Defining a symbol e that represents plus-minus-one, so that e^2 == 1 and e^3 == e

Question

I have a variable (say, e) that can assume only the values $1$ or $-1$. My calculations are extensive and if I introduce the symbol e to represent these possible values ($1$ or $-1$), I get an output with several powers of e.

I can, of course, after the calculation is done, simplify the result through a list that replace the even powers of e by $1$, and the odd powers by e. However, I would like instead to define a symbol e so that Mathematica understands (automatically, even in the middle of its calculations) that e^2 is 1 and e^3 == e (etc.).

In other words, I want Mathematica to treat e in the same manner as it treats the imaginary number I, except that e^2 == 1 instead of $-1$ (etc.) Is that possible in Mathematica or in Maple? Thanks.

score 12 · Accepted Answer · answered Sep 06 '17 at 02:19

12

A quick way is to use TagSet[] and TagSetDelayed[] to teach Power[] how to deal with your special symbol:

e /: e^0 = 1;
e /: e^1 = e;
e /: e^(n_Integer) := e^Mod[n, 2]

Thus:

Table[e^k, {k, 0, 5}]
   {1, e, 1, e, 1, e}

answered Sep 06 '17 at 02:19

J. M.'s missing motivation

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Thank you very much for your elegant solution J. M. ! Someone knows how to do a similar thing on Maple? Thank you in advance. – R. Vieira Sep 06 '17 at 19:54
This really isn't the place to ask that, if you don't mind reading this site's name again more carefully. – J. M.'s missing motivation Sep 06 '17 at 20:45
Oh! Sorry for that. Anyway, thank you again, your answer was very helpful. – R. Vieira Sep 12 '17 at 02:06
If you believe this has resolved your question, please click on the check mark to the left of my answer (under the arrows), so that the system no longer counts it as unanswered. – J. M.'s missing motivation Sep 12 '17 at 02:17

Defining a symbol e that represents plus-minus-one, so that e^2 == 1 and e^3 == e

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