1. ## Complex Number questions

There might be a couple, but this is the first...
$\text{For each z in the set of complex numbers, show there are unique real numbers}a\text{and}b\text{such that}z=a+bw$

2. ## Re: Complex Number questions

Originally Posted by BenHowe
There might be a couple, but this is the first...
$\text{For each z in the set of complex numbers, show there are unique real numbers}a\text{and}b\text{such that}z=a+bw$
What is w? There won't exist such real a and b if w is real, but if w is non-real, then there will.

3. ## Re: Complex Number questions

Originally Posted by InteGrand
What is w? There won't exist such real a and b if w is real, but if w is non-real, then there will.
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4. ## Re: Complex Number questions

Originally Posted by BenHowe
There are various ways to show it. It comes out essentially immediately with a bit of Linear Algebra knowledge.

5. ## Re: Complex Number questions

Uh I haven't done any linear algebra yet,

6. ## Re: Complex Number questions

I'm also struggling with using field and order axioms to prove stuff like given

$x+z=y+z\\\text{show}x=y$

7. ## Re: Complex Number questions

Originally Posted by BenHowe
I'm also struggling with using field and order axioms to prove stuff like given

$x+z=y+z\\\text{show}x=y$
Add the negative of z to both sides and use associativity of addition.

8. ## Re: Complex Number questions

How do you like refer to the order/field axioms

9. ## Re: Complex Number questions

Originally Posted by BenHowe
How do you like refer to the order/field axioms
Depends which ones. You can say things like "associativity of addition" etc.

10. ## Re: Complex Number questions

Ahh gotcha tbh I worked it out. Thanks for your help. The thing I don't understand still are partial orders and how u'd use it for a q like this. Also what the curved inequality signs mean.

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11. ## Re: Complex Number questions

Originally Posted by BenHowe
Ahh gotcha tbh I worked it out. Thanks for your help. The thing I don't understand still are partial orders and how u'd use it for a q like this. Also what the curved inequality signs mean.

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$\noindent That curved inequality sign is being used to denote a partial order (\geq ) on the given set.$

12. ## Re: Complex Number questions

Originally Posted by InteGrand
There are various ways to show it. It comes out essentially immediately with a bit of Linear Algebra knowledge.
$\noindent Here is one way to show a and b are unique. Let u = a + b \omega. If a and b are not unique, then it is possible to write this complex number as u = \alpha + \beta \omega. Here a,b,\alpha, \beta \in \mathbb{R} while \omega is the complex cube root of unity whose imaginary part is positive. Also, let \omega = w_1 + i w_2 where w_1 \in \mathbb{R} and w_2 > 0.$

$\noindent So$

$u = a + b\omega = a + b(w_1 + i w_2) = (a + bw_1) + i b w_2$

$\noindent and$

$u = \alpha + \beta\omega = \alpha + \beta (w_1 + i w_2) = (\alpha + b\beta w_1) + i \beta w_2$

$\noindent Thus$

$(a + bw_1) + i bw_2 = (\alpha + \beta w_1) + i \beta w_2.$

$\noindent On equating real and imaginary parts we have$

$a + b w_1 = \alpha + \beta w_1 and b w_2 = \beta w_2.$

$\noindent From b w_2 = \beta w_2 we see that (b - \beta) w_2 = 0. But since w_2 > 0 this can only be true provided b = \beta. It therefore follows that a = \alpha showing a and b are unique.$

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