ACT Math · Complex numbers

What is (6+2i)/(i)?

What is 6+2ii\dfrac{6+2i}{i}?

Answer and explanation

Answer: H: 2 - 6i

Multiply the numerator and denominator by ii: (6+2i)ii⋅i=6i+2i2i2=6i−2−1=2−6i\frac{(6+2i)i}{i\cdot i}=\frac{6i+2i^2}{i^2}=\frac{6i-2}{-1}=2-6i, choice H. Equivalently, since 1i=−i\frac{1}{i}=-i, the quotient is (6+2i)(−i)=−6i−2i2=−6i+2=2−6i(6+2i)(-i)=-6i-2i^2=-6i+2=2-6i. Choice G (2+6i2+6i) loses the sign of the imaginary part. Choice J (−2+6i-2+6i) has both signs reversed, which happens if you forget to divide by −1-1. Choice F (6+2i6+2i) is the original numerator and was not divided at all. Check by multiplying the answer by ii: (2−6i)i=2i+6=6+2i(2-6i)i=2i+6=6+2i. Dividing by ii is the same as multiplying by −i-i.

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