11. Graphing Complex Numbers

Products and Quotients of Complex Numbers

11. Graphing Complex Numbers

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

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To multiply complex numbers in polar form, multiply the r values and add the angles: $ r_1 \cdot r_2 $ and $ \theta_1 + \theta_2 $. For division, divide the r values and subtract the angles: $ \frac{r_1}{r_2} $ and $ \theta_1 - \theta_2 $. This process simplifies calculations, allowing for compact notation using $ r \text{cis} \theta $. Mastering these operations is essential for understanding complex numbers and their applications in various mathematical contexts.

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Products of Complex Numbers in Polar Form

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Alright. 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. Now 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 r1 and r2 values. So you start by multiplying the r values out in 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.

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. 3×2 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 I can see that the first angle is 15 degrees, and the second angle is 30 degrees. So we're going to have 15 plus 30 degrees. So all we're doing is adding θ1 and θ2, which are the two angles.

So we're going to 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 going to 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×cos(45°)+i×sin(45°). 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 rcosθ+isinθ, which is the typical thing we see for polar form, you can actually abbreviate this to r cis θ. 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 45°. So as you can see, we're able to write this in a much more simple 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 going to multiply the r values.

So we're going to have 4×5, which is 20. Now rather than writing out these cosines and sines, I'm actually going to abbreviate this all to be cis. I'm going to use this new notation we learned about because I think this is going to be a lot more simple. So what I'm going to have is cis, and remember we add the two angles. So what we're going to have is the first angle, which I can see is π6 radians, and I'm going to add this to the second angle, which I can see is π3 radians. So this would be the sum of the two angles, and that right there is the product.

So all I need to do from here is simplify what we have within this SIS notation. We're going to have π6 plus π3. Now what I need to do is get like denominators, and I can do that by taking this π3 and multiplying the top and bottom of the fraction by 2. Because what that's going to give us isπ6, which we have on the left side, plus, and we're going to have 2π6. 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 π plus 2π, which is 3π, 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 going to come out to 20 cis 12π, because that's the same thing as 1 half of π. So we have r values and add the angles, and that's going to 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.

Problem

Given $z_{1} = 5 (\cos \frac{π}{6} + i \sin \frac{π}{6})$ and $z_{2} = 3 (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})$ , find the product $z sub 1 ・ z sub 2$ .

$z_{1} ・ z_{2} = 15 C i S (\frac{11 π^{2}}{12})$

$z_{1} ・ z_{2} = 15 C i S (\frac{π}{2})$

$z_{1} ・ z_{2} = C i S (\frac{π^{2}}{2})$

$z_{1} ・ z_{2} = 15 C i S (\frac{11 π}{12})$

Problem

Given $z_{1} = \frac{2}{3} (\cos 25 ° + i \sin 25 °)$ and $z_{2} = \frac{5}{2} (\cos 15 ° + i \sin 15 °)$ , find the product $z sub 1 ・ z sub 2$ .

$z sub 1 ・ z sub 2 equals five thirds C i S of 375 degrees$

$z sub 1 ・ z sub 2 equals C i S of 40 degrees$

$z sub 1 ・ z sub 2 equals nineteen sixths C i S of 40 degrees$

$z sub 1 ・ z sub 2 equals nineteen sixths C i S of 375 degrees$

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Quotients of Complex Numbers in Polar Form

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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 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, is division. So rather than multiplying the r values, we would divide the r values. So r1 ÷ 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 going to 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 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 ÷ 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 gonna have the 2 angles subtracted. So you can see the first angle is π/3. So we're gonna have π/3, and then we're subtracting the angles, and that's gonna be minus π/9 radians. And that right there would be what we have. So let's just go ahead and simplify this. So we're gonna have 5/4 cis 2π9.

This is the solution right here, and that's how you can solve these types of problems. So I hope you found this video helpful. Thanks for watching, and let's move on and get some more practice.

Problem

Given $z_{1} = \frac{1}{5} (\cos \frac{π}{2} + i \sin \frac{π}{2})$ and $z_{2} = 5 (\cos \frac{π}{5} + i \sin \frac{π}{5})$ , find the quotient $the fraction with numerator z sub 1 and denominator z sub 2$ .

$\frac{z_{1}}{z_{2}} = \frac{1}{25} C i S (\frac{3 π}{10})$

$\frac{z_{1}}{z_{2}} = \frac{1}{25} C i S (\frac{5 π}{2})$

$\frac{z_{1}}{z_{2}} = - \frac{24}{5} C i S (\frac{3 π}{10})$

$\frac{z_{1}}{z_{2}} = - \frac{24}{5} C i S (\frac{5}{2})$

Problem

Given $z_{1} = 12 (\cos 30 ° + i \sin 30 °)$ and $z_{2} = 3 (\cos 50 ° + i \sin 50 °)$ , find the quotient $the fraction with numerator z sub 1 and denominator z sub 2$ .

$\frac{z_{1}}{z_{2}} = 4 C i S (20 °)$

$\frac{z_{1}}{z_{2}} = 9 C i S (340 °)$

$\frac{z_{1}}{z_{2}} = 9 C i S (20 °)$

$\frac{z_{1}}{z_{2}} = 4 C i S (340 °)$

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How do you multiply complex numbers in polar form?

To multiply complex numbers in polar form, you follow a straightforward process. First, multiply the magnitudes (r values) of the two complex numbers. Then, add their angles. The formula is:

$r_{1} \cdot r_{2}$ and $θ_{1} + θ_{2}$ .

For example, if you have two complex numbers $3 cis 15 °$ and $2 cis 30 °$ , their product is:

$6 cis 45 °$ .

