Complex numbers
Adding, multiplying and dividing complex numbers, the modulus, quadratic equations with a negative discriminant, the polar form and De Moivre's formula.
How to solve it
A complex number $a + bi$ is a point in the plane, and $i$ is a number with $i^2 = -1$. You treat it like a letter in algebra, except that $i^2$ is replaced by $-1$ at once. Multiplying by a complex number rotates and stretches: the moduli multiply and the angles add.
Step by step
- Adding and multiplying: expand as usual and replace $i^2$ by $-1$. Powers of $i$ repeat every four: $i^4 = 1$.
- Dividing: multiply the top and the bottom by the conjugate of the denominator ($a + bi \to a - bi$). The denominator becomes the real number $a^2 + b^2$.
- A quadratic with $D < 0$: $\sqrt{D} = i\sqrt{|D|}$, and the roots are conjugates.
- Large powers: go to the polar form $r(\cos\varphi + i\sin\varphi)$ and use De Moivre's formula.
Common mistakes
- Forgetting that $i^2 = -1$: $(3 - 2i)^2 = 9 - 12i + 4i^2 = 5 - 12i$, not $13 - 12i$.
- Multiplying only the denominator by the conjugate: that changes the fraction. The numerator is multiplied too.
- Taking the argument without minding the quadrant: for $-1 + i$ the angle is $\frac{3\pi}{4}$, not $-\frac{\pi}{4}$.
- Finding one root of $z^2 = w$. There are two, opposite to each other.