I get an imaginary result. I was hoping the imaginary result, if cubed, would give me -1, but it got worse.
-1^(1/3).n()
-1.00000000000000
(-1)^(1/3).n()
0.500000000000000 + 0.866025403784439*I
The power operator has higher priority than the unary negation.
So in the first case you are really doing "minus (1 to the 1/3)", and not "(minus 1) to the 1/3".
For the real n-th root, use real_nth_root.
sage: real_nth_root(-1, 3)
-1
Regarding cubing the cube root:
sage: a = (-1)^(1/3)
sage: a
(-1)^(1/3)
sage: a^3
-1
Using the numerical approximation of the cube root:
sage: aa = a.n()
sage: aa
0.500000000000000 + 0.866025403784439*I
sage: aa^3
-1.00000000000000 + 3.88578058618805e-16*I
So, very close to -1, with tiny imaginary part from rounding errors.
$x^3+1=0$ has three roots in $\mathbb{C}$. Therefore it could be argued that (-1)^(1/3) is in fact a set of three numbers ; our habit of using this notation to denote one of these numbers
is not consistent, and
does not explicitly specify which of these numbers it denotes
But this is far to be the sole inconsistency in our mathematical notational conventions... Equating $\sqrt [3]{-1}$ to $-1$ is just a habit...
Asked: 2020-09-17 23:44:47 +0100
Seen: 257 times
Last updated: Sep 18 '20
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