Wikipedia:Reference desk/Archives/Mathematics/2012 February 5

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February 5

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If I can trisect an angle, why can't i accurately divide a piece of paper into 3 equal parts with no instruments of measurement?--92.25.103.212 (talk) 01:18, 5 February 2012 (UTC)

Can you trisect an angle? Haga's theorem in Mathematics of paper folding shows how to divide a line in three just using paper. Later in that article there is a bit on trisecting an angle. Dmcq (talk) 01:22, 5 February 2012 (UTC)

Name of 0

Is it nought, ought, zero, nothing, nil or O?--92.25.103.212 (talk) 01:32, 5 February 2012 (UTC)

It is nada or zilch. Oops I see 0 (number) doesn't mention nada. ;-) Dmcq (talk) 01:35, 5 February 2012 (UTC)

Yes. --COVIZAPIBETEFOKY (talk) 03:41, 5 February 2012 (UTC)

Only a few of those are used in math, though, and some are used in different fields of math than others. StuRat (talk) 04:51, 5 February 2012 (UTC)

As an Australian I use the expression "bugger all" a bit. I treat it as meaning a very small amount, but a colleague argued that it meant nothing. Maybe it's a variation on Sweet Fanny Adams, sometimes abbreviated to SFA, which could stand for "Sweet f... all", and sometime said without the "Sweet". HiLo48 (talk) 04:59, 5 February 2012 (UTC)

Don't forget love for tennis players. And American sports reports sometimes call it zip.    → Michael J Ⓣ Ⓒ Ⓜ 21:44, 7 February 2012 (UTC)

As L. Neil Smith put it, zero is the ultimate round number. --Trovatore (talk) 22:14, 7 February 2012 (UTC)

Reduced residue systems

Let r₁, ..., r_n be a reduced residue system modulo m, where n = φ(m). I'm trying to show that r₁^k, ..., r_n^k is a reduced residue system mod m if and only if gcd(k,n) = 1. The "if" component is easy enough, but it's the "only if" component that's stumping me. I'm looking for a solution which does not involve group theory, since this question comes before the introduction of groups in my textbook. Thanks in advance for the help. —Anonymous Dissident^Talk 10:13, 5 February 2012 (UTC)

Let's see. If gcd(k,n) is not equal to 1 then there is some s less than n such that sk is a multiple of n. And we know that r_iⁿ = 1 modulo m for each residue r_i. So r_i^sk = 1 modulo m for each residue r_i, and so each r_i^k is an sth root of 1 modulo m. But 1 can only have at most s distinct sth roots modulo m, and we know that s is less than n ... so what does this tell us about the values of r_i^k mod m ?

An example: take m=7 so n=6, and take k=4, so gcd(k,n)=2. If the r_i are [1,2,3,4,5,6] then the values of r_i⁴ modulo 7 are [1,2,4,4,2,1] so there only 3 distinct values of r_i⁴ modulo 7. Gandalf61 (talk) 12:07, 5 February 2012 (UTC)

That's a great approach. I got up to noticing that each r_i^k would have to be an sth root of 1 modulo m, but I didn't know how to show that leads to problems: I thought cases like m = 8, with r_i² = 1 (mod m) for all coprime r_i, would mean I'd have to examine certain cases. Thanks for the insight. —Anonymous Dissident^Talk 13:10, 5 February 2012 (UTC)

I've just realised there may be a flaw in your argument: Why must 1 have at most s s-th roots (mod m)? For example, there are 4 roots of x² = 1 (mod 12). —Anonymous Dissident^Talk 22:37, 5 February 2012 (UTC)