Roots of Complex Numbers (3 Viewers)

[Grey Council]

No, my teacher has not explained what the two components are. Is Component A questions 1-4, and B 5-8? (or something)?

Hmm

EDIT:
Did a search on google.

Component A (60%) is primarily concerned with the student's knowledge, understanding and skills developed in each Content Area listed in the syllabus.
Component B (40%) is primarily concerned with the student's reasoning, interpretative, explanatory and communicative abilities.

So I should try to master component A by doing the exercises in textbooks, and then try to do as many 'component B' AKA questions 7 and 8 (or something) as possible?

Hm, what you say makes a lot of sense.

 

[Grey Council]

You can forget the idea, if you ever had it, that all you require is a bit of natural genius and that then you can wait for inspiration to strike. There is simply no substitute for hard work and perseverance.

Oh right. Heheeh. As I said, I'll give it my best shot.

If you perservere as much as you seem to be at the moment, you will get gold.

You think so? I'm going to hold you to that.

oh, btw, seems Coroneos is out. I've been phoning him for a while now, no one is picking up in either of the phone lines. And it seems Dymocks IS closed.

ahh well, there isn't that much of a hurry, I suppose. I'll try Coroneos again tommorow, and Dymocks on the 3rd. After all, it IS new years day today, can't expect everyone to be doing businesss. I've printed out all of the exercises you recommended. Will start them soon.

Bleh, goddamn hot, innit? Lucky I have an air con in my room. hehe, parents want me to study so much, they have resorted to bribing. I now have a new table, a kick arse new chair, an air con, a new bed, and pretty much everything i need.

 

[Grey Council]

How long does it take you to get to school in the city from Mount Druit every day?

Hmm, I leave my house at exactly 7:17, reach school around 8:40.

I've actually misquoted Wiles (who is the best mathematician in the world)

Really? The best? nice.

 

[Grey Council]

MAD!

I just talked with coroneos. He picked up this time.

He is away, on holidays i think. I gave him my number, and he said he'll ring me back in a couple of days. hehe

Thanks doctor.

 

[Grey Council]

MAD!

rofl <-- you making fun of me? or is it an oldie being affected by the younger net generation?

Get his autograph? ok. I'll sell it. Any offers?

I'm not putting solutions back on the net because you will learn more from the struggle with a hard question than from the solution itself.

Ok. You have more experience than any of us, so I think i can live without solutions. Anyway, i think i've already said that its better for me if i dont have solutions, cause then when i get stuck I go off and do another subject, like english or economics.

BTW, is it possible for me to mark the coroneos exercises? I'm going through them, but unless I mark my work, I won't know what I'm doing wrong. Actually, stuff that. I'm buying the textbook(s) so i'll just work off those.
^__^

 

[Grey Council]

How do you simplify:
(e^i +e^-i)(e^i - e^-1)

I've seen e before, does it have any special rules? and how does the i as a power work?

Another thing:
Form the equation whose roots are 2 - root 3 and 2+root 3
I did (x-(2 + root 3))(x-(2-root 3))
Is this correct? Or is there a certain way to do these questions?

Last thing, I posted on the buy/sell forum for the coroneos books, I've managed to acquire both 4 unit books, the year 12 3u book and a book with sample papers, all by coroneos. Or i'll have them on sunday, anyway. $15 for the lot.

 

[ND]

Originally posted by GuardiaN
How do you simplify:
(e^i +e^-i)(e^i - e^-1)

I've seen e before, does it have any special rules? and how does the i as a power work?

I don't know whether they want you to use euler's equation, but anyway it's:

e^ix=cosx+isinx

Another thing:
Form the equation whose roots are 2 - root 3 and 2+root 3
I did (x-(2 + root 3))(x-(2-root 3))
Is this correct? Or is there a certain way to do these questions?

