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Polar & rectangular forms of complex numbers

Sal rewrites the complex number -3+2i (which is given in rectangular form) in its polar form. Created by Sal Khan.
Video transcript
Let's say that I have the complex number z, and in rectangular form we can write it as negative 3 plus 2i. So, first, let's think about where this is in the complex plane. So, this is our imaginary axis, and that is our real axis. And, let's see. The real part is negative 3. So we can go one, two ,three to the left of the origin, let me space that out a little bit more evenly, so one, two, three to the left of the origin, so negative 3. And then we have 2i. So we are gonna go up 2. So up 2 in the imaginary direction. So one, two, just like that. And so z is real part negative 3, imaginary part 2. Z would be right over here in the complex plane. So that is Z right over there. Now, what I want to think about if there are other ways to essentially specify the location of z. And in particular, if instead of giving the real and imaginary part, essentially the coordinates here, let's think about giving a direction and a distance to get to z. So, for example, we could give the distance from the origin to z , so let's call this distance r, and the distance by itself isn't enough to say where z is, you have to say in what direction we have to go the distance of r to get to z. In so to specify the direction, we will have to this angle theta in radians, this angle between the positive real axis and this line right over here, or the segment that connects the origin and z. So if someone gave you this angle and this distance, you could get to z. Now, what I want you to do right now is pause this video and see if you can find a relationship between r and theta and negative 3 and 2. In fact, even this complex number in retangular form, can you figure out what r and theta are? Well, let's think through this a little bit. And to help us that, let's remind ourselves of the unit circle definition of trig function, because we are going to use some trig functions to relate r, theta and 2, and negative 3. So, I am gonna construct a unit circle here, so construct a unit circle. So this right here is a unit circle, circle of ratio 1. So construct a unit circle, that is a unit circle. So, by definition, what are the horizontal and vertical coordinates of this point right over here, where this line intersects the unit circle? Well, this is forming an angle of theta, with the positive real axis, and so the horizontal coordinate over here, by definition, is going to be to cosine of theta. That is the unit circle definition of cosine of theta. And the vertical coordinate is going to be sine of theta. And so, what would, how would the horizontal and vertical coordinates of this point be? We obviously know their negative 3 and 2, but what would be in terms of cosine theta and sine theta? Well, look. This point right over here is ratio of 1 away from the origin. So this distance right over here, that distance right over here is 1. But now we are r away from the origin, we are r times this far. So for r times this far in that direction, we are going to be r times this far in the vertical direction, and r times this far in the horizontal direction. We are going to have to scale all of, we are going to scale by r. So the horizontal coordinate of this point right over here, instead of being cosine of theta, is going to be r times cosine of the theta. So this point right over here, which we know as negative three, is going to be equal to r cosine of theta. And by the same logic, this point over here, the vertical coordinate, we are gonna scale up sine theta by r, where r times this far. So this point right over here is going to be r sine theta, and we already know that it is equal to 2, r sine theta. So given that, can we now figure out what r and theta are? So, let's first think about what theta is. So, to do that, let's think about some of our trig functions. So one trig function that involves sine theta and cosine theta is tangent theta. So, for example, we could say tangent theta, tangent of our angle, tangent of theta is equal to sine of theta over cosine of theta. We could also multiply the numerator and the denominator by r, that won't change the value, so that is the same thing as r sine theta over r cosine of theta. And we know that r sine theta is going to be equal to 2 and we know that r cosine of theta is negative 3. So this whole thing is going to be negative 2/3. Another way of thinking about it is that tangent of theta is going to be the same thing as the slope of this line right over here. And what is the slope of that line? Well, if you start z and you want to go to the origin, we are going to go positive 3 in the x direction, and then you are going to go negative 2, or you could say positive 3 in the horizontal direction and then you go negative 2 in the vertical direction. So the slope, the change in vertical over the change in horizontal, is negative 2/3. But now we can use this to solve for theta, solve for theta, we just take the inverse tangent of both sides, and we get theta is equal to inverse tangent of negative 2/3, this thing right over here. So we copy and paste that. And we can get our calculator out to figure out what this actually is, so, let's turn it on, let me make sure that I am in radians mode, I am. And so I can take the inverse tangent of negative 2/3, which gets us negative 0,58800 on and on and on. So if we round to the nearest hundred of radian, it would be negative 0,59. So this is approximately negative 0,59. Now, is this the right angle? Is this the theta that we are looking for? Well, this theta, this negative 0,59, that is going to get us over here, that is this angle right over here. That is what the inverse tangent gave us. And it makes sense, because this ray is a continuation, of this ray right over here, together they would form a line that has the same slope. But that is not the theta that we are looking for, the theta we are looking for is going in the opposite direction, it is half way around the circle. So, if we think in terms of radians, it is going to be this angle plus π radians. So the theta that we are looking for isn't going to be what the calculator gives us for this, it is going to be this value plus π. So theta, the theta that we care about is going to be negative 0,59 roughly plus π. Let me say approximately, just because I have rounded right over here. So let's take the calculator out. So let's take our previous response and, that is just the previous answer plus π, cause we are gonna go the opposite direction. That gives us 2, roughly 2, 55 radians. So this is approximately 2,55 radians, it is our claim for what theta is. Now, does that make sense? That this angle right over here is 2,55 radians? Well, this angle, if we go straight up, that is π/2 radians. And π is 3,14, so that is going to be 1,57 right? 1, 5 something like that. And this is π radians, so that is 3,14159, keep going on and on and on. So 2,55 is indeed in the right quadrant. So that is right. This is going to be an angle of 2,55 radians. Now, we just have to figure out the length, what r is. And that ,we can just use the Pythagorean theorem for. We can set up a right triangle. We know that this distance right over here is 2, because this distance right over here is 2. And we know that this distance is, the coordinate here is negative 3, but the distance here is just 3. So we know from the Pythagorean theorem that r squared is equal to 2 squared plus 3 squared. Or that r is equal to square root of 4 plus 9, which is 13. And so, there we have, we have negative 3, so we know r is equal to square root of 13, and theta is 2,55 radians. So using that information, we can now rewrite z in terms of r and our theta. So now we can say that z is equal to, instead of negative 3, we can write it as r cosine theta. So we can rewrite it as square root of 13 times the cosine of theta, which is, we are saying that theta is roughly 2,55, if you round it to the nearest hundredth. Plus 2, we already know it is r sine of theta, so r, square of 13 and then, sine of theta, 2,55. And then all of this, we could say, all of this is going to be multiplied times i. And if we like to simplify a little bit, or we kind of really want to make clear what the r is here, we can fator it out, so we can say that z is equal to square root of 13 times cosine of 2,55, and once again 2,55 is an approximation, so maybe I should say approximately equal to, cosine of 2,55 plus, i will trow the i in front, plus i times sine of 2,55. And so when I rewrote z in this format right over here, it makes much clearer what the direction I have to go in this, the direction is 2,55 radians, counter clock wise from the positive real axis, and also says how far we need to go, I need to of square root of 13. So this is essentially makes the polar, and makes it clearer how we get there, in kinda of a more, I think we could say, polar mindset. And that is why this form of the complex number, writing in this way is called the retangular form. Well, writing this way is called polar form.