-1

I tried to calculate the following expression in GAP, but failed:

gap> E(-3);
Error, E: <n> must be a positive small integer (not the integer -3)
not in any function at *stdin*:49
type 'quit;' to quit to outer loop

So, how can I compute the expression like exp(2πi/-3) in GAP?

EDIT: Thank you, MathGeek, I figured out the following expression based on your clues:

gap> Cos(-2* FLOAT.PI/3);
-0.5
gap> Sin(-2* FLOAT.PI/3);
-0.866025

So, $e^{{2\pi i}/3}$ should be expressed as follows:

gap> [Cos(-2* FLOAT.PI/3), Sin(-2* FLOAT.PI/3)];
[ -0.5, -0.866025 ]

Regards, HZ

Hongyi Zhao
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    Please consult the people who know about this first, as it is a question about the documentation of a piece of software, and not a mathematical question. https://www.gap-system.org/Contacts/People/supportgroup.html – rschwieb May 14 '22 at 09:55
  • Use Euler's formula to reduce the problem to a positive argument. – Peter May 14 '22 at 10:56
  • @Peter Do you mean the method I described in the OP currently? – Hongyi Zhao May 14 '22 at 14:02
  • Basically what I meant. Note that if you have calculated the expression with the positive sign, say $e^{ix}=a+bi$ ($x$ positive real) , then you just have the conugate complex for the negative case : $e^{-ix}=a-bi$. This follows easily from the fact that the cosine is an even function and the sine an odd function. – Peter May 14 '22 at 14:08
  • But it seems the negative case can't be expressed as the cyclotomic polynomial of positive entries, so GAP can't calculate E(-3) directly. – Hongyi Zhao May 14 '22 at 14:33
  • In short : $e^{-ix}$ for positive real $x$ is just the conjugate complex of $e^{ix}$ – Peter May 14 '22 at 14:34
  • No, using floating point is not the same. Floating point numbers are not the same as real numbers. In fact floating point numbers do not even form a mathematical field, as the binary operations on them are not associative. They are also not exact: the values you computed via Cos etc. are only approximations of the real values by necessity. As such, these values are useless for e.g. working with representation. – Max Horn May 14 '22 at 23:41
  • So, this is why I ask: For the purpose of group representation, can we use only algebraic numbers without involving transcendental numbers to solve all problems? – Hongyi Zhao May 15 '22 at 00:13
  • For the purpose of representations of finite groups, we can always work in a number field, i.e. a finite degree extension of the rationals. – Max Horn May 16 '22 at 07:18
  • Do you mean: Polynomial based on E(n) is enough for the ultimate representations of finite groups in theory? – Hongyi Zhao May 16 '22 at 08:16

1 Answers1

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We can use Euler's formula for this. $$e^{ix}= cos(x)+isin(x)$$ We can input ${-2\pi}/3$ for $x$ to get the following: $$e^{{-2\pi i}/3}= cos({-2\pi}/3)+isin({-2\pi}/3)$$ Hopefully these inputs are valid in GAP.

MathGeek
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