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Questions and Answers
Write a polar form of the complex number $z = 3 + 4i$ with amplitude $\theta$ and modulus $r$.
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 $z = -2 - i$.
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 $z = 3 + 4i$?
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 $n = 3$.
Apply De Moivre’s theorem to find the expression for $z^n$ where $z = 4 + 3i$ and $n = 3$.
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What is the complex conjugate of the number $z = 2 - i$?
What is the complex conjugate of the number $z = 2 - i$?
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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.
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 $\bar{z} = 2 + i$.
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Description
Test your understanding of complex numbers with this crash course quiz on finding the Nth power, complex conjugates, modulus, and amplitude/argument. Sharpen your mathematical skills and master these fundamental concepts.
