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10.5 Polar Form of Complex Numbers

4 min read•june 18, 2024

Complex numbers expand our understanding of numbers beyond the real line. They combine real and imaginary parts, allowing us to represent points on a two-dimensional plane. This opens up new possibilities for solving equations and modeling real-world phenomena.

The of complex numbers offers a different way to represent these numbers. It uses and angle instead of real and imaginary parts. This form simplifies certain operations like multiplication and division, making it useful in various applications.

Representing Complex Numbers

Complex numbers on complex plane

is 2D coordinate system with real part on horizontal axis and imaginary part on vertical axis
$a + bi$ $a + bi$ plotted as point $(a, b)$ $(a, b)$ on complex plane
- Real part $a$ determines horizontal position
- Imaginary part $b$ determines vertical position
Example: Complex number $3 + 2i$ plotted at point $(3, 2)$ on complex plane

Absolute value of complex numbers

Absolute value () of complex number $z = a + bi$ is distance from origin to point $(a, b)$ on complex plane
Denoted as $|z|$ and calculated using formula: $|z| = \sqrt{a^2 + b^2}$
Absolute value always non-negative real number
Example: For complex number $3 + 4i$ , absolute value is $|3 + 4i| = \sqrt{3^2 + 4^2} = 5$
The absolute value is also known as the magnitude of the complex number

Polar Form of Complex Numbers

Complex numbers in polar form

Polar form of complex number $z$ $z$ is $z = r(\cos\theta + i\sin\theta)$ $z = r (cos θ + i sin θ)$ or $z = re^{i\theta}$ $z = r e^{i θ}$
- $r$ is absolute value (modulus) of $z$
- $\theta$ is argument (angle) in radians, measured counterclockwise from positive real axis
Polar form represents complex number as point $(r, \theta)$ in polar coordinates
To find polar form, use formulas:
- $r = \sqrt{a^2 + b^2}$
- $\theta = \tan^{-1}(\frac{b}{a})$ , with appropriate adjustments based on quadrant of $(a, b)$
The angle $\theta$ is also referred to as the phase of the complex number

Polar vs rectangular forms

Converting from polar form $z = r(\cos\theta + i\sin\theta)$ $z = r (cos θ + i sin θ)$ to $z = a + bi$ $z = a + bi$ :
- $a = r\cos\theta$
- $b = r\sin\theta$
Converting from rectangular form $z = a + bi$ $z = a + bi$ to polar form $z = r(\cos\theta + i\sin\theta)$ $z = r (cos θ + i sin θ)$ :
- $r = \sqrt{a^2 + b^2}$
- $\theta = \tan^{-1}(\frac{b}{a})$ , with appropriate adjustments based on quadrant of $(a, b)$
The polar form is also known as the trigonometric form of a complex number

Multiplication and division in polar form

Multiplication in polar form: If $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ $z_{1} = r_{1} (cos θ_{1} + i sin θ_{1})$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$ $z_{2} = r_{2} (cos θ_{2} + i sin θ_{2})$ , then:
- $z_1 \cdot z_2 = r_1r_2[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]$
Division in polar form: If $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ $z_{1} = r_{1} (cos θ_{1} + i sin θ_{1})$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$ $z_{2} = r_{2} (cos θ_{2} + i sin θ_{2})$ , then:
- $\frac{z_1}{z_2} = \frac{r_1}{r_2}[\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)]$
Multiplication and division in polar form simplify to operations on moduli and arguments

Powers and roots in polar form

To raise complex number $z = r(\cos\theta + i\sin\theta)$ $z = r (cos θ + i sin θ)$ to power $n$ $n$ :
- $z^n = r^n(\cos(n\theta) + i\sin(n\theta))$
To find $n$ $n$ -th roots of complex number $z = r(\cos\theta + i\sin\theta)$ $z = r (cos θ + i sin θ)$ :
- $\sqrt[n]{z} = \sqrt[n]{r}(\cos(\frac{\theta + 2k\pi}{n}) + i\sin(\frac{\theta + 2k\pi}{n}))$ , where $k = 0, 1, 2, ..., n-1$
- There are $n$ distinct $n$ -th roots of a complex number

Applications of De Moivre's theorem

's theorem states that for any complex number $z = r(\cos\theta + i\sin\theta)$ $z = r (cos θ + i sin θ)$ and any integer $n$ $n$ :
- $[r(\cos\theta + i\sin\theta)]^n = r^n(\cos(n\theta) + i\sin(n\theta))$
Simplifies computation of powers and roots of complex numbers
Useful for finding roots of unity and solving equations involving complex numbers

