1.4 Polar form of complex numbers

Complex numbers in polar form offer a powerful way to represent and manipulate numbers in the complex plane. By expressing numbers as a magnitude and angle, we can easily visualize their geometric properties and perform operations like multiplication and division.

Polar form simplifies certain calculations, especially those involving powers and roots of complex numbers. It provides a clear connection between algebra and geometry, making it easier to understand the behavior of complex numbers in various mathematical contexts.

Complex Number Forms

Converting from Rectangular to Polar Form

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• A complex number in rectangular form is written as $z = a + bi$
  • $a$ represents the real part
  • $b$ represents the imaginary part
  • $i$ is the imaginary unit, defined as $i^2 = -1$
• To convert from rectangular to polar form, use the following formulas:
  • Modulus: $r = \sqrt{a^2 + b^2}$
  • Argument: $\theta = \arctan(\frac{b}{a})$ or $\theta = \text{atan2}(b, a)$
    • The argument is expressed in radians
    • The $\text{atan2}$ function takes into account the quadrant of the complex number

Converting from Polar to Rectangular Form

• A complex number in polar form is written as $z = r(\cos(\theta) + i \sin(\theta))$ or $z = r \text{ cis}(\theta)$
  • $r$ is the modulus (magnitude) of the complex number
  • $\theta$ is the argument (angle) in radians
  • The $\text{cis}$ function is defined as $\text{cis}(\theta) = \cos(\theta) + i \sin(\theta)$
• To convert from polar to rectangular form, use the following formulas:
  • Real part: $a = r \cos(\theta)$
  • Imaginary part: $b = r \sin(\theta)$
• The polar form of a complex number is not unique
  • The argument can be expressed as $\theta + 2\pi k$, where $k$ is any integer
  • This is due to the periodicity of the trigonometric functions $\cos$ and $\sin$

Arithmetic with Polar Complex Numbers

Multiplication and Division

• Multiplication of complex numbers in polar form:
  • If $z_1 = r_1 \text{ cis}(\theta_1)$ and $z_2 = r_2 \text{ cis}(\theta_2)$, then $z_1 \times z_2 = r_1 r_2 \text{ cis}(\theta_1 + \theta_2)$
  • The moduli are multiplied, and the arguments are added
• Division of complex numbers in polar form:
  • If $z_1 = r_1 \text{ cis}(\theta_1)$ and $z_2 = r_2 \text{ cis}(\theta_2)$, then $z_1 \div z_2 = \frac{r_1}{r_2} \text{ cis}(\theta_1 - \theta_2)$
  • The moduli are divided, and the arguments are subtracted

Powers and Roots

• Powers of complex numbers in polar form:
  • If $z = r \text{ cis}(\theta)$, then $z^n = r^n \text{ cis}(n\theta)$ for any integer $n$
  • The modulus is raised to the power $n$, and the argument is multiplied by $n$
• Roots of complex numbers in polar form:
  • If $z = r \text{ cis}(\theta)$, then the $n$-th roots of $z$ are given by $w_k = \sqrt[n]{r} \text{ cis}\left(\frac{\theta + 2\pi k}{n}\right)$, where $k = 0, 1, \ldots, n-1$
  • There are $n$ distinct $n$-th roots of a complex number
  • The modulus is taken to the $\frac{1}{n}$ power, and the argument is divided by $n$ and added to $\frac{2\pi k}{n}$

Geometric Interpretation of Modulus and Argument

Modulus as Distance

• The modulus (magnitude) of a complex number represents the distance from the origin to the point representing the complex number in the complex plane
  • For a complex number $z = a + bi$, the modulus is given by $|z| = \sqrt{a^2 + b^2}$
  • The modulus is always a non-negative real number

Argument as Angle

• The argument (angle) of a complex number represents the angle between the positive real axis and the line segment connecting the origin to the point representing the complex number in the complex plane
  • The argument is typically expressed in radians
  • It can be visualized as a counterclockwise rotation from the positive real axis
  • The argument is given by $\arg(z) = \arctan(\frac{b}{a})$ or $\arg(z) = \text{atan2}(b, a)$
• The modulus and argument of a complex number provide a geometric interpretation of the number's position in the complex plane
  • The modulus determines the distance from the origin
  • The argument determines the angle of rotation from the positive real axis

Solving Equations with Polar Complex Numbers

Equating Moduli and Arguments

• When solving equations involving complex numbers in polar form, it is often necessary to equate the moduli and arguments of the complex numbers on both sides of the equation
  • This is because two complex numbers are equal if and only if their moduli are equal and their arguments differ by a multiple of $2\pi$
  • Equating the moduli results in an equation involving real numbers
  • Equating the arguments may result in multiple solutions due to the periodicity of the argument

Simplifying Powers and Roots

• For equations involving powers or roots of complex numbers, using the polar form can simplify the problem by applying the properties of exponents and logarithms
  • The multiplication of arguments in the polar form corresponds to the addition of exponents
  • The division of arguments in the polar form corresponds to the subtraction of exponents
  • Taking the logarithm of both sides of an equation can help solve for the argument or modulus

Consistent Form and Graphical Methods

• When solving equations with multiple complex numbers, it may be helpful to convert all numbers to either polar or rectangular form for consistency
  • This allows for easier comparison and manipulation of the numbers
  • Converting between forms may be necessary to solve certain parts of the equation
• Graphical methods, such as plotting complex numbers in the complex plane, can aid in understanding and solving equations involving complex numbers in polar form
  • Visualizing the moduli and arguments of the numbers can provide insights into the problem
  • Geometric transformations, such as rotations and dilations, can be applied to the complex plane to solve equations