Mathematics desk | ||
---|---|---|
< December 13 | << Nov | December | Jan >> | Current desk > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
(ab)m = ambm
I have questions regarding the validity of the above law of indices depending on what a,b,m are.
I'm trying to come up with the valid values that a,b,m can take so that the above law is always true for a given set of restrictions.
I'm trying to be as inclusive as possible, meaning that if something isn't wrong or results in something being wrong, then I won't exclude it. For example, if a complex number (not just a real number) also works in some situations of the above law, then I'll also try to include and describe those situations.
Note: For all cases, any situation that results in division by 0 (or 0 to a negative real number) and 00 are exceptions that have been noted. I'm not writing expressions that show I'm disallowing them so that I can reduce clutter in describing the different cases below.
Case 1:
If m ∈ {ℝ \ ℤ}, then a,b ∈ ℝ+
This is so we avoid the wrong results of 1 = -1 and i = -i (and other similar wrong results).
Case 2:
If m ∈ ℤ, then a,b ∈ ℂ
Apart from that note I made above, I don't see any problems with this case.
Case 3:
If m ∈ j⁄2k+1 where j,k ∈ ℤ, then a,b ∈ ℂ
The power is rational, and its denominator cannot be even because taking even roots of negative real numbers results in the same problems mentioned in Case 1.
So, my questions are:
1. I paraphrased Case 1 and 2 from this wikipedia article (http://en.wikipedia.org/wiki/Exponentiation#Failure_of_power_and_logarithm_identities). Is it correct for me to paraphrase it this way?
2. I'm not sure if Case 3 is correct though, and that is where I require help in checking its correctness. The reason I wanted another case (other than the 2 mentioned in that wikipedia article) is because (just from working a few examples out) there are some complex numbered bases that can be raised to a rational power (with an odd denominator) that satisfy this particular law of indices, so it feels too restrictive/un-inclusive if we don't cater to these cases.
Thanks. 175.156.52.140 (talk) 19:25, 14 December 2014 (UTC)
I've extended the case 1 function domain, since you're trying to to be maximally inclusive. The ℂ in the domain can be replaced with far more general objects in both cases (e.g. matrices, quaternions, etc.) if you wish to be maximally inclusive. —Quondum 01:49, 15 December 2014 (UTC)
Hi,
Can A and B run a Zero Knowledge Proof algorithm to learn whether A's or B's positive integer is larger (or if they are the same) without either of them learning (or giving away) any further information beyond this fact? They should not have to trust a third party, this is part of the field of mathematics called zero knowledge proofs. Thank you. 212.96.61.236 (talk) 20:28, 14 December 2014 (UTC)