Geometry of polar form operations

Multiplication by $z = r(\cos\theta + i\sin\theta)$ $z = r (cos θ + i sin θ)$ in polar form:
- Scales modulus by factor of $r$
- Rotates argument by angle of $\theta$ counterclockwise
Division by $z = r(\cos\theta + i\sin\theta)$ $z = r (cos θ + i sin θ)$ in polar form:
- Scales modulus by factor of $\frac{1}{r}$
- Rotates argument by angle of $-\theta$ (clockwise)
Raising complex number to power $n$ $n$ :
- Raises modulus to power $n$
- Multiplies argument by $n$
Taking $n$ $n$ -th root of complex number:
- Takes $n$ -th root of modulus
- Divides argument by $n$ and adds multiples of $\frac{2\pi}{n}$ to generate $n$ roots

Additional Concepts in Polar Form

Complex conjugate: For a complex number $z = a + bi$ , its complex conjugate is $\bar{z} = a - bi$
Principal argument: The unique value of $\theta$ in the interval $(-\pi, \pi]$ that represents the angle of a complex number in polar form
Unit circle: The circle with radius 1 centered at the origin, used to visualize complex numbers with magnitude 1 in the complex plane

Key Terms to Review (15)

Complex number: A complex number is a number of the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit with property $i^2 = -1$. Complex numbers can be represented in both rectangular and polar forms.

Complex plane: The complex plane is a two-dimensional plane used to represent complex numbers graphically. Each point on the plane corresponds to a unique complex number, with the horizontal axis representing the real part and the vertical axis representing the imaginary part.

De Moivre: De Moivre's Theorem states that for any real number $\theta$ and integer $n$, $(\cos(\theta) + i \sin(\theta))^n = \cos(n\theta) + i \sin(n\theta)$. It is used to raise complex numbers in polar form to a power.

De Moivre’s Theorem: De Moivre's Theorem states that for any complex number in polar form $r(\cos \theta + i\sin \theta)$ and integer $n$, $(r(\cos \theta + i\sin \theta))^n = r^n(\cos(n\theta) + i\sin(n\theta))$. It provides a way to raise complex numbers to powers and extract roots using trigonometric functions.

Descartes: René Descartes was a French philosopher and mathematician who made significant contributions to mathematics, particularly in the field of coordinate geometry. His work laid the foundation for the Cartesian coordinate system, which is essential in algebra and trigonometry.

Euler: Euler's formula, $e^{ix} = \cos(x) + i\sin(x)$, connects complex exponentials and trigonometric functions. It's fundamental in expressing complex numbers in polar form.

Gauss: Gauss, often referring to Carl Friedrich Gauss, made significant contributions to algebra and trigonometry, including methods for solving systems of linear equations. His Gaussian elimination method is a systematic approach for reducing a matrix to row echelon form.

Kronecker: The Kronecker Delta function, denoted as $\delta_{ij}$, is a function of two variables which equals 1 if the variables are equal and 0 otherwise. It is commonly used in various areas of mathematics including linear algebra and tensor analysis.

Magnitude: Magnitude represents the size or length of a vector. It is always a non-negative value and can be found using the Pythagorean theorem in two or three dimensions.

Modulus: The modulus of a complex number is its distance from the origin in the complex plane, represented as $|z|$. It is calculated using the formula $|z| = \sqrt{a^2 + b^2}$ where $z = a + bi$.

Nth root of a complex number: The nth root of a complex number is one of the n distinct complex numbers which, when raised to the nth power, gives the original complex number. It can be calculated using polar form and De Moivre's Theorem.

Polar form: Polar form represents complex numbers using a magnitude (r) and an angle ($\theta$). It is expressed as $z = r(\cos \theta + i \sin \theta)$ or $z = re^{i\theta}$.

Polar form of a complex number: The polar form of a complex number expresses it as $z = r(\cos(\theta) + i\sin(\theta))$ or $z = re^{i\theta}$, where $r$ is the magnitude and $\theta$ is the argument (angle). This form emphasizes the geometric interpretation of complex numbers.

Pythagoras: Pythagoras was an ancient Greek mathematician and philosopher who is best known for the Pythagorean theorem, a fundamental principle in geometry. His work has significant implications in various fields of mathematics, including algebra and trigonometry.

Rectangular form: Rectangular form of a complex number is expressed as $z = a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit. It represents the complex number in terms of its horizontal (real) and vertical (imaginary) components.

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