Modulus (absolute value) and argument (angle) of complex numbers
I have the complex number 3 minus 4i. I've plotted it on the complex plane. We see that the real part is 3, so we've gone 3 along the horizontal axis, or the real axis. And the imaginary part is negative 4, so we've gone down 4 along the vertical axis. So this right here is the point 3 minus 4i. Now what I want to think about is what the absolute value of 3 minus 4i is. And just as a reminder, absolute value literally means-- whether we're talking about a complex number or a real number, it literally just means distance away from 0. So the absolute value of 3 minus 4i is going to be the distance between 0, between the origin on the complex plane, and that point, and the point 3 minus 4i. So this distance right over here is going to be the absolute value of 3 minus 4i. So how can we think about that? Well, we could literally just set up a right triangle and then use the Pythagorean theorem. So let's think about it. If we wanted to set up a right triangle, the height here, the distance between 0 and negative 4, well, that distance is going to be 4. And then the base of this triangle, the distance between 0 and 3, is just going to be 3. And this is definitely a right angle. This is a horizontal line. This is a vertical line. We can now use the Pythagorean theorem to figure out the absolute value of 3 minus 4i. The distance between this point and 0-- it's the hypotenuse of this right triangle. So we just use the Pythagorean theorem. This side squared, 3 squared, plus this side squared, plus 4 squared, is going to be equal to the absolute value of 3 minus 4i squared, the absolute value squared. So 3 squared plus 4 squared, that's 9 plus 16, which is 25. So you get 25 is equal to the absolute value of 3 minus 4i squared. And we know if you take the absolute value of something, this is just a distance. It's going to be positive. So we want to take the positive square root, the principal square root, of both sides of this. And so we're going to be left with-- well, the principal square root, the positive square root of 25, is 5, is equal to the absolute value of 3 minus 4i. So another way of saying it, this thing right over here is going to be equal to 5. This distance right over here is equal to 5. Now, without having to draw it, one way you could just think about this is I'll have my real part. I have my imaginary part. I could literally take each of those parts, square them, take the sum, and take the square root. So another way of taking it, if you didn't want to visualize all this-- but this is really what we're doing. You could say, well, this is just going to be equal to take the real part squared. Take the imaginary part squared-- so let me write this. Add them together, and then take the square root, or the principal root. The principal root's just the positive square root. So that's going to be the square root of 9 plus 16, which, once again, is equal to 5.