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“God made the integers; all else is the work of man.” This rather famous quote by nineteenth-century German mathematician Leopold Kronecker sets the stage for this section on the polar form of a complex number. Complex numbers were invented by people and represent over a thousand years of continuous investigation and struggle by mathematicians such as Pythagoras , Descartes , De Moivre, Euler , Gauss , and others. Complex numbers answered questions that for centuries had puzzled the greatest minds in science.
We first encountered complex numbers in Complex Numbers . In this section, we will focus on the mechanics of working with complex numbers: translation of complex numbers from polar form to rectangular form and vice versa, interpretation of complex numbers in the scheme of applications, and application of De Moivre’s Theorem.
Plotting a complex number is similar to plotting a real number, except that the horizontal axis represents the real part of the number, and the vertical axis represents the imaginary part of the number,
Given a complex number plot it in the complex plane.
Plot the complex number in the complex plane .
From the origin, move two units in the positive horizontal direction and three units in the negative vertical direction. See [link] .
The first step toward working with a complex number in polar form is to find the absolute value. The absolute value of a complex number is the same as its magnitude , or It measures the distance from the origin to a point in the plane. For example, the graph of in [link] , shows
Given a complex number, the absolute value of is defined as
It is the distance from the origin to the point
Notice that the absolute value of a real number gives the distance of the number from 0, while the absolute value of a complex number gives the distance of the number from the origin,
Find the absolute value of
Given find
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