Complex number polar form intuition

We're asked to adjust the angle in radius of the orange plotted complex number-- so that's this right over here-- to match the blue plotted complex number negative 3.5 plus 6.06i. So this is this one right over here. So just to get our bearings, this vertical axis, this is the imaginary axis. This horizontal axis is the real axis. The real part of this number is negative 3.5, and so we've plotted that on the real axis-- negative 1, negative 2, negative 3.5. And then the imaginary part is 6.06. So we go up 1, 2, 3, 4, 5, 6, and then a very little bit to get to 6.06. So they've plotted it like that. And now they want us to think about it in kind of the polar form is one way of thinking about it. And so they tell us that we can adjust the angle, which is often called the argument when you're plotting up a complex number in this form, and the radius, which is often called the modulus. So let's change that. So let's see, if we increase the radius, so then the thing is going to go out a little bit. If we increase the angle, that's pi over 3. This is pi over 2. So we increase the angle to 2 pi over 3. And this seems to be the right direction, and we just have to increase the radius a little bit. So there we have a radius of 7, an angle of 2/3 pi. Now, let's verify for ourselves that this complex number and this complex number really are the same thing. We've already seen it right over here in this actual diagram. We've plotted it both in I guess you could say, rectangular form and polar form, although this is a complex axis, and this is a real axis. So let me copy and paste the information that we have here. So let me just copy and paste all of that. And then I can put it on my little scratch pad. So let me throw it right over there, and let's actually verify that. So here we have an argument, or an angle I guess I could say. So this angle, often referred to as phi when you're dealing in the complex plane, this is equal to 2/3 pi. And this is, of course, in radians. And the modulus, which is really the distance from the origin to the complex number, this right over here, by just manipulating that little tool, we saw that to be 7. So let's verify that this is really the same complex number as negative 3.5 plus 6.06i. Well, we can represent this as 7-- this is the modulus, or the radius. 7 times e to the 2/3 pi i, which we know from Euler's identity is the exact same thing. This is equal to-- and this is one of really the most mind-blowing things in all of mathematics. This is the same thing as 7 times cosine of 2/3 pi plus i sine of 2/3 pi. And they already wrote that down for us right over there. Now let's verify what cosine of 2/3 pi is and what sine of 2/3 pi is. So if this is 2/3 as an angle-- and now you can almost even think of the unit circle. So this is 2/3 pi. So actually, just let me just draw a unit circle like this because I don't want us to confuse just our traditional rectangular coordinates with what we're doing on the polar axis. So this isn't a polar-- we're not graphing imaginary numbers right over here. This is just the traditional unit circle. This is our x-axis this is our y-axis. 2/3 pi is the same thing as 120 degrees. So it looks just like what we did over here, so 2/3 pi. So let me draw it as neatly as I can, which isn't that neatly. So that angle is 2/3 pi. Our cosine of 2/3 pi is going to be our x-coordinate here. And sine of 2/3 pi is going to be our y-coordinates over here. So what's our x-coordinate? Well, if this right over here is 120 degrees, then this is 60 degrees. This is a 30-60-90 triangle. And we know this is a unit circle. The radius is 1. So the 30-degree side will have length 1/2. So this coordinate right over here is going to be negative 1/2. And then the side opposite the 60-degree side is square root of 3 times the 1/2. So this right over here is going to be square root of 3 over 2. And you could verify that on a calculator, if you like. So cosine of 2/3 pi is negative 1/2. So this is going to be equal to 7 times negative 1/2. That's cosine of 2/3 pi plus i times the square root of 3 over 2. So let's verify that that's actually the same thing as that over there. Well, 7 times negative 1/2 is negative 3.5 plus i. I'll just write it lik e7 square root of 3 over 2i. So we already see that the real parts are definitely the same. It's negative 3.5. And I'll just get a calculator out to evaluate this and see if it is roughly 6.06. So let me get my trusty TI-85 out. And then I want to evaluate 7 times the square root of 3, and then that divided by 2 is indeed 6.06, if we round to the nearest hundredths. So this is approximately equal to negative 3.5 plus 6.06i, which is exactly the complex number that we started with.