Chapter 9: Polar Coordinates and Complex Numbers (2024)

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    The picture on the left below is a photo of a fern leaf. The picture on the right is a computer-generated image called the Barnsley fern. It is named after the British mathematician Michael Barnsley, who developed it in the 1980’s.

    Chapter 9: Polar Coordinates and Complex Numbers (2)

    The Barnsley fern is an example of a fractal. A fractal is an infinitely complex pattern created by iteration: repeating a process over and over in an ongoing feedback loop. You can watch a Barnsley fern being created at

    https://www.youtube.com/watch?v=iGMGVpLMtMs

    One of the most famous and intriguing fractals is the Mandelbrot set, named after the French mathematician Benoît Mandelbrot, who lived from 1924 to 2010. The Mandelbrot set looks like a cardioid, or heart-shaped figure, studded with circles all around its boundary.

    Chapter 9: Polar Coordinates and Complex Numbers (3)

    Like the Barnsley fern, we can create the Mandelbrot set by repeatedly evaluating a simple equation over and over. This equation uses complex numbers.

    The plane of complex numbers is similar to a two dimensional coordinate system. Every point on the plane is represented by a complex number \(c = x + iy\), and we perform the following steps for every point:

    Starting with \(z_1 = 0\), we create an infinite sequence of complex numbers \(z_n\) according to the rule

    \(z_{n+1} = (z_n)^2 + c\)

    If this sequence increases to infinity (it diverges), we color the point \(c\) white. If the sequence does not increase beyond a certain limit (it is bounded), we color the point black.

    We create a sequence for every point \(c\) in the plane. The collection of all the black points is the Mandelbrot set. It sits in the portion of the plane where \(x\) is between −1 and 2, and \(y\) is between −1.5 and 1.5. Outside the set, all the values of \(c\) cause the sequence to go to infinity, and the color of the point is proportional to the speed at which the sequence diverges.

    Because the calculations must be performed thousands or millions of times, we need computers to study them. The higher the number of iterations, the better the quality of the image produced. Not coincidentally, the Mandelbrot set was discovered in 1980, shortly after the invention of the personal computer.

    One of the intriguing features of fractal images is self-similarity. If we zoom in on a fractal, we see the same pattern repeated again and again, sometimes with interesting variations. Here is a portion of the boundary of the Mandelbrot set, blown up to many times its original size. You can see the copy of whole set embedded in the image.

    Chapter 9: Polar Coordinates and Complex Numbers (4)

    You can find a ”Mandelbrot Zoomer” at

    https://mathigon.org/course/fractals/introduction

    Here are some more fractals, called Julia sets:

    Chapter 9: Polar Coordinates and Complex Numbers (5)

    Fractal patterns seem familiar because the laws that govern the creation of fractals are found throughout the natural world. Tree branches, rivers, ice crystals, and seashells all form in fractal shapes. You can see some examples at

    https://www.treehugger.com/earth-matters/wilderness-resources/blogs

    • 9.1: Polar Coordinates
      This section introduces polar coordinates, explaining the relationship between polar and rectangular coordinates, and how to convert between them. It covers plotting points using polar coordinates, interpreting angles and radii, and understanding the use of polar equations. The section also provides examples and exercises to help solidify the concepts and explore applications in graphing and real-world contexts.
    • 9.2: Polar Graphs
      This section covers polar graphs, focusing on how to plot equations in the polar coordinate system. It explains common polar graph shapes, such as circles, limaçons, rose curves, and lemniscates, and provides techniques for sketching these graphs. The section also includes methods for converting polar equations to rectangular form and vice versa, along with examples to demonstrate these processes. Practical applications and exercises help reinforce understanding.
    • 9.3: Complex Numbers
      This section introduces complex numbers, covering their standard form 𝑎 + 𝑏 𝑖, where 𝑖 is the imaginary unit. It explains operations with complex numbers, including addition, subtraction, multiplication, and division. The section also covers how to represent complex numbers graphically in the complex plane and discusses the polar form of complex numbers, including how to convert between rectangular and polar forms. Practical examples and exercises reinforce these concepts.
    • 9.4: Polar Form of Complex Numbers
      This section covers the polar form of complex numbers, explaining how to represent complex numbers using magnitude and angle. It details the conversion between rectangular and polar forms, and the use of De Moivre's Theorem for powers and roots of complex numbers. The section includes examples of multiplying and dividing complex numbers in polar form and provides exercises to practice these concepts.
    Chapter 9: Polar Coordinates and Complex Numbers (2024)

    FAQs

    What is polar form of complex numbers Class 9? ›

    We can write z = x + iy as z = r cosθ + ir sinθ = r (cosθ + i sinθ), which is called the polar form of complex number. In any other interval of length 2π, for example, consider the interval -π < θ ≤ π, then the value of θ is called the principal argument of z.

