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FP1 Complex Numbers Help

I don't understand what to do for question 9 part b?

Arg z = inverse tan of opp/adj so that should equal pie/4 in the example but for part a I got up to 1-0.5i as being the modulus and then from that I got 0.464 as being the the argument?? ImageUploadedByStudent Room1393778507.589995.jpg


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davros
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Original post by Saywhatyoumean
I don't understand what to do for question 9 part b?

Arg z = inverse tan of opp/adj so that should equal pie/4 in the example but for part a I got up to 1-0.5i as being the modulus and then from that I got 0.464 as being the the argument?? ImageUploadedByStudent Room1393778507.589995.jpg


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How can 1 - 0.5i be the modulus? The modulus is always a positive real number (or possibly 0 if z = 0).
Original post by Saywhatyoumean
...


For a complex number z=x+iyz = x + iy, the modulus is ∣z∣=x2+y2|z| = \sqrt{x^2 + y^2} and the argument is arg⁡(z)=tan⁡−1(yx)\arg(z) = \tan^{-1} \left( \dfrac{y}{x} \right).
You've been given the complex number:

z=4+3i2+4iz = \dfrac{4+3i}{2+4i}

so you might want to rearrange it into the form z=x+iyz = x+iy to make things a little easier :smile:
(edited 10 years ago)
Original post by Saywhatyoumean
I don't understand what to do for question 9 part b?

Arg z = inverse tan of opp/adj so that should equal pie/4 in the example but for part a I got up to 1-0.5i as being the modulus and then from that I got 0.464 as being the the argument?? ImageUploadedByStudent Room1393778507.589995.jpg


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∣z1z2∣=∣z1∣∣z2∣,arg⁡(z1z2)=arg⁡z1−arg⁡z2 |\frac{z_1}{z_2}|=\frac{|z_1|}{|z_2|}, \arg (\frac{z_1}{z_2})= \arg{z_1}- \arg{z_2}
Original post by davros
How can 1 - 0.5i be the modulus? The modulus is always a positive real number (or possibly 0 if z = 0).


Oh sorry I mean I got root 5/2 for the modulus after working out it was 1-0.5i in complex number form


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Original post by brianeverit
∣z1z2∣=∣z1∣∣z2∣,arg⁡(z1z2)=arg⁡z1−arg⁡z2 |\frac{z_1}{z_2}|=\frac{|z_1|}{|z_2|}, \arg (\frac{z_1}{z_2})= \arg{z_1}- \arg{z_2}


But I'm only given z??


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Original post by Khallil
For a complex number z=x+iyz = x + iy, the modulus is ∣z∣=x2+y2|z| = \sqrt{x^2 + y^2} and the argument is arg⁡(z)=tan⁡−1(yx)\arg(z) = \tan^{-1} \left( \dfrac{y}{x} \right).

You've been given the complex number:

z=4+3i2+4iz = \dfrac{4+3i}{2+4i}

so you might want to rearrange it into the form z=x+iyz = x+iy to make things a little easier :smile:


I did and after 'rationalising' I got 1-0.5i

But how do I use that to answer part b?


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Original post by Saywhatyoumean
I did and after 'rationalising' I got 1-0.5i


You haven't yet completed the first part, since you need to find the modulus of z=1−i2z = 1 - \frac{i}{2}

Original post by Saywhatyoumean
But how do I use that to answer part b?


You don't. You use the general complex number in terms of aa that they've already given you.

z=a+3i2+aiz = \dfrac{a+3i}{2+ai}

First, consider the difference of two squares in the denominator, just like you did in the first part.

Then use the fact that arg⁡(z)=tan⁡−1(yx)\arg(z) = \tan^{-1} \left( \dfrac{y}{x} \right).

We're looking for a value of aa such that arg⁡(z)=π4\arg(z) = \dfrac{\pi}{4}, so solve for aa.
(edited 10 years ago)
Original post by Saywhatyoumean
But I'm only given z??


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No you have two z1=a+3i, z2=2+aiz_1=a+3i,\ z_2=2+ai
Oh okay, thanks everyone


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So I end up with a quadratic and factorise this to end up with two values for a

I sub a back into the original z and with one answer I get pie/4 which is what I was trying to show :smile:


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