Welcome to the presentation on i and imaginary numbers.
So let me just start out with a definition.
i is equal to the square root of negative 1.
Or to view it another way, you could say that i squared
is equal to negative 1.
Now why is this special?
Well, we knew or we've learned that any number when you
square it is equal to a positive number, right?
If I had negative 1 times negative 1, that
equals positive 1.
You don't have to write the positive every time but that
equal positive 1 and so does 1 times 1, that
equals positive 1.
So if we think about the square root of a number, so far all
we've learned is taking the square roots of positive
numbers, and that makes sense to us because the notion of a
square root of a negative number didn't really
exist until now.
So what we've done is we've set this definition that the number
i, and i isn't a variable, it's an actual number.
It's value is equal to the square root of negative 1.
Now I won't go into all of the philosophical musings about
whether i, as a number i, or any imaginary numbers
actually exist.
Maybe I'll make another presentation on that.
But they exist enough to be very useful to many
engineers and physicists.
So I'll leave you with that, and I'll also just hint at --
well, I won't go into the whole e to the i pi
equals negative 1.
But that blows my mind, but I won't go into that.
And when you think about whether i really exists, you
should also think about whether anything really exists.
So, I've diversed for too long, so let's get back to what I was
saying before. i is equal to the square root of
negative 1, and i squared equals negative 1.
So let's think about the implications of this.
If I were to say -- well, i to the first power, just like
anything else, is equal to itself, right?
i squared, I've already said using this definition, i
squared is equal to negative 1.
i to the third power, well that would just equal i
squared times i, right?
And i squared is negative 1, so it would be negative 1 times i.
And that just equals negative i.
And i to the fourth would equal i to the third times i.
I'm just using my exponent rules here.
Well i to the third is negative i times i, right? i to the
third is negative i, and then we just kept that i.
Well what's negative i times i?
Well that's the same thing as negative 1 times
i times i, right?
And what's i times i?
Well, the definition.
i times i, i squared, is equal to negative 1.
So that equals negative 1 times negative 1, which equals 1.
Interesting.
Let me clean this up a little bit.
Actually let me start with i to the zero power.
i to the zero power, well we know anything to the zero
power is equal to 1.
So we'll keep that -- that still equals 1.
i to the first power is equal to i.
i squared by definition is equal to negative 1.
i to the third power, I just showed you, is equal to
negative i, and that makes sense because that's
just i squared times i.
And i to the fourth power equaled 1 again.
If I did i to the fifth power, well that's just equal to i to
the fourth power times i, right?
I'll write that down, i to the fourth times i.
i to the fourth is equal to 1, right?
This is equal to 1, so 1 times i is equal to i.
Do you see a pattern here?
i to the zero power equals 1.
i to the negative 1 power equals i.
i squared is equal to negative 1.
i to the third power equals negative i.
i the fourth equals 1 again.
So i to the fourth equals i to zero, and i to the one power
is equal to i to the fifth.
I think you'll find out that i to the zero, and you could try
this out if you don't believe me, i to the zero is equal to i
to the fourth, which equals i to the eighth, which equals i
to the twelfth, I think you see the pattern.
Any multiple of 4 equals 1.
And i to the first equals i to the fifth, equals i to the
ninth equals i to the thirteenth, equals i.
So that's i to any power that is a multiple
of 4 plus 1, right?
Because 5 is equal to 4 plus 1, nine is equal to 8 plus 1.
And we could do a similar pattern. i squared is equal to
i to the sixth, equals i to the tenth, and so on, and
that equals negative 1.
And finally i to the third is equal to i to the seventh,
which equals i to the eleventh, and so on, equals negative i.
So why is this useful?
We see a pattern.
It's a cycle of four.
In this pattern, if we look at this, we can use
this to determine what i to any power is.
So if I were to ask you what i to the hundredth power is,
well, you could just work it out, you could say, well that's
just equal to i times i to the ninety-ninth and
so far so down.
But if we use the cycle, we see that hundred is
a multiple of 4, right?
4 times 25 is 100.
So i to the hundredth will fall into this category,
this first one.
It's a multiple of 4.
So we know that i to the hundredth power is equal to 1.
Similarly, if I said i to the hundred and one power, that's
going to equal i, right?
Because that equals 100 plus 1.
So it puts you into this category.
Equal to a multiple of 4, 100 is a multiple of 4, and 101
is a multiple of 4 plus 1.
i to the hundred and second power, similarly would
equal negative 1.
i to the hundred and third power would equal negative i.
I hope you understand what I'm doing here, and all I did is I
defined i as the square root of negative 1, and then I
kept multiplying i to figure out a pattern.
I said i to the zero equals 1, i to the first equals i, i
squared equals negative 1, i to the third equals negative i,
and i to the fourth equals 1 again.
And the pattern repeated itself.
And then I used that pattern to be able to figure out i to any
power, even it's a very high number.
So a very simple way to think about it is if I had i to the
three hundred and twenty three.
What I do is I say if I were to divide 4 into 323,
what's the remainder?
Well, I know 4 goes into 320, right?
4 times 80 is 320.
So I know that when I divide 4 into 323, so 4 goes into 323 80
times with a remainder of 3, right?
And the remainder is what we care about.
And this number is actually called a modulus.
Maybe I'll do another module on modulus -- it's very important
actually in computer programming.
But since we know that when you divide this exponent by 4 the
remainder is 3, we can say that this is the same thing as i to
the third, which we've learned is negative i.
Similarly, if I said i to the five hundred and second
power, well I know 500 is divisible by 4, right?
4 times 125 is 500.
So the remainder is 2 if I were divide it by 4.
So I could say that this is the same thing as i squared.
And i squared we learned by definition is negative 1.
If I were to ask you i to the thirty-seven?
We know 36 was divisible by 4, so the remainder is 1.
So it would be i to the 1, which equals i.
Hopefully that gives you an indication of what i is.
It might have been confusing the first time because we're
dealing with a number that's "imaginary," and I'm teaching
the cycle property of it.
What you might want to do is review the video again, but
then after that you could just try the module on i, which
essentially just keeps working you through this
type of problem.
I hope you have fun.
Bye.