Glossary

10.3 Complex Numbers in Polar Form

2 min read•july 25, 2024

Complex numbers in offer a powerful way to visualize and manipulate numbers on the complex plane. By using and angle, we can easily perform operations like multiplication and division, which can be tricky in rectangular form.

Polar form shines when dealing with powers and roots of complex numbers. It allows us to use for exponents and find multiple solutions for roots, making it a versatile tool for solving complex equations.

Complex Numbers in Polar Form

Polar form of complex numbers

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Top images from around the web for Polar form of complex numbers

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  • Polar Form of Complex Numbers – Algebra and Trigonometry OpenStax View original

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  • Polar form representation r(cos⁡θ+isin⁡θ)r(\cos\theta + i\sin\theta) expresses complex numbers using magnitude and angle
  • Modulus rr measures distance from origin to point in complex plane (3 units)
  • θ\theta indicates angle from positive x-axis to point (45°)
  • Relates to rectangular form a+bi=r(cos⁡θ+isin⁡θ)a + bi = r(\cos\theta + i\sin\theta) connecting Cartesian and
  • r⋅cis(θ)r \cdot cis(\theta) provides compact way to write polar form where cis(θ)=cos⁡θ+isin⁡θcis(\theta) = \cos\theta + i\sin\theta

Rectangular vs polar form conversion

  • Rectangular to polar conversion calculates modulus r=a2+b2r = \sqrt{a^2 + b^2} and argument θ=tan⁡−1(ba)\theta = \tan^{-1}(\frac{b}{a})
  • Polar to rectangular conversion finds real part a=rcos⁡θa = r\cos\theta and imaginary part b=rsin⁡θb = r\sin\theta
  • Quadrant considerations require adjusting argument for quadrants II, III, and IV (add π or 2π)
  • Special cases include pure real numbers with θ=0\theta = 0 or π\pi (3+0i) and pure imaginary numbers with θ=π2\theta = \frac{\pi}{2} or 3π2\frac{3\pi}{2} (0+2i)

Multiplication and division in polar form

  • Multiplication in polar form multiplies moduli and adds arguments r1r2[cos⁡(θ1+θ2)+isin⁡(θ1+θ2)]r_1r_2[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]
  • Division in polar form divides moduli and subtracts arguments r1r2[cos⁡(θ1−θ2)+isin⁡(θ1−θ2)]\frac{r_1}{r_2}[\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)]
  • Simplifies complex arithmetic compared to rectangular form especially for repeated multiplications or divisions

Powers and roots using polar form

  • De Moivre's Theorem for powers raises modulus to power and multiplies argument by power rn(cos⁡(nθ)+isin⁡(nθ))r^n(\cos(n\theta) + i\sin(n\theta))
  • use formula rn(cos⁡(θ+2πkn)+isin⁡(θ+2πkn))\sqrt[n]{r}(\cos(\frac{\theta + 2\pi k}{n}) + i\sin(\frac{\theta + 2\pi k}{n})) with k=0,1,2,...,n−1k = 0, 1, 2, ..., n-1
  • Applies to solving complex equations and finding multiple solutions in trigonometric equations (cubic roots of unity)

Key Terms to Review (11)

Argument: In mathematics, particularly in the context of complex numbers, the argument refers to the angle formed between the positive x-axis and the line connecting the origin to a point representing a complex number in the polar coordinate system. This angle is crucial for expressing complex numbers in polar form and plays a significant role in operations like multiplication and finding roots of complex numbers.
Cis notation: Cis notation is a shorthand way to express complex numbers in polar form, specifically written as $$ ext{cis}( heta) = ext{cos}( heta) + i ext{sin}( heta)$$. This notation simplifies the representation of complex numbers and is particularly useful when working with multiplication, division, and roots of complex numbers. It connects closely with polar coordinates, making it easier to perform operations involving angles and magnitudes.
De Moivre's Theorem: De Moivre's Theorem states that for any real number $$ heta$$ and any integer $$n$$, the complex number in polar form can be expressed as \( r(\cos(n\theta) + i\sin(n\theta)) \). This theorem connects the powers and roots of complex numbers to trigonometric functions, providing a powerful way to compute these operations efficiently. Understanding De Moivre's Theorem is essential for finding the nth roots of complex numbers and expresses how to multiply complex numbers in polar form using their magnitudes and angles.
Electrical Engineering: Electrical engineering is a branch of engineering that deals with the study, design, and application of equipment and systems that use electricity, electronics, and electromagnetism. This field encompasses various aspects of power generation, control systems, telecommunications, and circuit design, making it essential in both industrial and consumer technology applications.
Magnitude: Magnitude refers to the size or length of a quantity, often represented numerically. In the context of mathematics, especially with complex numbers and vectors, magnitude gives us an essential understanding of how far a point is from the origin in a coordinate system or how strong a vector is in terms of its direction and length. It’s a fundamental concept that connects various mathematical tools and applications, influencing calculations and interpretations across different areas.
Nth roots of complex numbers: Nth roots of complex numbers are the complex numbers that, when raised to the power of 'n', yield a given complex number. This concept is crucial when working with complex numbers in polar form, as it allows for the identification of all possible solutions to the equation z^n = r(cos(θ) + i sin(θ)). Understanding nth roots involves recognizing how the polar representation simplifies calculations and leads to multiple distinct roots evenly spaced around a circle in the complex plane.
Polar coordinates: Polar coordinates are a two-dimensional coordinate system that represents points in the plane using a distance from a reference point and an angle from a reference direction. This system is particularly useful for visualizing and solving problems related to circular motion and periodic phenomena, connecting seamlessly with concepts like complex numbers and vector operations.
Polar form: Polar form is a way to represent complex numbers using the magnitude (or modulus) and angle (or argument) rather than the standard rectangular form. This representation makes it easier to perform multiplication, division, and exponentiation of complex numbers by utilizing trigonometric functions. By expressing complex numbers in polar form, one can simplify calculations involving De Moivre's Theorem and find nth roots more efficiently.
Rotation: Rotation is the act of turning an object around a fixed point or axis, which can be expressed in terms of angles measured in degrees or radians. In the context of the polar coordinate system, rotation helps to describe how points are positioned and can significantly affect the appearance and properties of graphs and complex numbers. Understanding rotation is crucial for analyzing transformations and the relationships between different polar equations and complex numbers.
Scaling: Scaling refers to the process of resizing or transforming figures while maintaining their proportions and relationships. In the context of complex numbers in polar form and polar equations, scaling can adjust the magnitude of these numbers or the distance from the origin in their graphical representation, which impacts how these elements relate to one another in their respective systems.
Signal Processing: Signal processing is the analysis, interpretation, and manipulation of signals to enhance their quality or extract useful information. This concept is crucial in various applications, such as telecommunications, audio processing, and image processing, as it allows for the transformation of raw data into a more meaningful format. Understanding complex numbers in polar form is essential in signal processing since signals can often be represented as vectors in a complex plane, enabling more effective analysis and filtering techniques.
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