In this unit, went to look at how to multiply 2 complex
numbers together. And multiplying two complex numbers
simply requires us to be able to multiply out brackets to collect
together like turns, and to remember that I is the special
number whose property is that I squared is equal to minus one.
So let's look at how that all works in an example.
So here we have two complex numbers 4 + 7 I.
And 2 + 3 I and we're going to do is going to multiply these
two complex numbers together.
And the first thing we do is we just multiply out the brackets
so we each term in the first bracket must multiply each term
in the SEC bracket.
So we have 4 * 2 which is 8.
Four times plus three I, which is plus 12 I.
Plus Seven I times two is
plus 49. And plus Seven I times plus three. I is
21. I squared.
Now we see straight away that we would be able to combine these
two terms because they're both terms with eyes in them.
So the the front is going to stay the same plus 12 I plus
14. I gives us plus 26 I.
Now over this term, on the end, which is 21, I squared. Now we
have to remember what we know about I. I is the square root of
minus one or said another way I squared is equal to minus one.
So in this term 21 I squared we can replace the isquared by
minus one. So that's plus 21 times minus one.
Now 21 times minus one is minus 21, so we have 8 - 21. You can
combine those two terms.
8 - 21 is
minus 13. Plus the 26 I was already there.
And so our answer, when we multiply these two complex
numbers together, is this new complex number minus 13 + 26? I
OK? We're going to look at another example, two different
complex numbers. This time the complex numbers minus 2 + 5. I
first one on 1 - 3 I is the second one, exactly the same
principles before, but we have to be a bit more careful 'cause
we got lots of minus signs
floating about. So we multiply out the brackets minus 2 * 1.
Is minus 2.
Minus two times minus three I gives us plus 6I.
Plus 5I Times one gives us
plus 5I. And plus 5I Times minus three I years, minus
15 I squared.
Now we combine together our items so we have minus 2 + 6
I plus 5I, giving us plus 11 I.
And in this term, remember that I squared is minus one, so this
is minus 15 times minus one, which is plus 15.
And so the final thing we do is combine the minus two and the
plus 15 to get plus 13 and then plus 11 I.
And so our answer, we multiply these two complex numbers
together is the complex number 13 + 11 I.
So that's how we multiply together to complex numbers
in the next unit. We're going to look at a property that
complex numbers have called the complex conjugate.