BAM!!! Mr. Tarrou. How are you doing today? I am going to be talking about imaginary numbers.
Like they hardly even exist but yet we still need them in mathematics. Now imaginary numbers
really in a textbook are called Complex Numbers. They have two parts, they have a real part
and an imaginary part...or a plus b times i. Now what do I mean by imaginary. What does
that even mean, or come up in math? Maybe if you have started solving parabolas, or
doing the quadratic formula, you might have noticed that sometimes you don't get an answer.
You try and do the square root and you end up with a negative number inside your radical.
That is one example where imaginary numbers come in to play. The square root of negative
1 is equal to "i" Now there are some things in math that we teach as a rule that I can
do a pretty good job of teaching my students like why that works or how. This is one is
a little bit tough for that, so we are just going to let this letter "i" stand for the
square root of negative one. i for imaginary! Why is it imaginary? Because if you square
root negative one, or you try to come up with a square root, there are not two numbers that
you multiply by themselves to get you negative one. The square root of 4 is 2. Why? Because
2 times 2 is 4. But what two real numbers can you multiply together to get negative
one, they don't exist. Negative one times negative one is positive one. Maybe it is
the square root of negative one is positive one. Well one times one is still positive
one. You cannot take an even root of a negative number. So we must give it this imaginary
value if i. Well one fact that we are going to use quite a bit today is about i squared.
Now i squared. i squared is just as it says, i times i or the square root of negative one
squared. What happens when you square a square root, don't they cancel out? They do so those
are going to cancel out and you get that i squared is equal to negative one. So i is
the square root of negative one, but then i squared is negative one. This is how we
can use these imaginary numbers in math calculations because as we manipulate it more and more
sometimes we are able to cancel out that square root that we have and actually get a real
number back that we can use in a later calculation. Ok, i to the third is i times i squared. We
just said that i squared is what, i squared is negative one...so i to the third is negative
i. i to the forth is i squared times i squared which is negative one times negative one which
is positive one. Now I am giving you these four values or powers of i because in your
book you may be using those for some questions. Now I am not going to use the third and forth
powers for what I am about to run through for my students in PreCalculus...and this
also applies to an algebra 2 class. We are going to be focusing on i is the square root
of negative one and i squared then becomes negative one, not the square root but just
negative one. We are not doing this in my math class, but in case you are... if you
are trying to graph or draw where these imaginary numbers are on the x y axis. Well we don't
really have an x y axis if we are talking about imaginary numbers. No, we have a real
axis and an imaginary axis. Now you notice that I took out the x and put in real which
is the first number. I took out y and put in imaginary which is the coefficient of the
second term. So when you go to graph these, it is going to seem like are graphing a regular
point. 2 + 3i is over two and up three. -1 -4i is left and down, so negative one minus
four i is there. This is just 6i, where is the real part? Right there! No left right
movement, just go up six... 1,2,3,4,5,6 So there is a super quick example of how you
can graph these three complex values. Now let's get into what my kids are doing which
is just some basic algebraic manipulation that involve the value of i. Let me make sure
I have some time left. Ten minutes. Ok, let's run through them. I am going to go as quickly
as I can because I want to get a lot of examples done in the next ten minutes. We have got
3 plus 2i minus 4 plus 5i. I am just giving you some binomials, two complex numbers, and
asked you to basically simplify. You are going to do this just like you did your generic
little combining like terms and getting rid of parenthesis problems. Distribute to get
the negative through the parenthesis and look for like terms. Three and negative four is negative one. What about two and negative
five? Well two minus five is negative three. Real and imaginary. So this looks very similar
to basic algebra skills you have been using for a long time. Let's take a look at this
example. Negative three plus seven i squared. OHHHH how many times do I see kids mess this
up. This is a binomial squared. It is not negative three squared and seven squared.
no no no no... It is negative 3 plus 7i squared is negative 3 plus 7i times negative 3 plus
7i. Please take the time and save your teacher a lot of grief and actually distribute those
together so you have all the terms, you don't miss one. Negative 3 times negative 3 is 9.
Negative 3 and 7 is negative 21. And going to the next set of lines we have negative
21 again...i And then positive 49 i squared. Here we start, this is the only place where
there is a difference when you are working with complex numbers compared to just your
regular algebra. Instead of this being just like an x squared, it is i squared. We just
talked about...oops I erased it... We just talked about how i squared was equal to negative
one. So this i squared comes out and in its place goes negative one. Now we are going
to combine like terms. We have 9 and this is going to be negative 49, so negative 49
plus 9 is negative 40. The two like terms with i's, now they are not i squares so the
i's are going to stay there, negative 21 and negative 21 is negative 42 i. Now for proper
form I need to write this in the other way because the real should come before the imaginary.
