Products and Quotients of Complex Numbers - Online Tutor, Practice Problems & Exam Prep

To multiply complex numbers in polar form, multiply their magnitudes (r values) and add their angles. This can be expressed as \( r_1 \cdot r_2 \) and \( \theta_1 + \theta_2 \), resulting in \( r \text{ cis } \theta \). For division, divide the magnitudes and subtract the angles: \( \frac{r_1}{r_2} \) and \( \theta_1 - \theta_2 \). This process simplifies calculations and enhances understanding of complex number operations, crucial for mastering advanced mathematical concepts.

Products of Complex Numbers in Polar Form

Alright. So in the past couple of videos, we've been spending some time talking about complex numbers in polar form, which looks something like that. Now in this video, we're going to be learning how we can actually do operations on complex numbers, like, say, find the product of 2 complex numbers. Now this might sound like it's going to be really complicated if we have to keep things in polar form, but don't sweat it because it turns out there's actually a very straightforward process for doing this. And once you're familiar with what this equation looks like, I think you're going to find these problems are super straightforward. This is a skill you're going to need to have in this course, so let's just go ahead and jump right into things.

So let's say we have this example, we're asked to find the product of these two complex numbers, and notice that they're both written in polar form. Rather than converting these to rectangular form and then multiplying and converting back, there's actually a much simpler way to do this. All you need to do to multiply two complex numbers in polar form is to multiply the r values. So you start by multiplying the r's out front and then adding the angles. Once you've done this, you just plug everything into this equation, and you'll get your answer. So let's go ahead and try that with the example we have here.

So I'm first going to multiply the 2 r values. I can see that the r value out front here is 3, and the r value out front there is 2. 3 times 2 is 6, so that's what we get. Now we're going to have 6 times the cosine of the 2 angles added together. Now you can see that the first angle is 15 degrees, and see that the second angle is 30 degrees. So we're going to have 15 plus 30 degrees. So all we're doing is adding theta 1 and theta 2, which are the two angles. So we're gonna have the cosine of 15 degrees plus I times the sine of the two angles added together, which is 15 plus 30 degrees.

So we're gonna have this, and then this right here is how we can multiply these two complex numbers in polar form. So let's go ahead and simplify what we have. We have 15 degrees plus 30 degrees, which comes out to 45 degrees. So this is all going to come out to 6 times the cosine of 15 plus 30, which is 45 degrees, plus I times the sine of 45 degrees. And this right here is the product of the 2 complex numbers. So as you can see, once you know this equation, it's a very straightforward process. Just plug in the numbers and simplify.

Now I will say there is something that can get kind of tedious with this, though, and that's writing out the cosines and sines every single time. But it turns out there actually is a shortcut for doing this. So whenever you see this r cos theta + i sin theta, which is the typical thing we see for polar form, you can actually abbreviate this to r cis theta. This is just a more simple and compact way of writing this. So rather than writing our solution as 6 and then with the cosines and sines, we can actually write this whole thing as 6 cis, our angle, which is 45 degrees.

So as you can see, we're able to write this in a much simpler way if we use this kind of cis notation here rather than just writing the cosines and sines every time. So hopefully, this process seems pretty straightforward. But to really make sure we understand this new notation and the process of multiplying complex numbers, let's actually try one more example.

So in this example, we're asked to find the product of the complex numbers below. Now as you can see, we have these two complex numbers, and both of them are in polar form. Now what I'm going to do first is the process that we use for multiplying these two complex numbers up here, so I'm gonna multiply the r values. So we're going to have 4 times 5, which is 20. Now rather than writing out these cosines and sines, I'm actually going to abbreviate all this to -sis. I'm gonna use this new notation we learned about because I think this is gonna be a lot simpler.

So what I'm going to have is cis, and remember we add the two angles. So what we're gonna have is the first angle, which I can see is pi over 6 radians, and I'm going to add this to the second angle, which I can see is pi over 3 radians. So this would be the sum of the 2 angles, and that right there is the product. So all I need to do from here is simplify what we have within this cis notation.

