In this video I'm going to try to motivate the study of complex
numbers by explaining how we can find the square root of a
negative number. Before we do that, let's record some facts
about real numbers.
On the diagram here, I've drawn what we call a real number line.
And every real number has its place on this line. Now I've
marked the whole real numbers from minus nine up to plus nine,
so all the positive numbers to the right hand side. The
negative numbers are to the left hand side. Every real number has
its place on this line, so the integers, positive integers,
negative inches, integers are here. We could also put the
fractions on as well. So for example the real number minus
1/2 would lie somewhere in here.
Decimal numbers, like 3.5 would be somewhere in there.
And even numbers like pie with some, some place someone here as
well. So as pie is going to be in there somewhere. So the point
is that all real numbers have their place on this real number
line. Let's look at what happens when we square any real number.
Suppose we take the number 3 and we square it.
When we square 3, remember we're multiplying it by itself, so 3 *
3 the answer is 9.
What about if we take the number minus three and square that?
Again, when we square it, we multiplying the number by
itself, so it's minus 3 multiplied by minus three.
And here, if you recall that multiplying a negative number by
a negative number yields a positive result, the answer
minus three times minus three is plus 9 positive 9.
Now the point I'm trying to make is, whenever you Square a
number, be it a positive number or a negative number. The answer
is never negative.
In fact, unless the answer, unless the number we started
with zero, the answer is always going to be positive. You can't
get a negative answer by squaring a real number.
Now, over the years, mathematicians found this
shortcoming a problem and they decided that will try and work
around that by introducing a new number, and we're going to give
this new number at the symbol I, and I'm going to be a special
number that has this property that when you square it, the
answer is minus one.
So I is a special number such that the square of I I squared
is minus one. Now that clearly is a very special number because
I've just explained that when you square any positive or
negative real number, the answer can never be negative. So
clearly this number I can't be a
real number. What it is, it's an imaginary number. We say I is an
imaginary number. Now that might seem rather strange. When you
first meet, it's starting to deal with imaginary numbers, but
it turns out that when we progress a little further and we
do some calculations with this imaginary number, I lots of
problems in engineering and physics and applied mathematics
can be solved using this imaginary number I.
Now using I, we can formally write down the square root of
any negative number at all. So supposing we want to write down
an expression for the square root of minus nine square root
of a negative number.
What we do is we write the minus nine in the following way. We
write it as plus 9 multiplied by
minus one. And then we split this product as follows. We
split it as the square root of
9. Multiplied by the square root of minus one.
So the square root of 9. We do know the square root of 9 is 3
and the square root of minus one is going to be I because I
squared is minus one. So I is the square root of minus one.
The words now we can formally write down the square root of
minus nine is 3 times I.
Let's give you another example. Suppose we wanted the square
root. Of minus Seven, we do it in exactly the same way we split
minus 7 into 7 times minus one, and we write it as the square
root of 7 multiplied by the square root of minus one.
Now the square root of 7. We can't simplify, will just leave
that in this so called surd form square root of 7 and the square
root of minus one.
We know is I.
So the square root of minus Seven. We can write as the
square root of plus Seven times. I, so the introduction of this
imaginary number I allows us to formally write down an
expression for the square root of any negative number.
Now using this imaginary number I, we can do various algebraic
calculations, just as we would with normal algebra. Let me show
you a couple of examples. Supposing were asked to
calculate I cubed.
Or I cubed we can write as I squared multiplied by I.
We already know that I squared is minus one, so I squared
becomes minus 1 multiplied by I.
Which is just minus one times I or minus I so we can simplify
the expression I cubed in this way just to get the answer.
Minus I let's look at another one, supposing we have an
expression I to the four, we could write that as I squared
multiplied by I squared.
And in each case I squared is minus one. So here we have minus
1 multiplied by minus one and minus one times minus one is
plus one. So I to the four is plus one and in the same way we
can start to simplify any expression that involves I and
powers of I. You'll see this imaginary number I
in lots more calculations in the following videos.