ACT Math · Complex numbers

What is |3 - 4i|?

What is ∣3−4i∣|3 - 4i|?

Answer and explanation

Answer: H: 5

The modulus of a+bia+bi is the distance from the origin in the complex plane: a2+b2\sqrt{a^2+b^2}. For 3−4i3-4i it is 32+(−4)2=9+16=25=5\sqrt{3^2+(-4)^2}=\sqrt{9+16}=\sqrt{25}=5, choice H. This is the familiar 3-4-5 right triangle. Choice J (77) adds the absolute values of the parts instead of using the Pythagorean theorem. Choice G (11) is 3−43-4 with the sign dropped, and choice F (−1-1) is 3−43-4. A modulus is a distance, so it can never be negative, which rules out choice F immediately.

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