# Thread: Complex numbers and polar form

1. ## Complex numbers and polar form

Let z = 3 - 3i. Express the following complex numbers in polar form.
(i) z
(ii) z^4
(iii) 1/z

With parts (ii) and (iii) of the question, would z be different from being 3 - 3i? So, for (ii), it would be (3 - 3i)^4 as being z and from that, you would have to change that result into polar form. - Is that right? However, when I was working part (i), I became a bit puzzled when I realised that theta was equal to -pi/4 or 7pi/4. So, I'm unsure as to which one to use in the overall answer.
z = 3- 3i
= √(3^2) + (-3^2)
= √9 + 9
= √18
= 3√2
theta = tan^-1 (-3/3) = tan^-1 (-1) = -pi/4 or 7pi/4
z = 3√2 cis (-pi/4) = 3√2 (cos pi/4 - isin pi/4)
(But then again, couldn't the answer also be: 3√2 (cos 7pi/4 + isin 7pi/4) ?)

I'm not sure if my working is right with that either. I tried with part (ii), with getting the answer (3√2)^4 cis (-pi/4 • 4) = 324 cis (-pi) = 324 (cos pi - i sin pi) ... although, as with part (i), theta was -pi or 7pi so, I'm unsure with which theta you end up using (if that makes any sense).

As for part (iii), I'm completely clueless with how to answer that question. Although, is it right to answer the questions with z = 3√2? (As seen above, that is how I did the working out for part (ii).)

If anyone could help me with my working out and with answering all/either three parts of the questions, it would be extremely appreciated. Thanks!

2. Originally Posted by cottontails
Let z = 3 - 3i. Express the following complex numbers in polar form.
(i) z
(ii) z^4
(iii) 1/z

With parts (ii) and (iii) of the question, would z be different from being 3 - 3i? So, for (ii), it would be (3 - 3i)^4 as being z and from that, you would have to change that result into polar form. - Is that right? However, when I was working part (i), I became a bit puzzled when I realised that theta was equal to -pi/4 or 7pi/4. So, I'm unsure as to which one to use in the overall answer.
z = 3- 3i
= √(3^2) + (-3^2)
= √9 + 9
= √18
= 3√2
theta = tan^-1 (-3/3) = tan^-1 (-1) = -pi/4 or 7pi/4
z = 3√2 cis (-pi/4) = 3√2 (cos pi/4 - isin pi/4)
(But then again, couldn't the answer also be: 3√2 (cos 7pi/4 + isin 7pi/4) ?)

I'm not sure if my working is right with that either. I tried with part (ii), with getting the answer (3√2)^4 cis (-pi/4 • 4) = 324 cis (-pi) = 324 (cos pi - i sin pi) ... although, as with part (i), theta was -pi or 7pi so, I'm unsure with which theta you end up using (if that makes any sense).

As for part (iii), I'm completely clueless with how to answer that question. Although, is it right to answer the questions with z = 3√2? (As seen above, that is how I did the working out for part (ii).)

If anyone could help me with my working out and with answering all/either three parts of the questions, it would be extremely appreciated. Thanks!
For part (i), both answers are correct. However, if the principle argument is to be used, only one of them is correct. You need to go back and check what defintion of principle argument is being used so that you can choose between them.

For part (iii), use deMoivre's Theorem with n = -1.

3. I plotted z = 3 - 3i on the complex plane and it is within the fourth quadrant. Is it still correct to go by "ASTC" so thereby, the negative answer (-pi/4) would have to be the right value of theta to use?

4. Originally Posted by cottontails
Let z = 3 - 3i. Express the following complex numbers in polar form.
(i) z
(ii) z^4
(iii) 1/z

With parts (ii) and (iii) of the question, would z be different from being 3 - 3i? So, for (ii), it would be (3 - 3i)^4 as being z and from that, you would have to change that result into polar form. - Is that right?
That is a very strange way of expressing yourself. In all 3 problems z is the number you are gien, 3- 3i. In (ii) the number you want to put in polar form is not "z" but $z^4$.
However, once you have done (i) and know the polar form form z, say $z= re^{i\theta}$ then $z^4= r^4e^{4i\theta}$ and $1/z= (1/r)e^{-i\theta}$.

