Subtracting complex numbers

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So we've plotted two complex numbers right over here on this

argand diagram. And what I think about is, what if we define a third

complex number c as being equal to the complex

number a minus the complex number b.

What is c going to look like if we were to write it as

just a sum of its real and imaginary parts? And what would it look like

on this argand diagram? So let's just first think about

this. So c is (let me do that in yellow colour) equal to

a which is the same thing as six plus two i minus b.

Now what is b?

b is two minus i.

So when we're subtracting anything, we can distribute this negative sign

and so this would be the same thing. This would be equivalent to

six plus two i

minus two and then

distributing the negative on a negative i, that's going to be a positive i,

plus i just like that. And now this is going to be equal to

(I'll do it right over here), this is going to be equal to, we can add

the real parts right over here. We have six minus two and so that's going to be four.

And then we can add, then we can add the two i

to the i, which is going to be three i

plus four plus three i,

and that is the complex number c.

Now let's see why that actually makes sense when we view,

when we visualise a and b as vectors.

So this is b, but we're essentially, when we do a minus b, that's essentially the same thing

as a plus negative b.

So what's negative b going to be equal to?

Negative b is going to be equal to

negative two plus i. I just multiplied all

the terms by negative one. So what does that look like? Let's see, negative two

plus i gets us right over there.

And we see, we've essentially just flipped

this vector over the origin. So this right over here

is negative b, that is negative b.

And we've already seen, if you're adding two things, we're going to add a to negative b.

So we've plotted a, so that its tail is at its origin

its head is right over here. And then we could take the tale of

negative b, put it right over here and then if you just shift it over,

it goes two to the left and one up.

It gets to that point right over here so that

this vector is another way to visualise

negative b as a vector. So where our new

head is once we put the tail of negative b here

is going to be the head of our vector c.

So c is going to be this vector right over here,

c is going to be this vector right over here if we visualize

this vector right over here. And we see that is

one, two, three, four. That is four plus

one, two, three i. We see that right over there.

So on one level it's kind of straight forward, manipulating expressions

and just recognizing that you can only add the imaginary parts to the imaginary parts

and the real parts to the real parts. And if you want to visualize it on an argand diagram

it really is analogous to... Subtracting b from a

is equivalent to adding the negative of b to a.