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Number and Algebra - Complex Numbers (Cartesian, Polar, Euler Forms)

Grade 12IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The Imaginary Unit and Cartesian Form: A complex number is defined as z=a+biz = a + bi, where i=−1i = \sqrt{-1} and a,b∈Ra, b \in \mathbb{R}. In the complex plane, also known as the Argand diagram, aa represents the horizontal displacement (Real axis) and bb represents the vertical displacement (Imaginary axis), positioning the number as a point (a,b)(a, b) or a vector from the origin.

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The Complex Conjugate: For a complex number z=a+biz = a + bi, its conjugate is z∗=a−biz^* = a - bi. Geometrically, this is a reflection of the point zz across the horizontal real axis, maintaining the same real value but reversing the sign of the imaginary component.

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Modulus and Argument: The modulus ∣z∣=r|z| = r is the magnitude or distance of the complex number from the origin in the Argand diagram. The argument θ=arg⁡(z)\theta = \arg(z) is the angle measured counter-clockwise from the positive real axis to the vector zz. Principal arguments are typically restricted to the range −π<θ≤π-\pi < \theta \le \pi or 0≤θ<2π0 \le \theta < 2\pi.

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Polar (Trigonometric) Form: Complex numbers can be expressed as z=r(cos⁡θ+isin⁡θ)z = r(\cos \theta + i \sin \theta), often abbreviated as r cis θr \text{ cis } \theta. This form highlights the rotation (θ\theta) and scaling (rr) aspect of the number, useful for multiplication and finding powers.

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Euler Form: Based on the identity eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos \theta + i \sin \theta, a complex number is written as z=reiθz = re^{i\theta}. This representation treats the complex number as a point on a circle of radius rr, making it exceptionally efficient for operations involving exponents and circular motion.

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Multiplication and Division in Polar/Euler Forms: When multiplying two numbers z1z_1 and z2z_2, you multiply their moduli (r1r2r_1 r_2) and add their arguments (θ1+θ2\theta_1 + \theta_2). When dividing, you divide the moduli (r1r2\frac{r_1}{r_2}) and subtract the arguments (θ1−θ2\theta_1 - \theta_2). Geometrically, multiplication corresponds to a simultaneous rotation and stretch/compression.

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De Moivre's Theorem: For any integer nn, [r(cos⁡θ+isin⁡θ)]n=rn(cos⁡nθ+isin⁡nθ)[r(\cos \theta + i \sin \theta)]^n = r^n(\cos n\theta + i \sin n\theta). This theorem allows for the rapid calculation of high powers of complex numbers by scaling the modulus to the nn-th power and rotating the argument nn times.

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Roots of Complex Numbers: To find the nn-th roots of a complex number, we solve zn=wz^n = w. This results in nn distinct roots, all having the same modulus rn\sqrt[n]{r} and arguments separated by 2πn\frac{2\pi}{n}. Visually, these roots form the vertices of a regular nn-gon centered at the origin in the Argand diagram.

📐Formulae

i2=−1i^2 = -1

∣z∣=r=a2+b2|z| = r = \sqrt{a^2 + b^2}

θ=arctan⁡(ba)\theta = \arctan(\frac{b}{a}) (adjusted for quadrant)

z=a+biz = a + bi (Cartesian)

z=r(cos⁡θ+isin⁡θ)z = r(\cos \theta + i \sin \theta) (Polar)

z=reiθz = re^{i\theta} (Euler)

z⋅z∗=∣z∣2=a2+b2z \cdot z^* = |z|^2 = a^2 + b^2

z1z2=r1r2ei(θ1+θ2)z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}

z1z2=r1r2ei(θ1−θ2)\frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}

(reiθ)n=rneinθ(re^{i\theta})^n = r^n e^{in\theta} (De Moivre's Theorem)

💡Examples

Problem 1:

Convert the complex number z=1−i3z = 1 - i\sqrt{3} from Cartesian form to Polar and Euler forms.

Solution:

Step 1: Calculate the modulus rr. r=(1)2+(−3)2=1+3=4=2r = \sqrt{(1)^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 Step 2: Calculate the argument θ\theta. Since a>0a > 0 and b<0b < 0, the point is in the 4th quadrant. θ=arctan⁡(−31)=−π3\theta = \arctan(\frac{-\sqrt{3}}{1}) = -\frac{\pi}{3} Step 3: Write in Polar form. z=2(cos⁡(−π3)+isin⁡(−π3))z = 2(\cos(-\frac{\pi}{3}) + i \sin(-\frac{\pi}{3})) Step 4: Write in Euler form. z=2e−iπ3z = 2e^{-i\frac{\pi}{3}}

Explanation:

We first identify the coordinates (1,−3)(1, -\sqrt{3}) to find the distance from the origin (modulus) and the direction angle (argument). Because it lies in the fourth quadrant, the argument is negative π3\frac{\pi}{3}.

Problem 2:

Given z=2eiπ4z = \sqrt{2} e^{i\frac{\pi}{4}}, calculate z8z^8 using De Moivre's Theorem.

Solution:

Step 1: Apply the power to the Euler form. z8=(2eiπ4)8z^8 = (\sqrt{2} e^{i\frac{\pi}{4}})^8 Step 2: Distribute the power to the modulus and the exponent. z8=(2)8⋅ei(8⋅π4)z^8 = (\sqrt{2})^8 \cdot e^{i(8 \cdot \frac{\pi}{4})} Step 3: Simplify the modulus. (2)8=(21/2)8=24=16(\sqrt{2})^8 = (2^{1/2})^8 = 2^4 = 16 Step 4: Simplify the argument. ei(2π)e^{i(2\pi)} Step 5: Convert back to Cartesian form if needed. z8=16(cos⁡2π+isin⁡2π)=16(1+0i)=16z^8 = 16(\cos 2\pi + i \sin 2\pi) = 16(1 + 0i) = 16

Explanation:

Using Euler form is the most efficient way to find powers. We raise the magnitude to the power of 8 and multiply the angle by 8. Since an angle of 2π2\pi is coterminal with 00, the resulting number lies on the positive real axis.