Tuesday, 12 August, 2008

Conjugate And Its Applications

There is a special term in algebra called "Conjugate". Its power is not to be undermined.

What is this Conjugate?

Let's me start with an example.

A maths equation can be expressed as Z = a + b.

The conjugate to this Z is "a - b". Let us express this conjugate as conj(Z).

Look closely, we can see that the conjugate is just negating on of the term in Z, that is, the "b" term.

Note:

  • We can also negate the "a" term but keeping the "b" term untouched.

  • The rule is that only one term is reversed in sign.

  • The term to be negated is selected on a case-by-case condition to suit the application and objective.


Another example:
S = A + C + F
Conj(S) = (A + C ) - F
Similarly it can be, Conj(S) = A - (C + F)
Again to emphasize, to which conj(S) to use is entirely on its objective of usage.

More Example:
Y= E - G
Conj(Y) = E + G
It can be also Conj(Y) = -E - G.

The message is that we can select one term (or a group of terms) to be reversed in sign.
This is conjugating.

Its Application

After knowing what is "Conjugate", let us look at how to apply it.

Let Z = a + b.

If we multiply Z with its conjugate, that is, (Z) x Conj(Z), we will get (a + b) (a - b).

And (a + b)(a - b) = a2 - b2, which is a special product in algebra.

The usefulness of this application is more obvious in Complex Number computation.

Let H = a + ib

The conjugate is conj(H) = a - ib.

Their product is (H) x conj(H) = (a + ib) (a - ib) ==> a2 - (ib)2

Rewriting the above, we get a2 - i2 b2.

Note from complex number understanding, i2 = -1.

Therefore a2 - i2 b2 = a2 - (-1) b2 = a2 + b2 .

What happened to the result of the product?

We can see that the application of conjugate multiplication results in converting a Complex Number (a + ib) into a real number (without any "i") !

This knowledge when applied properly is very powerful in the Division operation of Complex Number.

As a rule on Conjugate Multiplication of Complex Number:
Z x Conj(Z) = a2 + b2


However, care has to be taken when applying Conjugate.

Common Mistake in Conjugate

To perform a conjugate, we only negate one term (or one group of terms).
Mistake 1: Z = a + ib ==> Conj(Z) = -a - ib (wrongly done!) ==> Both terms negated !

Complex Number Conjugate Multiplication results in a2 + b2.
Mistake 2: Z x Conj(Z) = a2 - b2 ==> wrong sign for the second term !
This was confused with the normal algebra special product (a + b)(a - b) = a2 - b2.

Why such mistake?
It is due to the "i" symbol in Complex Number that actually causes the sign to be negated.
(i2 = -1). The only solution to this mistake is a full awareness of the "i" symbol and understanding what you are doing.

Just a littel "i" can cause so much trouble, right?
On the other hand, what so difficult about handling this "i"? It is just a little "i", that's all.
You have a choice of which thinking and approach you want.
A wise man however will go for a positive one.

:-)

3 comments:

Anonymous said...

Wow that has made it a LOT clearer!! Why can't teachers teach like this =S... They make everything so confusing! D=

Anonymous said...

Great help!

EeHai said...

Nice to knw that this post helps and reduces the confusion for my readers.