9709 · 3.9
Complex numbers flashcards
Revision flashcards for Cambridge 9709 Complex numbers (syllabus 3.9). Flip, recall, then mark a real past-paper question.
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What is the imaginary unit, $i$?
$i$ is defined as the square root of -1, so $i^2 = -1$. It allows us to find solutions to equations like $x^2 + 4 = 0$.
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What is the Cartesian form of a complex number?
$z = x + iy$, where $x$ is the real part, $\text{Re}(z)$, and $y$ is the imaginary part, $\text{Im}(z)$.
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What is an Argand diagram?
A two-dimensional plane used to plot complex numbers. The horizontal axis is the real axis (Re) and the vertical axis is the imaginary axis (Im).
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Define the modulus of a complex number $z = x + iy$.
The modulus, $|z|$, is the distance from the origin to the point $(x, y)$ on the Argand diagram. It is calculated as $|z| = \sqrt{x^2 + y^2}$.
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Define the argument of a complex number $z = x + iy$.
The argument, $\arg(z)$, is the angle $\theta$ that the line connecting the origin to $z$ makes with the positive real axis, measured anticlockwise. $\tan(\theta) = y/x$.
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What is the principal argument?
The specific value of the argument that lies in the range $-\pi < \arg(z) \le \pi$. Calculators often give this value, but you must check the quadrant.
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What is the polar form of a complex number?
$z = r(\cos \theta + i \sin \theta)$, where $r$ is the modulus and $\theta$ is the argument.
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What is the exponential form of a complex number?
$z = re^{i\theta}$, where $r$ is the modulus and $\theta$ is the argument in radians. This is a compact version of the polar form.
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What is a complex conjugate?
The conjugate of $z = x + iy$ is $z^* = x - iy$. On an Argand diagram, it's a reflection in the real axis. Key property: $z z^* = |z|^2$.
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State De Moivre's Theorem.
For any integer $n$, $(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)$. In exponential form: $(e^{i\theta})^n = e^{in\theta}$.
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Trap: When finding the argument of $z = -1 - i$, my calculator gives $\tan^{-1}(-1/-1) = \pi/4$. Is this correct?
No. The point $(-1, -1)$ is in the third quadrant. The principal argument should be in $(-\pi, \pi]$. The correct argument is $-\frac{3\pi}{4}$. Always sketch an Argand diagram to check the quadrant.
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How do you multiply two complex numbers in polar form, $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$?
Multiply the moduli and add the arguments: $z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}$.