You can
put this solution on YOUR website!I have been asked to simplify an expression containing complex numbers and I am not sure how to do it. Please help me!
-3i/5 + 4i Translation: -3i divided by 5 + 4i Thanks a lot. I am not doing
that well in Algebra Two right now.
COMBINE ALL TERMS WITH i ,TAKING i AS A COMMON FACTOR.
COMBINE TERMS NOT CONTAINING i IN THE USUAL MANNER.
THAT IS THE WAY TO GO ABOUT..
HERE WE HAVE ONLY TERMS CONTAINING i..SO...WE GET...
i(-3/5 +4)=i{(-3+4*5)/5}= i(-3+20)/5=i17/20......OR.......17i/20...AS IT IS USUALLY WRITTEN.
You can
put this solution on YOUR website!-3i/(5 + 4i)
=[(-3i)X(5-4i)]/[(5+4i)(5-4i)]
(mulitplying the nr and dr by the comlex conjugate of (5+4i)
=-3i(5-4i)/[5^2+4^2] [ (aib)(a-ib) = a^2+b^ ]
=-3(5i-4i^2)/(25+16) [(drawing the i in) meaning to say mulitplying every term inside the bracket with this i that was outside)]
=-3(5i+4)/41 ( using i^2 = -1)
= -3(4+5i)/41 or
Answer: -3i/(5 + 4i)= -3(4+5i)/41
Note: Caution! If you read this problem in a casual manner without proper focussing the numbers 4nd 5 will start dancing in front of your eyes!
While the given dr was (5+4i) a portion of the nr is (4+5i)
Note: Why should we multiply the nr and dr by (5-4i)
(5-4i) is the complex conjugate of (5+4i)
which when multiplied with (5+4i) makes it real.
So making the dr real is the idea.
Why should we make the dr real?
We need to simplify the given fraction (which is one complex number divided by another complex number) and simplification should lead to the final presentation in the form of a complex number (m+in) where m and n are real (of course m and n can be fractions and need not always be integers!) and hence the need to make the dr real.
Can we not expand (5+4i)(5-4i) directly?
You may and you definitely can
(5+4i)(5-4i)
= 5X(5-4i)+4i(5-4i)
=25-20i+20i-16i^2
=25-16i^2 ( since -20i+20i= 0)
=25+16 (since i^2=-1)
=41
Then why did we apply formula?
Our idea is to simplify the given quotient and if we indulge in side track activities like the above expansion there is a likelihood of forgetting what exactly our job is. One other thing is consumtion of time and extra labour.
Instead of this simple time tested direct expansion you may recognise a junior class formula namely (p+q)(p-q) = p^2-q^2 and hence
(a+ib)(a-ib) = (a)^2-(ib)^2 = a^2-(i^2b^2) = a^2-(-b^2) = a^2+b^2
Under the DO IT YOURSELF ACTIVITY you may practise multiplications:
1)(2+3i)(2-3i), 2)(11-i)(11+i) 3)(7+5i)(7-5i) etc and gain some speed in seeing, feeling and handling complex numbers in algebraic operations and every time you come across i^2 your putting its value (-1) will hasten your work elsewhere when you don't have to pause and think!
If you give yourself this mind gym activity then it will be very easy for you to remember that (aib)(a-ib) = a^2+b^2
Remark: There is no short cut to success. Methodical hard work combined with logical analysis and a positive attitude will make your mind develop the skill of becoming focussed and you will automatically eliminate the not so relevant or not so important details and concentrate on matters that matter. Lo! and behold!. Success comes gushing in!
NOTHING IS IMPOSSIBLE.
Do you feel confident now?
Don't you worry.
Every problem has a solution.
Every dark cloud has a silver lining.
This very algebra which is boggling you at the moment might as well
become your strong tool if you will and wish it to be. Why NOT?
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