Dividing Complex Impedances and Checking a Radical Claim
An electronics lab is reducing two complex impedances. The equivalent impedance is Z = (3 + 4i)/(1 − 2i) ohms, where i^2 = −1. One student rationalizes by multiplying numerator and denominator by (1 + 2i) but reports Z = 1 + 2i. Another student gets a real denominator but is unsure whether the numerator should be −5 + 10i or 11 + 10i, because of a sign error with i^2. A third student claims the result is real because 'the imaginary parts cancel in the denominator.' Separately, a homework problem asks for √(−4)·√(−9). A fourth student writes √(−4)·√(−9) = √36 = 6 and concludes the product is positive real. The instructor wants the group to compute Z in a + bi form, check its modulus, and settle the radical claim without a calculator.
1. Compute Z = (3 + 4i)/(1 − 2i) in a + bi form. Show the rationalization step and identify the source of the sign error.
Read the worked answer
Multiply numerator and denominator by the conjugate 1 + 2i: Z = (3 + 4i)(1 + 2i) / [(1 − 2i)(1 + 2i)]. The denominator is 1^2 − (2i)^2 = 1 − 4i^2 = 1 + 4 = 5. The numerator is 3 + 6i + 4i + 8i^2 = 3 + 10i − 8 = −5 + 10i. Therefore Z = (−5 + 10i)/5 = −1 + 2i. The student who got 1 + 2i made a sign error in the real part. The value 11 + 10i comes from incorrectly using i^2 = +1 in 8i^2, which would give 3 + 10i + 8 = 11 + 10i. The correct rule is i^2 = −1.
2. Find |Z| and verify that the modulus of the quotient equals the quotient of the moduli for these numbers.
Read the worked answer
For Z = −1 + 2i, |Z| = √[(−1)^2 + 2^2] = √(1 + 4) = √5. Separately, |3 + 4i| = √(9 + 16) = 5 and |1 − 2i| = √[1^2 + (−2)^2] = √5. The quotient of the moduli is 5/√5 = √5, which matches |Z|. This is expected because for complex numbers with a nonzero denominator, |w/z| = |w|/|z|. The denominator 1 − 2i is nonzero, so the property applies.
3. Evaluate √(−4)·√(−9) correctly and explain why the step √(−4)·√(−9) = √36 is invalid.
Read the worked answer
Write each principal square root in terms of i: √(−4) = 2i and √(−9) = 3i. Their product is (2i)(3i) = 6i^2 = −6, not 6. The rule √a·√b = √(ab) is valid when a and b are nonnegative real numbers. It does not extend to negative radicands because the principal square root of a negative number is imaginary. The computation √36 = 6 corresponds to √4·√9, not to √(−4)·√(−9). The correct product is negative real, −6.