Complex Numbers Crash Course Quiz

Complex Numbers in Polar Form

• The polar form of a complex number $z = a + bi$ is given by $z = r(\cos\theta + i\sin\theta)$, where $r$ is the modulus and $\theta$ is the amplitude.
• To find the polar form of $z = 3 + 4i$, we need to calculate the modulus $r$ and amplitude $\theta$.

Modulus of a Complex Number

• The modulus of a complex number $z = a + bi$ is given by $r = \sqrt{a^2 + b^2}$.
• The modulus of $z = 3 + 4i$ is $r = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$.

Amplitude of a Complex Number

• The amplitude of a complex number $z = a + bi$ is given by $\theta = \tan^{-1}(\frac{b}{a})$.
• The amplitude of $z = -2 - i$ is $\theta = \tan^{-1}(\frac{-1}{-2}) = \tan^{-1}(\frac{1}{2})$.

De Moivre's Theorem

• De Moivre's theorem states that for a complex number $z = r(\cos\theta + i\sin\theta)$, $z^n = r^n(\cos(n\theta) + i\sin(n\theta))$.
• Applying De Moivre's theorem to $z = 4 + 3i$ with $n = 3$, we need to find the polar form of $z$ first.

Complex Conjugate

• The complex conjugate of a complex number $z = a + bi$ is given by $\bar{z} = a - bi$.
• The complex conjugate of $z = 2 - i$ is $\bar{z} = 2 + i$.