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How do you divide complex numbers in polar form?

Dividing complex numbers in polar form is similar to multiplication but with opposite operations. First, divide the magnitudes (r values) of the two complex numbers. Then, subtract their angles. The formula is:

$\frac{r_{1}}{r_{2}}$ and $θ_{1} - θ_{2}$ .

For example, if you have two complex numbers $6 cis 45 °$ and $3 cis 15 °$ , their quotient is:

$2 cis 30 °$ .

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

The shortcut notation for complex numbers in polar form is $r cis θ$ . This is a more compact way of writing the standard form $r (\cos θ + i \sin θ)$ . For example, instead of writing $6 (\cos 45 ° + i \sin 45 °)$ , you can write $6 cis 45 °$ .

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Why is it useful to use polar form for complex numbers?

Using polar form for complex numbers is useful because it simplifies the process of multiplication and division. In polar form, you can multiply complex numbers by multiplying their magnitudes and adding their angles. Similarly, you can divide complex numbers by dividing their magnitudes and subtracting their angles. This is much simpler than converting to rectangular form, performing the operation, and converting back. The compact notation $r cis θ$ also makes calculations more straightforward.

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How do you convert a complex number from rectangular form to polar form?

To convert a complex number from rectangular form (a + bi) to polar form (r cis θ), follow these steps:

1. Calculate the magnitude (r) using the formula:

$r = \sqrt{a^{2} + b^{2}}$ .

2. Calculate the angle (θ) using the formula:

$θ = \tan - 1 (\frac{b}{a})$ .

For example, for the complex number 3 + 4i, the magnitude is:

$r = \sqrt{3 +^{42}} = 5$ .

The angle is:

$θ = \tan - 1 (\frac{4}{3}) \approx 53.13 °$ .

So, the polar form is:

$5 cis 53.13 °$ .

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

Products of Complex Numbers in Polar Form

Video transcript

Given $z_{1} = 5 (\cos \frac{π}{6} + i \sin \frac{π}{6})$ and $z_{2} = 3 (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})$ , find the product $z sub 1 ・ z sub 2$ .

Given $z_{1} = \frac{2}{3} (\cos 25 ° + i \sin 25 °)$ and $z_{2} = \frac{5}{2} (\cos 15 ° + i \sin 15 °)$ , find the product $z sub 1 ・ z sub 2$ .

Quotients of Complex Numbers in Polar Form

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Given $z_{1} = \frac{1}{5} (\cos \frac{π}{2} + i \sin \frac{π}{2})$ and $z_{2} = 5 (\cos \frac{π}{5} + i \sin \frac{π}{5})$ , find the quotient $the fraction with numerator z sub 1 and denominator z sub 2$ .

Given $z_{1} = 12 (\cos 30 ° + i \sin 30 °)$ and $z_{2} = 3 (\cos 50 ° + i \sin 50 °)$ , find the quotient $the fraction with numerator z sub 1 and denominator z sub 2$ .

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How do you multiply complex numbers in polar form?

How do you divide complex numbers in polar form?

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

Why is it useful to use polar form for complex numbers?

How do you convert a complex number from rectangular form to polar form?

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

Products of Complex Numbers in Polar Form

Video transcript

Given z1=5(cos⁡π6+isin⁡π6)z_1=5\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right) and z2=3(cos⁡3π4+isin⁡3π4)z_2=3\left(\cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4}\right), find the product z1・z2z_1・z_2.

Given z1=23(cos⁡25°+isin⁡25°)z_1=\frac23\left(\cos25\degree+i\sin25\degree\right) and z2=52(cos⁡15°+isin⁡15°)z_2=\frac52\left(\cos15\degree+i\sin15\degree\right), find the product z1・z2z_1・z_2.

Quotients of Complex Numbers in Polar Form

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Given z1=15(cos⁡π2+isin⁡π2)z_1=\frac15\left(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}\right) and z2=5(cos⁡π5+isin⁡π5)z_2=5\left(\cos\frac{\pi}{5}+i\sin\frac{\pi}{5}\right), find the quotient z1z2\frac{z_1}{z_2}.

Given z1=12(cos⁡30°+isin⁡30°)z_1=12\left(\cos30\degree+i\sin30\degree\right) and z2=3(cos⁡50°+isin⁡50°)z_2=3\left(\cos50\degree+i\sin50\degree\right), find the quotient z1z2\frac{z_1}{z_2}.

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Here’s what students ask on this topic:

How do you multiply complex numbers in polar form?

How do you divide complex numbers in polar form?

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

Why is it useful to use polar form for complex numbers?

How do you convert a complex number from rectangular form to polar form?

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Given $z_{1} = 5 (\cos \frac{π}{6} + i \sin \frac{π}{6})$ and $z_{2} = 3 (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})$ , find the product $z sub 1 ・ z sub 2$ .

Given $z_{1} = \frac{2}{3} (\cos 25 ° + i \sin 25 °)$ and $z_{2} = \frac{5}{2} (\cos 15 ° + i \sin 15 °)$ , find the product $z sub 1 ・ z sub 2$ .

Given $z_{1} = \frac{1}{5} (\cos \frac{π}{2} + i \sin \frac{π}{2})$ and $z_{2} = 5 (\cos \frac{π}{5} + i \sin \frac{π}{5})$ , find the quotient $the fraction with numerator z sub 1 and denominator z sub 2$ .

Given $z_{1} = 12 (\cos 30 ° + i \sin 30 °)$ and $z_{2} = 3 (\cos 50 ° + i \sin 50 °)$ , find the quotient $the fraction with numerator z sub 1 and denominator z sub 2$ .