ACT Math: Master Complex Numbers with the FOIL Shortcut

Complex numbers are written as a+bi, where i=√(-1) and i²=-1. To add or subtract, combine real parts and imaginary parts separately: (3+2i)+(5-4i)=(3+5)+(2i-4i)=8-2i. To multiply, use FOIL and remember i²=-1: (3+2i)(5-4i)=3(5)+3(-4i)+2i(5)+2i(-4i)=15-12i+10i-8i². Since i²=-1, we have -8i²=-8(-1)=8. So (3+2i)(5-4i)=15-12i+10i+8=23-2i. The key insight is that i²=-1 always converts a product with i² back into a real number.

Practice the pattern: (a+bi)(c+di)=ac+adi+bci+bdi²=ac+adi+bci-bd=(ac-bd)+(ad+bc)i. Once this is automatic, you multiply complex numbers the same way as any binomial; the trick is just remembering to replace i² with -1 at the end.

Mistake 1: Forgetting that i²=-1 and leaving i² in your final answer instead of converting it. Mistake 2: Mixing up the real and imaginary parts of the final answer. Mistake 3: Forgetting that you cannot simplify √(-16) the same way as √16; you must write it as 4i. Mistake 4: Making sign errors when subtracting imaginary parts. Always pause after multiplication and check: Did I replace every i² with -1 before simplifying?

Double-check by plugging your final answer into a calculator (most scientific calculators have complex number modes). If your hand calculation matches the calculator, your method is sound.

Problem 1: (4+3i)+(2-5i). Problem 2: (6-2i)-(3+4i). Problem 3: (2+i)(3-i). Problem 4: (1+2i)². Problem 5: (5-3i)(-1+2i). Answers: 1) 6-2i. 2) 3-6i. 3) 6-2i+3i-i²=6+i+1=7+i. 4) 1+4i+4i²=1+4i-4=-3+4i. 5) -5+10i+3i-6i²=-5+13i+6=1+13i. For each, verify that you applied FOIL, replaced i² with -1, and combined like terms correctly.

If you missed any, redo them and identify which step broke. Most errors happen at the i² replacement step, so that is where you should focus your checking routine.

Complex number problems test your understanding of algebraic operations and the definition of i. These appear 0-1 times per test but are worth the preparation because once you own FOIL, the problem is straightforward. Unlike some advanced topics, complex number questions are never tricky; they are just applications of the method.

This week, solve one set of 5 complex number problems daily. Time yourself; each problem should take under two minutes. By test day, complex number operations will be automatic, and you will spend your mental energy on harder topics.