So we're going to have pi over 6 plus pi over 3. Now what I need to do is get like denominators, and I can do that by taking this pi over 3 and multiplying the top and bottom of the fraction by 2. Because what that's gonna give us is pi over 6, which we have on the left side, plus, and we're going to have 2 times pi, which is 2 pi, over 3 times 2, which is 6. Now we have common denominators, so I can just add the numerators.

So we're going to have 20 cis, and then we're going to have the 2 angles added. Well, we have pi plus 2 pi, which is 3 pi, and then that's over 6. Now what I can actually do is simplify this, because 3 over 6 is the same thing as 1 half. This actually reduces. This whole thing is gonna come out to 20cis, and then we're going to have pi over 2, because that's the same thing as 1 half of pi. So we have pi over 2, and that right there is the product of our 2 complex numbers. So as you can see, using this cis notation can make this process a lot simpler, And when it comes to multiplying the numbers, you just need to remember to multiply the r values and add the angles, and that's gonna give you your solution.

So that's how you can solve these types of problems. Thanks for watching. Let's go ahead and move on and get some more practice.

Given $z_1=5\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right)$ and $z_2=3\left(\cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4}\right)$, find the product $z_1・z_2$.

• A. $z_1・z_2=15CiS\left(\frac{11\pi^2}{12}\right)$
• B. $z_1・z_2=15CiS\left(\frac{\pi}{2}\right)$
• C. $z_1・z_2=CiS\left(\frac{\pi^2}{2}\right)$
• D. $z_1・z_2=15CiS\left(\frac{11\pi}{12}\right)$

Given $z_1=\frac23\left(\cos25\degree+i\sin25\degree\right)$ and $z_2=\frac52\left(\cos15\degree+i\sin15\degree\right)$, find the product $z_1・z_2$.

• A. $z_1・z_2=\frac53CiS\left(375\degree\right)$
• B. $z_1・z_2=CiS\left(40\degree\right)$
• C. $z_1・z_2=\frac{19}{6}CiS\left(40\degree\right)$
• D. $z_1・z_2=\frac{19}{6}CiS\left(375\degree\right)$

Quotients of Complex Numbers in Polar Form

Welcome back, everyone. So recall in the last video how we learned about how we can take products of complex numbers in polar form, and the way we do this is by multiplying the r values and then adding the angles together, and then putting them all together in this equation. Well, in this video, we're now going to learn how we can do what is in essence the opposite of multiplying 2 complex numbers in polar form, which is going to be finding the quotient or dividing complex numbers. Now something you might be happy to hear about is even though this process may sound scary, it's actually very straightforward if you know how to multiply complex numbers. And this is a skill you're going to need to have in this course, so let's just jump right into an example to see how we can do this. So let's say we have these two complex numbers right here, and we're asked to find their quotient, which is the same thing as dividing them. Now, the nice thing about dividing complex numbers in polar form is that it's literally the same idea as multiplication, but with opposite operations. So let's think about this for a second. If I have to divide these complex numbers, well, what do we do when multiplying them? Well, multiplying complex numbers, we multiplied the r values. So, the opposite of multiplication, the way I like to think of it, it's division. So rather than multiplying the r values, we would divide the r values. So r1 divided by r2. And recall that for the multiplication or the product, we add the angles. So when it comes to dividing, we subtract the angles. So literally, it's just opposite operations. You divide the r values and subtract the angles rather than multiplication and addition. So it's straightforward. So let's go ahead and do it. So what I'm first going to do is divide the r values. So we're gonna have 6 divided by 3, that's the 2 r values, which is 2. And then that's going to be multiplied by we have the cosines and sines, but recall that we can actually write this in a more simple way. We can write this as cis. Right? That's a more compact way to write this that we learned in the previous video. So, we're going to have cis, the two angles subtracted. Well, I can see that the first angle is 45 degrees and that's going to be minus the second angle which I can see is 15 degrees. So we're going to have 45 minus 15 which is 30. So we're going to have 2 cis 30 degrees. And this right here is the solution to our problem. So that's how you can divide 2 complex numbers or find their quotient. Again, it's the same equation as multiplication, just with opposite operations. Now let's go ahead and try another example to really make sure that we have this down. So what I'm going to do is use this situation where we're told these are the two complex numbers that we have z1 and z2, and we're asked to find their quotient. So the first thing I'm going to do is I'm going to divide the r values out in front. So to find z1 divided by z2, we're going to first divide r1 and r2. r1 I can see is 5, and r2 is 4. So that's going to be what we have out in front, and then we're going to have this cis because that's the same thing as the cosines and sines, and we're going to have the 2 angles subtracted. So you can see the first angle is pi over 3. So we're going to have pi over 3, and then we're subtracting the angles, and that's going to be minus pi over 9 radians. And that right there would be what we have. So let's just go ahead and simplify this. So we're going to have 5 fourths cis, and we're going to have pi over 3 minus pi over 9. Now what I want to do is get like denominators. I can do that by taking the first fraction that I see and multiplying the top and bottom by 3. What that's going to give us is 3 times pi, which is 3 pi, divided by 3 times 3, which is 9. Then we won't have minus pi over 9 that stays the same. So this is what we have in the parentheses. Now from here, we're going to have 5 fourths cis, and then we're going to have 3 pi minus pi which is 2 pi divided by the common denominator of 9. So this is the solution right here, and that's how you can solve these types of problems. So, hope you found this video helpful. Thanks for watching, and let's move on and get some more practice.

