Let z = 2 - 3i and w = 4 + i. What is the real part of z / w?
Show the answer and explanation
Correct answer
B. 5/17
Principle or equation
To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator: (a+bi)/(c+di) = ((a+bi)(c-di))/(c^2+d^2).
Why this answer is correct
z/w = (2 - 3i)/(4 + i). Multiply by conjugate (4 - i): numerator = (2 - 3i)(4 - i) = 8 - 2i - 12i + 3i^2 = 8 - 14i - 3 = 5 - 14i. Denominator = 4^2 + 1^2 = 17. So z/w = (5 - 14i)/17. Real part = 5/17.
Example
For (1 + i)/(1 - i), multiply by (1 + i) to get (1 + 2i + i^2)/(1+1) = (2i)/2 = i, real part 0.
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