So negative 40 minus 42 i. BAM!!! Alright. Moving on. Did I wake you up? Let's take a
look at this. The square root of negative 81 times the square root of negative 32. You
cannot work with radicals that have negative numbers inside of them. If you just multiply
these together and get a positive whatever, I can't do that in my head, you are going
to get a number that is much too big...excuse me, not too big, you will have a sign error.
Your answer...you will believe your answer should be positive but you will see in a minute
it is actually supposed to be negative. If you have negatives inside your square root
symbols, you need to pull those out as i's before you start. Now 81, the square root
of 81 is what? It is 9. And what is the square root of negative one? Because the 81 is negative.
The square root of negative one is i. Times.... Well what is going on over here? I am going
to break this up a bit more just to make sure you know what I am doing. I have got negative
one, so the square root of negative one. And 32 won't square root but there is a perfect
square in 32. I am going to reduce this. There is a 16 times 2. So the negative one I am
going to leave separate. Square root of 16 and the square root of 2. The square root
of negative one is i and the square root of 16 is 4. So we are going to have 9i times
4i times the square root of 2. 9 times 4 is 36 i squared and the square root of 2. Now
if I had left this the way it was and just multiplied, my answer would be positive 36
square root of 2. But if I take the i out, like I am supposed to, i squared is what?
Yeah it is. Negative one. So negative 36 square root of 2. BAM!!! Alright. Let's take a look
at another example. Let's take a look at 4 - 3i over 2-5i. Simplify this. Well it looks
pretty simple already. If this were an x instead of an i we would be done. But i is a square
root...right? i is the square root of negative one. You cannot leave a radical in the denominator
if you are writing your final answer. So this, though it does not look like it, needs to
be rationalized. We need to multiply by something that will get my square root of negative one,
my i, out of the denominator. You do that when you have binomials by multiplying by
what is called a conjugate. That means that you simply change the sign, the middle sign,
and multiply that to the top and bottom of the fraction. So we are going to get 4 times
2 is 8. 4 times 5 is positive 20i. Negative 3 and 2 is negative 6i. Negative 3 and 5 is
negative 15 i squared. All over 2 times 2 is 4. 2 times positive 5 is 10i. Negative
5 and 2 is negative 10i. You might not want to show this work because you know the middle
term is going to cancel out, but I am teaching so I have to. Negative 5 and positive 5 is
negative 25 i squared. So we have got the positive and negative 10 canceling out. Because
of space, I am not going to write a new line, but what is i squared? Negative 1. What is
i squared...negative one. So we have 8 and that is positive 15 right...so 8 plus 15 is
23. What do we have here? We have 20 minus 6 is 14 i. All over 4 minus...not minus...that
is plus, so 4 and positive 25 is 29. Now this is not done. Couples numbers have a real and
an imaginary part. I am simply going to split this apart and give each fraction it's own
denominator of 29. BAM!!! Got it done. I have 2.5 minutes. Let's see if we can try and solve
a quadratic formula...shall we...or quadratic equation. Last example. Let's see if we can
solve this equation. Don't forget when you solve an equation your are trying to find
out where that equation crosses the x axis. We are trying to find out when y equals 0,
where is this graph crossing the x axis. Well this is a parabola. There is only one squared
term. And it might factor, but I want to show you the use of the quadratic formula which
is negative b plus or minus the square root of b squared minus 4ac all over 2a. When you
have an equation solved for 0 and your exponents count down, this is a, this is b, and this
is c. So let's plug it in and see what happens. The opposite of b plus or minus the square
root of b squared minus 4 times a times c all over 2 times a. Now you might notice every
time I do substitution I use parenthesis. That will save you your butt when it comes
to not making sign errors later on. Especially if you are using your TI-83 and TI-84's and
any kind of scientific calculator that allows you to type in the syntax. You want to be
careful with your signs of course and make sure that your answers are right, parenthesis
will help that happen. So we have negative 12 plus or minus the square root of 144, 4
times 2 is 8 and 8 times 20 is negative 160. All of this over 2 times 2 which is 4. This
comes out to be negative 12 plus and minus the square root of negative 16 all over 4.
I am running out of space. Look I am trying to square root a negative 16. The square root
of 16, I can do it is 4. But the negative comes out to be imaginary so we get negative
twelve plus or minus 4i over 4.