Given $z_1=\frac15\left(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}\right)$ and $z_2=5\left(\cos\frac{\pi}{5}+i\sin\frac{\pi}{5}\right)$, find the quotient $\frac{z_1}{z_2}$.

• A. $\frac{z_1}{z_2}=\frac{1}{25}CiS\left(\frac{3\pi}{10}\right)$
• B. $\frac{z_1}{z_2}=\frac{1}{25}CiS\left(\frac{5\pi}{2}\right)$
• C. $\frac{z_1}{z_2}=-\frac{24}{5}CiS\left(\frac{3\pi}{10}\right)$
• D. $\frac{z_1}{z_2}=-\frac{24}{5}CiS\left(\frac52\right)$

Given $z_1=12\left(\cos30\degree+i\sin30\degree\right)$ and $z_2=3\left(\cos50\degree+i\sin50\degree\right)$, find the quotient $\frac{z_1}{z_2}$.

• A. $\frac{z_1}{z_2}=4CiS\left(20\degree\right)$
• B. $\frac{z_1}{z_2}=9CiS\left(340\degree\right)$
• C. $\frac{z_1}{z_2}=9CiS\left(20\degree\right)$
• D. $\frac{z_1}{z_2}=4CiS\left(340\degree\right)$

Here’s what students ask on this topic:

How do you multiply complex numbers in polar form?

To multiply complex numbers in polar form, you multiply their magnitudes (r values) and add their angles. This can be expressed as:

$r_1\cdot r_2$ and ${\theta }_1+{\theta }_2$, resulting in $r\mathrm{cis}\theta$. This process simplifies calculations and enhances understanding of complex number operations.

How do you divide complex numbers in polar form?

To divide complex numbers in polar form, you divide their magnitudes (r values) and subtract their angles. This can be expressed as:

$\frac{r_1}{r_2}$ and ${\theta }_1-{\theta }_2$, resulting in $r\mathrm{cis}\theta$. This method is straightforward and mirrors the multiplication process but with opposite operations.

What is the shortcut notation for complex numbers in polar form?

The shortcut notation for complex numbers in polar form is $r\mathrm{cis}\theta$. This is a more compact way of writing the typical polar form $r\cos \theta +i\sin \theta$. Using this notation simplifies the process of multiplying and dividing complex numbers in polar form.

Why is it easier to multiply and divide complex numbers in polar form?

It is easier to multiply and divide complex numbers in polar form because the operations are simplified to basic arithmetic. For multiplication, you multiply the magnitudes and add the angles. For division, you divide the magnitudes and subtract the angles. This avoids the more complex algebraic manipulations required in rectangular form and provides a clearer geometric interpretation.

Can you provide an example of multiplying two complex numbers in polar form?

Sure! Let's multiply $3\mathrm{cis}15°$ and $2\mathrm{cis}30°$. First, multiply the magnitudes: $3\cdot 2=6$. Then, add the angles: $15°+30°=45°$. The result is $6\mathrm{cis}45°$.