However, when I was working part (i), I became a bit puzzled when I realised that theta was equal to -pi/4 or 7pi/4. So, I'm unsure as to which one to use in the overall answer.
In the complex plane, those are exactly the same. Which you should use depends upon what "convention" your class is using. If you are writing all arguments between 0 and $2\pi$, use $7\pi/4$ if, instead, you are using the convention of expressing arguments between $-\pi$ and $\pi$, use $-\pi/4$.

z = 3- 3i
= √(3^2) + (-3^2)
= √9 + 9
= √18
= 3√2
theta = tan^-1 (-3/3) = tan^-1 (-1) = -pi/4 or 7pi/4
z = 3√2 cis (-pi/4) = 3√2 (cos pi/4 - isin pi/4)
(But then again, couldn't the answer also be: 3√2 (cos 7pi/4 + isin 7pi/4) ?)
Yes, either of those is mathematically correct. Which you should use depends, as I said, on what convention your class is using.

I'm not sure if my working is right with that either. I tried with part (ii), with getting the answer (3√2)^4 cis (-pi/4 • 4) = 324 cis (-pi) = 324 (cos pi - i sin pi) ... although, as with part (i), theta was -pi or 7pi so, I'm unsure with which theta you end up using (if that makes any sense).

As for part (iii), I'm completely clueless with how to answer that question. Although, is it right to answer the questions with z = 3√2? (As seen above, that is how I did the working out for part (ii).)

If anyone could help me with my working out and with answering all/either three parts of the questions, it would be extremely appreciated. Thanks!

5. Here are some general tricks to help on (iii).

For all nonzero complex numbers $\dfrac{1}{z} = \dfrac{{\overline z }}{{\left| z \right|^2 }}$.

And the conjugate of $\text{cis}(\theta)$ is just $\text{cis}(-\theta)$.

Thus if $z=r(\text{cis}(\theta))$ then $\dfrac{1}{z}=\dfrac{\text{cis}(-\theta)}{r}$.

6. Originally Posted by cottontails
I plotted z = 3 - 3i on the complex plane and it is within the fourth quadrant. Is it still correct to go by "ASTC" so thereby, the negative answer (-pi/4) would have to be the right value of theta to use?
"ASTC"? You mean in which quadrant the trig functions are positive? Since you are "going the other way", finding the argument (angle) from the trig function, that is not relevant. As both mr fantastic and I have said, $-\pi/4$ and $7\pi/4$ are both correct and which of them you use depends upon the convention your class is using. If you don't remember your teacher explaining that ask your teacher.

(Strictly speaking, since you can always add multiples of $2\pi$ without changing the angle, such things as $8\pi- \pi/4= \frac{31}{4}\pi$ would also be possible but the two commonly used conventions are, as I said before, the argument between 0 and $2\pi$ or the argument between $-\pi$ and $\pi$.

7. With my textbook (and using part (i) as the example, for this case) - they would plot 3 - 3i on the complex plane. Hence, arg(3-3i) would be an angle in the fourth quadrant. Apparently by looking at it plotted on the complex plane, you are able to "easily distinguish between right and wrong answers".So, going by if you were to plot it on the complex plane, would you then go by the 'size' of the angle it makes and allowing whatever is the closest match out of the two to be theta? However, if that sort of thinking can even be correct then, I would assume it would again be a struggle with knowing which value of theta to choose for parts (ii) and (iii) as I'd imagine them being difficult to plot onto the complex plane.

8. I asked my friend in my maths tutorial about it and she also said the same thing about the principal argument (which I had completely forgotten about when I attempted the question). So from there, I was able to figure out what the right angles were. Thanks everyone for your help though!