Complex Numbers Crash Course Quiz

5 Questions

Write a polar form of the complex number $z = 3 + 4i$ with amplitude $\theta$ and modulus $r$.

The polar form of the complex number $z = 3 + 4i$ with amplitude $\theta$ and modulus $r$ is $5(\cos \theta + i \sin \theta)$.

Calculate the amplitude of the complex number $z = -2 - i$.

The amplitude of the complex number $z = -2 - i$ is given by $\theta = \tan^{-1}\left|\frac{-2}{-1}\right| = \tan^{-1}(2)$. Therefore, the amplitude is $\theta = \frac{\pi}{4}$.

What is the modulus of the complex number $z = 3 + 4i$?

The modulus of the complex number $z = 3 + 4i$ is given by $|z| = \sqrt{3^2 + 4^2} = 5$.

Apply De Moivre’s theorem to find the expression for $z^n$ where $z = 4 + 3i$ and $n = 3$.

Using De Moivre’s theorem, we have $z^n = (4 + 3i)^3 = (5 \text{ cis} \theta)^3 = 125 \text{ cis} 3\theta$.

What is the complex conjugate of the number $z = 2 - i$?

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

Study Notes

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.