Video Transcript
If 𝑎 plus 𝑏𝑖 is equal to
negative three minus five 𝑖 over negative three plus five 𝑖. Is it true that 𝑎 squared plus 𝑏
squared equals one?
We’ve been given an equation
involving complex numbers. Remember, a complex number is of
the form 𝑎 plus 𝑏𝑖. 𝑎 is known as the real component
of the complex number. Whereas 𝑏 is known as its
imaginary component. It’s quite clear that to be able to
work out whether 𝑎 squared plus 𝑏 squared is equal to one, we need to work out
what 𝑎 and 𝑏 are equal to. So what we’re going to do is we’re
going to simplify the quotient. We’re going to divide negative
three minus five 𝑖 by negative three plus five 𝑖.
So how do we divide complex
numbers? Well, to divide complex numbers, we
multiply both the numerator and denominator of the fraction by the complex conjugate
of the denominator. Sometimes denoted 𝑧 star or 𝑧
bar, the complex conjugate of the number 𝑎 plus 𝑏𝑖 is 𝑎 minus 𝑏𝑖. Essentially, you change the sign of
the imaginary part. The denominator of our fraction is
negative three plus five 𝑖. Changing its sign, we find its
complex conjugate to be equal to negative three minus five 𝑖. We’re therefore going to multiply
both the numerator and denominator of this fraction by negative three minus five
𝑖.
Let’s begin with the numerator. We’ll do this in the little space
on the right. Notice that this expression looks a
lot like the product of two binomials. And in fact, we do treat it a
little like one. We distribute the parentheses
either using the FOIL method or the grid method. Let’s use the FOIL method. F stands for first. We multiply the first term in the
first expression by the first term in the second. Negative three times negative three
is nine. O stands for outer. We multiply the outer term in each
expression. Negative three multiplied by
negative five is positive 15. So get positive 15𝑖. Then I stands for inner. We multiply the inner terms. And when we do, we get one more
15𝑖. Finally, L stands for last. We multiply the last two terms. A negative multiplied by a negative
is a positive. So we get 25𝑖 squared.
But remember, 𝑖 is the number
which is the solution to the equation 𝑥 squared equals negative one. We can say that 𝑖 squared equals
negative one. So the numerator becomes nine plus
15𝑖 plus 15𝑖 plus 25 times negative one. Well, that last term is of course
just negative 25. And so we collect like terms. And we see that the numerator is
equal to negative 16 plus 30𝑖.
So what about the denominator? We’ll use the same method to
distribute our parentheses. And it should become clear quite
quickly why we actually multiply the denominator by its complex conjugate. Remember, F stands for first. We get nine. We multiply the outer terms to get
plus 15𝑖. Then, the inner terms to get minus
15𝑖. And finally, the last terms to get
negative 25𝑖 squared. We can now see that 15𝑖 minus 15𝑖
is zero. And negative 25 times negative one
is positive 25. So we’re going to calculate nine
plus 25, which is 34. Okay, great! Our fraction becomes negative 16
plus 30𝑖 over 34.
By separating the real and
imaginary part of our fraction and dividing each bit by 34, we find 𝑎 plus 𝑏𝑖 is
equal to negative eight over 17 plus 15 over 17𝑖. 𝑎 is the real part of our complex
number. So it must be equal to negative
eight over 17. 𝑏 is the imaginary part. It’s the coefficient of 𝑖. So it’s 15 over 17. All that’s left is to substitute
these two numbers into 𝑎 squared plus 𝑏 squared and see if we actually get
one. That’s negative eight over 17
squared plus 15 over 17 squared. That gives us 64 over 289 plus 225
over 289, which is indeed equal to one.
So, yes, it is indeed true that 𝑎
squared plus 𝑏 squared equals one, in this case.
