value of (-1)**(2*n)

[Chris Smith]

If n is an unbounded integer (is there such a thing?) then which rule applies to the expression (-1)**(2*n): (-1)**oo = nan or (-1)**even = 1?

[Chris Smith]

http://boards.straightdope.com/sdmb/showthread.php?t=715451 may apply: integers have a finite number of digits

[Rathmann]

Well, couldn't you use a limit argument?

(-1)**(2*n) is 1 for any finite n, hence the limit of this expression as n increases without bound is also 1.

If your idea of unbounded integer has something to do with limits, then this argument might apply.

[Joachim Durchholz]

Am 18.09.2014 um 05:20 schrieb Chris Smith:
> If n is an unbounded integer (is there such a thing?) then which rule
> applies to the expression (-1)**(2*n): (-1)**oo = nan or (-1)**even = 1?

It all depends on how you define "unbounded integer".

From a programmer's perspective, an "unbounded integer" is just what a
mathematician calls an "integer", and an "integer" is what a
mathematician calls an "integer modulo 2**number_of_bits".
No mysteries involved.
In this case, an even power of -1 is 1, -1 ** oo is still undefined
(because oo is neither even nor odd), and -1 ** even is the set {1} if
"even" is the set of all even natural numbers and you define ** over
sets in the "obvious" way.

If you mean things like 12345... and 10000... - well, again it depends
on how you interpret them. If they're sets with some formula defining
their members, then you can either define the usual set extension and
reason over them, or you can define something unusual and reason about
the outcome of exponentiation again.

Nothing special here, it all depends on what an "unbounded number" *is*,
and once you nailed that down, the answer comes naturally.

[Aaron Meurer]

I'd say 'integer' should imply 'bounded'.

Aaron Meurer

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