I want to prepare you for something crazy that might happen when you're
dealing with square roots or any kind of roots.
Consider this equation. Let's try and solve it.
Well,
I guess the first thing we do is divide both sides by 5 to try and isolate that
x squared. Wouldn't we order to isolate the X itself
That's not too bad and we get x squared equals negative 1.
Ok, straightforward. Now, how do we get rid of that two? We use the rooter. Remember ?
Because if I do it on one side I have to do it on the other.
Well, on the left side, the square root of x squared is X; on the right side
the square root of negative 1...
Uhoh, obviously this can happen but we can't have the square root of a negative
number. So now what are we going to do?
Well, as you can see this can happen.
So, so that we can at least discuss the answers to such an equation we're going
to introduce something different.
We're going to introduce, i.
Nice to meet you, i.
i is the square root of negative one. We call him i because
he's imaginary. Of course we know the square root of a negative number doesn't
exist.
Ok, well i is the square root of negative one.
Now consider that up till now we've been dealing with numbers that we want to say
are real. They actually exist if we could write them down.
We call them rationals, didn't we? Fractions we could write down and
repeating decimals we could represent and we could write them down so we call
those rationals. Ok.
Some we couldn't actually write down but they were real.
They were irrational real numbers. For instance, the square root of two. Recall
we couldn't really write it down.
It just went on and on and there was no pattern.
pi, for instance. It actually goes on forever and ever and ever.
So we call that irrational but both rational and irrational are real
numbers; they actually exist.
Okay.
Imaginary numbers do not exist but we're going to discuss them.
So imaginary numbers involve, of course, what we just told you, i. They're not real.
Ok. Which means of course that they don't actually exist but they seem to crop up
in these equations as we saw at the beginning.
So any number of the form a plus bi,
what I mean is, 3 plus 5i
or 3-1i or three - i, (of course that's the same as 3-1i)
is gonna have to be part real
that's the a part 1 and part imaginary and anything that's part real and part
imaginary we're going to call complex. That's a complex number.
Now if you're truly dealing with complex numbers,
consider the exponent of i will have to be one because i is the square root of
negative one so if we square it, in other words if the exponent isn't one...
The basic definition of square roots, if we square a square root what will we get?
i squared will be negative 1 and then we won't have a complex number anymore, will
we? [Just when I think you said the stupidest thing ever you keep talking!]
Consider all the other powers of i.
Okay, of course if there's a one we don't have to pay any attention.
i squared would be negative 1 as we just showed you.
I to the third. Let's break it up.
I to the third would be an i squared times i. Well, i squared is negative 1
and i is on the first is just i. So i to the third would actually equal
a negative one i.
And that would of course make it complex because that the exponent would
turn back into a 1, wouldn't it?
i to the fourth is i squared times i squared. Now i to the fourth is very
special because i squared is negative 1 and each occurrence of that if you have
a negative 1 times negative 1 I'm going to get a positive one so anytime
you have i to the fourth you're going to end up with a positive one and usually
you, of course, you can throw out a one when you're multiplying.
Okay, so keep an eye out for these and we'll simplify them to the numbers that
they are equal to. Let's mix these facts with the rules of exponents and I want
you to simplify this expression.
i to the hundred and two minus i to the 49th. Holy mackerel!
Well, [Are you pondering what I'm pondering? ] I always say look for fours
because remember what i to the fourth is. It's one so anytime you have a night of
the 4th or a factor of i to the fourth you can throw it out.
Well, i to the 100 and two is really i to the hundred times I to the two,
isn't it?
Because we add exponents and i to the hundredth is a multiple of four. It's
i to the fourth to the 25th. Because you know your rules of exponents, with a
parentheses there you multiply, so this portion of it is just 1 and 1 to any
power is 1 so now you've broken it down to just really
i to the second it is and that's minus 1.
Okay, now let's try and get this one down. Once again,
let's look for fours. What do you smell?
i to the 48th. Well that's going to become a one because it's i to the
fourth to the 12 and i to the fourth
is your buddy.
Remember we're looking for i to the 4th and, and 1 to any power is just 1.
Okay, so this, all this simplified, using your rules of exponents it's a good
exercise. Let's do -1-i.
Let me give you another. Let's simplify this:
i to the 72. Looking for i to the fourths and it goes evenly in doesn't
it?
So that portion of it, i to the fourth to the 18th because four times 18 is 72 is
one to the 18th which is what?
Just one and this portion of it...
Well, let's take the biggest multiple of four out and we'll get,
ok, i to the fourth to the six.
Once again 1 to any power, do you recall what i squared is? Its negative 1. So all
this becomes, what all this becomes, ok and that's the answer.
he
It's a good exercise in the use of i and the rules of exponents.
Now unless you square it, i can be treated like any variable. For instance,
in this problem all we're going to do is add like terms 3 and negative 2
become one and negative 4i and negative 3i added become negative 7i. instance we
don't have an i squared or I should say since we do have an i to the 1th power
we're all done. Ok, so just treat it like any other variable unless you get a
power higher than one on your i.
Good use of review of dealing with variables.
Remember when we try to subtract things, we end up changing the sign of the whole
second expression so I change that to addition and change the signs of
everybody in the second group and now it just becomes adding like terms again.
Three and two
become 5 and minus 4i and positive 3i
become a negative 1i and of course we only write the negative i.
Ok, I really think most of the reason we cover complex numbers is to review all
the other rules. For instance, in this one, we're going to review distribution. 5i times
4 is just 20i.
5i times what I would say is negative 7i becomes negative 35i squared. Now, wait
a minute,
i squared is negative 1. Ok, we have to always have i to the first power so
we'll simplify that and we'll get negative 35 times negative 1 or ...
And now we have our answer.
Multiplication is sometimes going to lead us to these i Squared's. Now this
one is kind of a FOIL problem.
Let's take the two and distribute him to both of the second expressions. 6i and
4i and now we're done with the two and then we'll take the 2i
and multiply him or distribute him to the two in the second expression. That's
going to be six i
I like to write it underneath his buddy because they're getting together
later and 4i squared.
Now, wait a minute, he adds, he just comes dow;n the four and 6i become 10i
but here we have an i squared.
Yeah.
Remember i isn't negative 1; i squared is negative one so we'll have a negative
foru there and then he can get together with the six and we'll have a final
answer.
Okay?
Do you remember when we simplified radicals?
Well, we broke them into the perfect squares
and then what was left over,
the garbage...In this case
the largest perfect square is nine.
That leaves me with a 2 and this radical simplified becomes 3
radical 2.
What are we gonna do here? We've got an imaginary number because we've got the
square root of a negative number.
Well, pretty much the same thing.
We're going to use our perfect squares but this time we're going to throw in
somebody else.
Ok? You see there's a new one in the middle. We've got the perfect squares and
the new one is the imaginary part. We still have the leftovers and they're
still last as appropriate.
In this case the perfect square is nine. The imaginary, imaginary part, of course
is the square root of minus one and the garbage is the same so the answer is
three i square root of two. Pretty much the same thing except
that you take out the negative 1 or the square root of negative one and you get i.
We'll do one more.
The perfect part, the imaginary part
and the garbage. Now what are they?
Yeah.
The perfect part of the island
the imaginary part,
and the leftovers. Think about it.
The perfect part 25, the imaginary part of the island
it's always the same and the leftover.
Now it's always three steps.
Five, i, radical 3.
It's pretty much,
pretty much, the same thing isn't it?
Except we have that imaginary part that we have to take out because it's a
negative square root.
Ok, it's time to go do your homework and please do it right away.
Don't wait to the last minute!