 But take a complex number say z equals x plus yi. We like to use x for the real part and y for the imaginary part because you often like to think of complex numbers as points in the complex plane, the horizontal being the real part and the vertical being the imaginary part. Well, given these points in the plane, we can measure their distance from the origin in the plane and this is commonly referred to as the modulus of a complex number. Sometimes it's called absolute value because we actually use the same notation to describe the modulus of the complex number as we do, as we describe the absolute value of a real number. And in fact, when we realize that real numbers are just special types of complex numbers, this really is extending the usual notion of absolute value. So the modulus of a complex number is the distance that complex number is from the origin in the complex plane. And this follows from the usual Pythagorean equation, if we think of our point plotted like say in the following way, we have some complex point right here, its coordinates would be x comma y, then the distance from the origin right here, which is just the complex number zero, we can associate to it a right triangle where x is the horizontal distance, y is the vertical distance and by the Pythagorean equation, if we call the hypotenuse r, the hypotenuse which will be what we define to be the modulus of the complex number, it will be the square root of x squared plus y squared, again by the usual Pythagorean theorem right there. So if we take some complex numbers for example, let's start off with the complex number five i, this is actually a purely imaginary number. In this situation, the modulus of five i, this would be the square root of zero squared plus five squared, this will just be the square root of 25 squared, sorry, the square root of 25, which is five squared. So that's a five there. And so if your number was purely imaginary, that is it lives on the imaginary axis, the y axis, then you're just measuring how far up and down you are, and it doesn't matter whether you're above or below. On the other hand, if you take a real number like seven, then you're gonna get the square root of seven squared, which in this case the square of 49 would just give you back a seven again. And so if you have a real number, it lives on the x axis, whether to the left or the right, then you're just measuring your distance from the origin, the distance from zero on this, this will always be a positive number. So again, the complex modulus of a real number will just give you back the absolute value of the real number. If you take the modulus of an purely imaginary number, it'll just give you back the absolute value of the imaginary part. But if you have a proper complex number, it's neither real nor imaginary, it has a little bit of both there. You can see what happens is you get the square root of three squared, which was the real part, plus four squared, which was the imaginary part. That gives you a nine plus 16, which is 25, inside of a square root, you're gonna get five as well. And so the complex number three plus four i is five units away from the origin in the complex plane. Related to this is the idea of the argument of a complex number, which is essentially the, it's the angle that the complex number forms with the x axis. So kind of redrawing the picture we had before, if we have some complex number x comma y. So this, when we say of course x comma y as a point, we're really thinking of it as the complex number x plus yi. So we see that the distance, let's say this complex number is z, the distance to the point z is the modulus of z and associated to this is that right triangle for which the horizontal distance will be the real part of z. And the vertical distance will be the imaginary part of z. So we have these relationships going on there, but then as it's trigonometry, you might wanna think about the angle. The angle theta here is what we call the argument of z. It's the angle that the complex number forms with the positive real axis, the positive x axis in the situation. And so the argument essentially gives the direction of the complex number is in the plane right there. And if you know the three sides of the complex number, that is the three sides of the triangle associated with this complex number, there's a lot of ways of computing this. The angle theta, we could do a tangent ratio. So notice that tangent of theta will equal y over x, the imaginary part over the real part. So we could do a tangent ratio. You could also do, I mean, if you know the modulus, you could also do sine of theta, which we call y over r, the complex modulus. You could do a cosine ratio, theta equals x over r. I mean, heck, you could even do cotangent secant, cosecant, whichever you wanna do, doesn't really matter. Oftentimes you would then compute the argument using like our tangent of y over x here. Be cautious though, because if you use your calculator using like our tangent, it's only gonna give you the reference angle. It'll give you something in the first quadrant or in the, what's the other one gonna be, in the second quadrant for tangent. You might have to, if you're in like the third or fourth quadrant, you might have to just use that as the reference angle with your calculator, but a little bit of trigonometry does us a lot of good in this situation. So we can calculate the modulus using the Pythagorean equation. We can calculate the argument using Socatowa, whichever of those angles you prefer, doesn't really matter. As long as we have the modulus, we could do all of them. And so imagine we have some type of complex number, which I'm writing in vector form right now. z equals x plus yi. Well, because of the trigonometry in play right here, the x-quadron is just r times cosine of the argument. So r is the modulus and theta is the argument of the complex number. x is just r cosine theta, and y equals r sine theta. This is just a parametric representation of these complex numbers. And so then if we go to the complex number z, it has the standard form, or sometimes called the Cartesian form of the complex number is x plus yi. If its modulus is r and its argument is theta, then x plus yi becomes r cosine theta plus r sine theta. But there's also this imaginary unit i that's sticking around from before, right? Well, recognizing that there's a factor of r, we can multiply it out. And we can write this as r times cosine theta plus i sine theta. And so we see this right here. This is commonly referred to as the, so this z equals r cosine theta plus i sine theta is commonly referred to as the trigonometric form of the complex number. Sometimes it's called the polar form, similar to the polar coordinates that we've seen previously. So we have the trigonometric form or polar form as opposed to the standard form of the complex number, maybe called the Cartesian form, or maybe called the rectangular form of the complex number. And there are advantages of using one over the other. We're gonna see very quickly that, although it's very easy to add and subtract complex numbers using rectangular coordinates, it's gonna be much easier to multiply and divide complex numbers when you do it in polar form. And this is gonna come from Euler's formula.