    How to solve the polar form of complex numbers? ›

    Equation of Polar Form of Complex Numbers

    The polar form of a complex number z = x + iy with coordinates (x, y) is given as z = r cosθ + i r sinθ = r (cosθ + i sinθ). The abbreviated polar form of a complex number is z = rcis θ, where r = √(x2 + y2) and θ = tan-1 (y/x).

    How to write numbers in polar form? ›

    In addition to the Cartesian form, z=a+bi z = a + b i , complex numbers can also be written in trigonometric polar form z=r(cosθ+isinθ) z = r ( cos ⁡ θ + i sin ⁡ where r is the modulus and θ is the argument of the number, in radians.

    How to convert polar to rectangular complex numbers? ›

    If you want to go from Polar Coordinates to Cartesian Coordinates, that is just: (r*cos(θ), r*sin(θ)) . And since the rectangular form of a Complex Number is a + bi , just replace the letters: a + bi = r*cos(θ) + r*sin(θ)i ← The right-hand side is a Complex Number in Polar Form.

    What is the formula for polar coordinates? ›

    Polar Coordinates Formula

    (r, θ+2πn) or (-r, θ+(2n+1)π), where n is an integer. The value of θ is positive if measured counterclockwise. The value of θ is negative if measured clockwise. The value of r is positive if laid off at the terminal side of θ.

    What is the formula for multiplying complex numbers in polar form? ›

    It turns out to be super easy to multiply complex numbers in polar form. Just multiply the magnitudes r, and add the angles, using the fact that (cos(x) + i sin(x)) (cos(y) + i sin(y)) = cos(x+y) + i sin(x+y).

    How to divide polar form? ›

    Multiplication and division of complex numbers in polar form. Note that to multiply the two numbers we multiply their moduli and add their arguments. To divide, we divide their moduli and subtract their arguments.

    How to find roots in polar form? ›

    There are n distinct nth roots and they can be found as follows:.
    1. Express both z and w in polar form z=reiθ,w=seiϕ. ...
    2. Solve the following two equations: rn=s. ...
    3. The solutions to rn=s are given by r=n√s.
    4. The solutions to einθ=eiϕ are given by: nθ=ϕ+2πℓ,forℓ=0,1,2,⋯,n−1. ...
    5. Using the solutions r,θ to the equations given in (6.3.
    Sep 16, 2022

    How do you convert to polar? ›

    3 and Example 8.3. 4. To convert from rectangular coordinates to polar coordinates, use one or more of the formulas: cosθ=xr, sinθ=yr, tanθ=yx, and r=√x2+y2.

    What does i equal in algebra 2? ›

    Learn about the imaginary unit, "i", a unique number defined as the square root of -1. It's a key part of complex numbers, which are in the form a + bi. The powers of "i" cycle through a set of values.

    What is cis in math? ›

    cis is a mathematical notation defined by cis x = cos x + i sin x, where cos is the cosine function, i is the imaginary unit and sin is the sine function. x is the argument of the complex number (angle between line to point and x-axis in polar form).

    What is polar form of complex number also known as? ›

    The polar form of a complex number is a different way to represent a complex number apart from rectangular form. Usually, we represent the complex numbers, in the form of z = x+iy where 'i' the imaginary number. But in polar form, the complex numbers are represented as the combination of modulus and argument.

    What is the polar form of 3 i? ›

    Thus, the complex number 3 + i in the polar form is 2 [ cos ( π 6 ) + i sin ( π 6 ) ] .

    What is the polar form of the complex number 9i? ›

    To evaluate the power, we first write the complex number in polar form. Since 9i has no real part, we know that this value would be plotted along the vertical axis, a distance of 9 from the origin at an angle of π2. This gives the polar form: 9i=9eπ2i.

    What is 0 4i in polar form? ›

    In polar coordinates, the complex number z=0+4i can be written as z=4(cos(π2)+isin(π2)) or 4cis(π2).

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