What is the absolute value of: (1 + i) (6 + 2i)/ (4 + i) (9 + 3i)  

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to find the absolute value of (1 + i) (6 + 2i)/ (4 + i) (9 + 3i). First we convert it to a form x+iy and then use the formula sqrt(x^2 + y^2) to determine the absolute value.

(1 + i)(6 + 2i)/ (4 + i) (9 + 3i)

=> (1 + i)(6 + 2i)(4 - i)(9 - 3i)/(4 + i)(9 + 3i)(4 - i)(9 - 3i)

=> (1 + i) (6 + 2i) (4 - i) (9 - 3i)/ (4^2 + 1^2) (9^2 + 3^2)

=> (6 + 6i + 2i^2 + 2i) (36 – 12i – 9i + 3i^2)/ (16 +1) (81 + 9)

=> (4 + 8i) (33 – 21i)/17*90

=> [4*33 + 8*33i – 84i – 21*8*i^2] / 17*90

=> [300 + 180i]/17*90

=> [10/51 + 2i/17]

The absolute value is sqrt [(10/51) ^2 + (2/17) ^2]

=> 0.2286

The absolute value is 0.2286

sewbsquared's profile pic

sewbsquared | High School Teacher | (Level 1) eNoter

Posted on

I would simplify the fraction first, by multiplying the numerator and denominator separately.  Don't forget to use FOIL when multiplying 2 complex numbers!!

(1+i)(6+2i) = 6+2i+6i+2i^2 =6 + 8i -2 = 4 + 8i

(4+i)(9+3i) = 36+12i+9i+3i^2 = 36 + 21i -3 = 33 +21i

You can't reduce, there's no matching factors in numerator & denominator.  I'm going to factor, just so I don't have to use large numbers.  If you use a calculator, you may just use the numbers there...

4(1+2i)/3(11+7i)

I need to multiply the numerator and denominator by the conjugate of the denominator, so it goes to a whole number.  The conjugate of (11+7i) is (11-7i)

4(1+2i)(11-7i) = 4(11-7i+22i-14i^2) = 4(11+15i+14) = 4(25+15i)

3(11+7i)(11-7i) = 3(121-49i^2) = 3(121+49) = 3(170)

I'm going to factor my numbers again, so I can reduce the fraction:

4(5)(5+3i)/3(10)17    = 20(5+3i)/30(17)  = 2(5+3i)/3(17)

My simplified fraction would be (10 + 6i)/51.  Now we can apply the absolute value of a complex number:

sqrt[(100+36)/51] = sqrt (136/51).  It now depends on what your teacher accepts as an answer.  The real answer is irrational, and if your teacher accepts approximations to a certain decimal point, use your calculator and do it.

If your teacher accepts nothing less than an exact answer, you have to rationalize the denominator by multiplying numerator and denominator by sqrt 51.  Then the exact answer is:

(sqrt 6936)/51.

 

Good luck!

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