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The "What?" -- and "So What?" -- of the Complex Conjugate

Date: 01/12/2011 at 07:37:02
From: Terri
Subject: Trying to define "mutual complex conjugates"

My book lists some "basic concepts for complex numbers." One of the
concepts states:

   "x + yi and x - yi are mutual complex conjugates."

Their example reads:

   "8 + 3i and 8 - 3i are mutual complex conjugates."

I see that the only difference between the two expressions is the positive
and negative signs, but I still don't know what a conjugate IS. Therefore,
I don't know how to actually use or apply this information.

I do understand what a complex number is. And this book shows examples of
mutual additive inverses, which I also understand.

I am doing this studying on my own to prepare for a math test, so I have
no teacher to ask; and the "math people" I personally know do not know
what it means, either.

Date: 01/12/2011 at 09:40:10
From: Doctor Rick
Subject: Re: Trying to define 

Hi, Terri.

I thought at first that your issue was with the word "mutual," since that
is the part of this term that I don't see much. However, since you
understand the concept of mutual additive inverses, I suppose you do
understand that this statement ...

   "x + yi and x - yi are mutual complex conjugates"
... means that x - yi is the complex conjugate of x + yi, and that x + yi
is the complex conjugate of x - yi. (The word "mutual" refers to something
that is true of each of two or more parties in respect to the other.)

The way to find the complex conjugate of a complex number is to negate the
imaginary part of the number. That's what the statement says: if you've
got a number of the form x + yi, then the complex conjugate of that number
is x - yi. (Left unstated: x and y are real numbers.) For instance, for
the number 8 + 3i, x is 8 and y is 3, so the complex conjugate is 8 - 3i.
I think you're saying that you get this, too.

Perhaps what you're missing is the "so what?" I'll show you one important
fact about complex conjugates that makes them useful enough to have a name.

Take the example: 8 + 3i and 8 - 3i. What is their product?

   (8 + 3i)(8 - 3i) = 8*8 + 8(-3i) + 8(3i) + (3i)(-3i)
                    = 64 - 24i + 24i + 9
                    = 73

The product of these two complex numbers is a real number! 

The same is true of any pair of mutual complex conjugates:

   (x + yi)(x - yi) = x^2 + y^2

One use of this fact is in dividing one complex number by another. For

   2 + 5i   (2 + 5i)(8 - 3i)   31 + 34i
   ------ = ---------------- = -------- = (31/73) + (34/73)i
   8 + 3i   (8 + 3i)(8 - 3i)      73

I multiplied the numerator and denominator by the complex conjugate of the
denominator; this made the denominator real, so I could distribute it
across the sum, writing the result as the sum of a real part and a
pure-imaginary part.

Does that help?

- Doctor Rick, The Math Forum 

Date: 01/12/2011 at 13:44:29
From: Terri
Subject: Thank you (Trying to define "mutual complex conjugates")

Dr. Rick,

Yes, that helped so much!

And you had it exactly right when you suggested that perhaps my underlying
question was "what's the point?" So not only did you help me with this
particular question, I now know how to ask this type of question with
other topics, as well! (In fact, I have another one I'm sending in about
absolute value....)

Thank you so much for taking time out of your day to do this. I greatly
appreciate it!

Associated Topics:
High School Imaginary/Complex Numbers

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