z = x + iy Form (2 Viewers)

[XcarvengerX]

Just did my Extension 2 exam today. Question 1 killed me.
Let A = 3 + 4i. Write the following in z = x + iy form.
a. A square
b. 1 over A
c. square root of A
d. 2 to the power of A

Part a and b are done easily.
a. Just expand, then collect like terms.
b. Just multiply by A's conjugate to rationalise the denominator, then separate the real from the imaginary.

However, I got confused on part c and d. Part c, I tried to do it like part b, but can't seem to form x + iy. I have i all over the place. Then, I have no idea for part d. Any help would be appreciated.

 

[Riviet]

Let sqrt(3+4i) = a+ib
3+4i = (a+ib)²
=a²-b²+2abi
equate real and imaginary parts,
a²-b²=3
2ab=4
Solve these two equations simultaneously.

Not sure about d either.

 

[YBK]

what do u mean two to the power of A

wait, i understand now.. let me try doing it..

bah, i can't do it... 4u test tomorrow !!!

 

[Riviet]

2³⁺⁴ⁱ=8.2⁴ⁱ
=8.(16)ⁱ
=8.e^log_e16i
=8.e^i.log_e16
=8cis(ln16), where ln16 is the angle "phi", using Euler's formula.

 

[Trev]

2³⁺⁴ⁱ
2³.(2⁴)ⁱ
8.16ⁱ

 

[Trev]

You're not taught Euler's formula though, are you?

 

[Riviet]

I don't think it's in the syllabus, but I know the teacher can put in anything that's out of the syllabus in a half yearly or assessment...

 

[XcarvengerX]

Thanks Riviet. But I think there is another alternative method like using conjugate pair or something like that. OK, but your method is simpler and neater anyway. Thanks.

What is Euler's formula?
And Trev, in your answer, is your x = 0?

 

[Riviet]

Euler's formula is this:

We don't need to know how to derive it and it is not in the syllabus.

 

[JulieClark]

Well I'm glad that it isn't. my teacher is a real stickler for the syllabus when it comes to exams. i have my ext 2 paper tomorrow, but i'm not really sure how many of the derivations of the conics formulas i'm supposed to know - or am i just supposed to memorize all of them? my teachers away, you see...

 

[Trev]

And Trev, in your answer, is your x = 0?

It looks like my x would be 0 but as Riviet showed using Euler's, it further simplifies so the x is the real part of 8cis(ln16).

I learnt Euler's theorem for the first time today (well I would have if I had gone to my lecture) in ELEC1000 (intro. to electrical engineering); wierd that they did it in this subject before we did it in my MATH1051 subject.

 

[Raginsheep]

Complex numbers are used to express currents in electronics meaning that if anyone starts bitching to you about the stupidity of "imaginary" numbers, tell them to play with the "imaginary" currents

 

[housemouse]

Well I'm glad that it isn't. my teacher is a real stickler for the syllabus when it comes to exams. i have my ext 2 paper tomorrow, but i'm not really sure how many of the derivations of the conics formulas i'm supposed to know - or am i just supposed to memorize all of them? my teachers away, you see...

For my exam, I just memorised all the formulas for the ellipse, hyperbola and rectangular hyperbola. However, depending on the difficulty of your exam, you may be asked to show something like PS = e(1 - acos theta)

 

[_ShiFTy_]

Well I'm glad that it isn't. my teacher is a real stickler for the syllabus when it comes to exams. i have my ext 2 paper tomorrow, but i'm not really sure how many of the derivations of the conics formulas i'm supposed to know - or am i just supposed to memorize all of them? my teachers away, you see...

I wouldnt bother to memorise the equations...just know how to derive them

 

[XcarvengerX]

2³⁺⁴ⁱ=8.2⁴ⁱ
=8.(16)ⁱ
=8.e^log_e16i
=8.e^i.log_e16
=8cis(ln16), where ln16 is the angle "phi", using Euler's formula.

Hey Riviet, can you finish your working out? What a weird angle... Is it the same for degree and radian mode?

While you are there, mind telling me what is gradient mode in calculator for?

 

[Raginsheep]

Complex numbers works with radians mostly.

 

[XcarvengerX]

Complex numbers works with radians mostly.

But cos(ln16) and sin(ln16) is the same both in degree and radian mode, isn't it? The difference between degree and radian is only in pi.

 

[Raginsheep]

Umm......no.
If it was in degree mode, then the angle would be ln16 degrees which is something like 2.7725......degrees.
2.7725... degrees = something like 0.0483 radians.

 

[XcarvengerX]

Yeah, I just checked it and it is really different...

Then what is gradient mode for?

 

[Riviet]

8cos(ln16)+8isin(ln16)

I would just leave it there, the decimal approximations don't make it any simpler.