Well yeh that's right, but the best way to do it here would be to remember that the equation is in the form:

x^2-(sum of roots)x+(product of roots)

now sum of roots = 4, product of roots = 4-3=1

so eqn is:
x^2-4x+1

 

[wogboy]

And if you're interested in the proof for Euler's formula (not that you need to know it):

let z = cos@ + isin@
dz/d@ = -sin@ + icos@
= i(cos@ + isin@)
dz/d@ = iz
d@/dz = -i/z
@ = -i * ln(Az)
i@ = ln(Az)
Az = e^i@
when @ = 0, z = 1 ---> A = 1
z = e^(i@)

e^(i@) = cos@ + isin@

 

[Grey Council]

hrm, I think you either didn't answer what I wanted, or what you said passed over my head.

I get the roots of the equation bit. But the question:
(e^i +e^-i)(e^i - e^-1)

I think I have to multiply it out. Is this possible?

 

[ND]

You can multiply it out, but it doesn't come to anything nice. Even using Euler's there's nothing nice. I dunno what they're after. Where did this question come from?

 

[Supra]

im an idiot, but isnt it diff of 2 squares???i dunno

 

[ND]

Nope; (e^i +e^-i)(e^i - e^-1).

 

[wogboy]

Expand it out & apply Euler's formula to simplify the expression.

(e^i +e^-i)(e^i - e^-1)
= ( cos(1) + isin(1) + cos(-1) + isin(-1) ) * (e^i - e^-1)
= 2cos(1) * (e^i - e^-1)
= 2cos(1) * ( cos(1) + isin(1) ) - ( 2cos(1) )/ e
= 2cos^2(1) - (2/e)cos(1) + i*2cos(1)sin(1)
= 2cos^2(1) - (2/e)cos(1) + isin(2)
= cos(2) + 1 - (2/e)cos(1) + isin(2)

Note that all sin() and cos() are in radians.

 

[Supra]

Originally posted by ND
Nope; (e^i +e^-i)(e^i - e^-1).

lol
i told u i was an idiot

 

[Grey Council]

Its in the coroneos exercises, you find them on the Doctor's site.

To be more exact, its exercise 2, question 2 vii.
Question 2 says simplify:
i) (-3+3i)^2
ii) (-3+3i)^4
vii) (e^i +e^-i)(e^i - e^-1) <--------------- this is wrong. The question is:
(e^i +e^-i)(e^i - e^-i) <---- Supra is right. It is the difference of two squares, but what i can't figure out is how does i as a power work out? and, as i said, i have seen e before, but i don't really know what e is. I know that e has some special rules, i was wondering if it applies here (i don't know the rules, just wondering if some do apply).

Once again, it is my bad. typo. heeheh, sorry ND and wogboy, really sorry.

 

[ND]

Originally posted by GuardiaN
Its in the coroneos exercises, you find them on the Doctor's site.

Doctor?

(e^i +e^-i)(e^i - e^-i) <---- Supra is right. It is the difference of two squares, but what i can't figure out is how does i as a power work out? and, as i said, i have seen e before, but i don't really know what e is. I know that e has some special rules, i was wondering if it applies here (i don't know the rules, just wondering if some do apply).

Have a go at multiplying it out, you'll be pleasantly surprised. There are special rules with e^i; Euler's equation, as posted above, but you don't need to use it here.

 

[Grey Council]

lol, i got kinda scared and a bit mystified. I'll go and do that now.

BTW, Doctor = Dr Buchanan.

And that Wiles guy proved Fermat's last theorem? o!

No wonder you think he is the best mathematician in the world.

 

[ND]

Originally posted by GuardiaN

BTW, Doctor = Dr Buchanan.

Oh, i was under the impression that "dr" stood for "Derek Robert".

 

[Grey Council]

you serious? i thought he was a doctor. LOL

 

[Supra]

Originally posted by ND

Have a go at multiplying it out, you'll be pleasantly surprised. There are special rules with e^i; Euler's equation, as posted above, but you don't need to use it here.

i use cambridge and i cant find euler in the complex no. section, if u used cambridge can u plz tell me where i can learn it from??my teacher hasnt taught us euler