iNumbers

Learning Focus

Add, subtract, multiply, and simplify complex numbers $a+bi$.

Big idea

• $i^2=-1$; powers cycle: $i,\,-1,\,-i,\,1$ with period 4.
• Add/subtract real and imaginary parts separately.
• Multiply by FOIL, then replace $i^2$ with $-1$.
• Conjugate of $a+bi$ is $a-bi$, and $(a+bi)(a-bi)=a^2+b^2.$

Worked example

$(2+i)(3-i)=6-2i+3i-i^2=6+i+1=7+i.$

Takeaways

• Treat $i$ like a variable, then use $i^2=-1$.
• Conjugate products are real.
• For $i^n$, reduce $n$ mod 4.