ACT Math: Solve Complex Numbers and Imaginary Units Systematically

A complex number has form a+bi, where a is real and bi is imaginary (i=√-1). Key rule: i^2=-1. To add/subtract complex numbers, combine real parts and imaginary parts separately. (3+2i)+(1+4i)=(3+1)+(2i+4i)=4+6i. To multiply, use FOIL and simplify using i^2=-1. (2+3i)(1+i)=2(1)+2(i)+3i(1)+3i(i)=2+2i+3i+3i^2=2+5i+3(-1)=2+5i-3=-1+5i. The key step is remembering i^2=-1; students who forget this make careless errors throughout.

More complex example: (3+2i)^2=(3+2i)(3+2i)=9+6i+6i+4i^2=9+12i+4(-1)=9+12i-4=5+12i. Division requires multiplying by the conjugate. To divide (2+3i)/(1+i), multiply numerator and denominator by the conjugate of denominator (1-i): [(2+3i)(1-i)] / [(1+i)(1-i)]. Numerator: 2+(-2i)+3i+(-3i^2)=2+i-3(-1)=2+i+3=5+i. Denominator: 1-i^2=1-(-1)=2. Result: (5+i)/2=5/2+i/2.

Mistake 1: Forgetting that i^2=-1. If you treat i as a variable, errors cascade. Mistake 2: Mixing up real and imaginary parts. The real part of 3+2i is 3, not 2. Mistake 3: Incorrect conjugate. The conjugate of a+bi is a-bi (sign change only on imaginary part). Mistake 4: Arithmetic errors in FOIL combined with i substitution. Double-check your middle terms when multiplying. After every complex number calculation, verify your result by checking that i^2 was replaced with -1 and simplified correctly.

Self-check: If your final answer has i^2 still in it, you forgot to simplify. If you have terms like 2i^3 or 5i^4, simplify: i^3=i^2⋅i=-1⋅i=-i, i^4=(i^2)^2=(-1)^2=1.

Problem 1: (2+3i)+(4+2i). Problem 2: (5+i)-(2+3i). Problem 3: (2+i)(3+2i). Problem 4: (3-2i)^2. Problem 5: (1+i)^2. Problem 6: (4+2i)/(1+i). Problem 7: i^3. Problem 8: (2+3i)(2-3i). For each, show FOIL or division steps, replace i^2 with -1, and simplify completely. Do this twice this week; complex number fluency develops quickly with repetition.

Answers: 1) 6+5i. 2) 3-2i. 3) 6+4i+3i+2i^2=6+7i-2=4+7i. 4) 9-12i+4i^2=9-12i-4=5-12i. 5) 1+2i+i^2=1+2i-1=2i. 6) Conjugate: (4+2i)(1-i)/[(1+i)(1-i)]=[(4-4i+2i-2i^2)]/2=[(4-2i+2)]/2=(6-2i)/2=3-i. 7) i^3=i^2⋅i=-i. 8) (2+3i)(2-3i)=4-9i^2=4-9(-1)=4+9=13 (real number; conjugates multiply to give real).

Complex number questions appear on about 40% of ACT Math sections and are often in the harder question range. Students who master them gain a significant edge. Spending one hour drilling complex number operations adds 1 point to your score by making these otherwise tricky problems routine.

Commit to the eight-problem drill twice this week. By test day, complex number arithmetic will feel as natural as regular multiplication and division.