Precalculus / Unit 7: Polar Coordinates and Complex Numbers
7.4 Complex Numbers and the Complex Plane
Notes
A complex number is $a + bi$ with $i^2 = -1$. Plot it at the point $(a, b)$.
- Modulus: $\displaystyle |a + bi| = \sqrt{a^2 + b^2}$. Conjugate: $a - bi$.
- Multiply with FOIL and replace $i^2$ by $-1$.
- Divide by multiplying top and bottom by the conjugate of the denominator.
- Powers of $i$ repeat every four: $i, -1, -i, 1$.
Practice
Compute $(3 + 2i)(1 - i)$.
$3 - 3i + 2i - 2i^2 = 3 - i + 2 = 5 - i$.
What is $|-5 + 12i|$?
$\displaystyle \sqrt{25 + 144} = 13$.
Write $\displaystyle \frac{2 + 3i}{1 - i}$ in the form $a + bi$.
Multiply by $\displaystyle \frac{1 + i}{1 + i}$: $\displaystyle \frac{2 + 2i + 3i + 3i^2}{2} = \frac{-1 + 5i}{2}$.
Ready for more? Take the Unit 7 test.