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Sal: Let's now think about what it means to add and subtract complex numbers. Let's say I have a complex number A and let's say that it's equal to three plus two I. And let's say I have complex number B that is equal to negative one minus three I. Let's think about what it means to add A and B. Let's say I have a third complex number that is C, that is defined as A plus B. I encourage you to pause this video and think about it on your own what A plus B is going to be. What we can do is, this is going to be equal to three plus two I, that's A. Three plus two I. Plus B, which is plus negative one minus three I. We could write it like that if we want to really be clear what B is and what A is. How would we actually add these two together? We can't add a real part to a complex part, that's why this is- Or we can't add a real part to an imaginary part I should say, that's why this is about as simplified as you can get. But I can add real part to real part and imaginary part to imaginary part. I can add the three to the negative one to get three plus negative one is two, so I get a real part now of two. Then I can add the two imaginary parts. If I have two I minus three I, that's going to be negative one I, or I could say that's going to be negative I. Just like that I added the two real parts, added the two imaginary parts and I got two minus I. We could also visualize this on the complex plane. You could do that using an Argand diagram. In an Argand diagram we represent each of these complex numbers as vectors on the complex plane. That's our imaginary axis and this is our real axis. A is three plus two I, so one, two, three along the real axis and then two along the imaginary axis. The imaginary part is two, two I. Three plus two I gets us right over there. I'm going to represent that as a vector, as a vector where its tail is at the origin and its head is at the coordinates three on the real axis, two on the imaginary axis. That right there on my Argand diagram is my representation of vector A. Now let's do the same thing for vector B. It is negative one along the real axis and negative three along the imaginary axis. One, two, three. This takes us right over there. If we represent it as a vector, our complex number B can be visualized this way. Now when we add A plus B you can add them the exact same way that you would add vectors. You take the tail of B, or you could, you essentially shift B over. You take the tail of B and you put it at the head of A, but it's the same thing. You go down three and you go to the left one. It's going to put your right over here. All I've done is shifted B over to this part so that its tail is at the head of A. It goes right over there. This is complex number B. This is B right over here. Notice, now if we start at the tail of A and go to the head of this shifted B this right over here is going to be vector C, this is going to be A plus B. This vector right over here is C. Notice C is two minus I. Two real part, negative one imaginary part. This right over here is C, which is A plus B. You were able to add these two complex numbers. You can visualize adding them the same way that you would visualize adding two vectors in the traditional coordinate plane. It makes complete sense because when you're adding these complex numbers you're keeping the real part separate and these are essentially the horizontal components of the vector, and you're keeping the imaginary part separate and that's essentially the vertical parts of these vectors.