Imaginary Numbers and Complex Operations: Mastering i and Basic Arithmetic

The imaginary unit i is defined by i^2=-1. All imaginary numbers are multiples of i, like 3i or 7i. Powers of i cycle every four: i^1=i, i^2=-1, i^3=-i, i^4=1, then i^5=i again. This cycle is the key to simplifying high powers of i quickly. To simplify i^47, divide 47 by 4: 47÷4=11 remainder 3, so i^47=i^3=-i. Learn this pattern immediately; it appears on nearly every SAT with complex numbers. Practice: what is i^100? 100÷4=25 remainder 0, so i^100=i^0=1. What is i^23? 23÷4=5 remainder 3, so i^23=i^3=-i. Master this shortcut first.

Complex numbers combine real and imaginary parts: 3+2i means 3 (real) plus 2i (imaginary). The SAT treats the real and imaginary parts separately in operations, so you add reals to reals and imaginary parts to imaginary parts, just like combining like terms.

Adding complex numbers works like combining like terms: (3+2i)+(4+5i)=(3+4)+(2i+5i)=7+7i. Combine real parts together and imaginary parts together; never mix them. Subtraction works the same way, remembering to distribute negative signs: (3+2i)-(4+5i)=(3-4)+(2i-5i)=-1-3i. This is straightforward algebra; the only difference is the presence of i. Most errors come from careless sign mistakes, not conceptual misunderstanding.

Build a three-step routine for every complex addition or subtraction: (1) group real parts, (2) group imaginary parts, (3) combine. Write out the grouping explicitly; do not try to do this in your head. "(5-3i)+(2+6i)=(5+2)+(-3i+6i)=7+3i." This clarity prevents sign errors.

Multiplying complex numbers uses FOIL just like binomial multiplication, with one extra step: simplify i^2 to -1. (2+3i)(4+5i)=2·4+2·5i+3i·4+3i·5i=8+10i+12i+15i^2=8+22i+15(-1)=8+22i-15=-7+22i. The key is remembering that i^2=-1, not leaving it as i^2 in your final answer. Many students forget this simplification step and lose points. After multiplying, always scan for i^2 terms and replace them with -1, then combine like terms.

Practice FOIL with five complex-number pairs daily, drilling the pattern: (a+bi)(c+di)=ac+adi+bci+bdi^2=ac+adi+bci-bd=(ac-bd)+(ad+bc)i. Know this formula by heart; it makes multiplication instantaneous.

Complex conjugates are pairs like (3+2i) and (3-2i). When you multiply a complex number by its conjugate, the imaginary parts cancel: (3+2i)(3-2i)=9-6i+6i-4i^2=9-4(-1)=9+4=13. The result is always a real number. The SAT uses this to divide complex numbers. To divide by a complex number, multiply both numerator and denominator by the conjugate. This converts the denominator to a real number, making division possible. For example, to compute (2+i)/(1+i), multiply by (1-i)/(1-i): [(2+i)(1-i)]/[(1+i)(1-i)]=[2-2i+i-i^2]/[1-i^2]=[2-i+1]/[1+1]=[3-i]/2=(3/2)-(1/2)i.

Master this conjugate technique: whenever you need to divide by a complex number, (1) identify the conjugate (flip the sign of i), (2) multiply numerator and denominator by it, (3) simplify the numerator (FOIL), (4) simplify the denominator (which becomes a real number), (5) separate real and imaginary parts. Practice five division problems daily until this becomes automatic.