Math Is Fun Forum

niharika_kumar

Member

From: Numeraland

Registered: 2013-02-12

Posts: 1,062

random questions

If n is a positive integer and

is a cube root of unity, then find the number of possible values of

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friendship is tan 90°.

phanthanhtom

Member

Registered: 2012-06-22

Posts: 290

Re: random questions

I do not recognise the code, but I'll assume its for omega^n.

Since omega is a cube root of unity, omega ^ 3 = 1. Therefore omega^n takes up to 3 values: omega, omega^2 (or 1/omega) and omega^3=1.

Since omega is not 1, omega, omega^2 and omega^3 have three distinct values, so there are exactly 3 values.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: random questions

hi Niharika,

Your Latex has an error so we're not seeing this question correctly.

After the 'e' you have a ^ and then a complicated expression. Powers will only display correctly if a single character follows or the power is enclosed in {} brackets.  I have tried putting in a pair and get this:

If n is a positive integer and

is a cube root of unity, then find the number of possible values of

Is that what you intended?  And is the 'matrix' just to get k combinations from n ?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

niharika_kumar

Member

From: Numeraland

Registered: 2013-02-12

Posts: 1,062

Re: random questions

yes bob this was what it meant.

---

friendship is tan 90°.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: random questions

The nth roots of unity always add up to 0.  This is because they are evenly distributed on a unit circle around the origin with conjugate pairs. eg.

I think that's a suitable starting point, but I'll have to have a longer think about how the combinations affect that.  I'd expect that the answer will depend on whether n has the form 3k, or 3k+1, or 3k+2.

Hopefully back later. 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: random questions

Binomial expansion:

So put a=1 and b=omega

So the expression becomes:

and using the roots of unity formula in my previous post

As omega can only take one of two values, it looks to me like the expression can only take one of two values.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob