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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 $\text{\hspace{0.17em}}a+bi\text{\hspace{0.17em}}$ is similar to plotting a real number, except that the horizontal axis represents the real part of the number, $\text{\hspace{0.17em}}a,\text{\hspace{0.17em}}$ and the vertical axis represents the imaginary part of the number, $\text{\hspace{0.17em}}bi.$
Given a complex number $\text{\hspace{0.17em}}a+bi,\text{\hspace{0.17em}}$ plot it in the complex plane.
Plot the complex number $\text{\hspace{0.17em}}2-3i\text{\hspace{0.17em}}$ 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] .
Plot the point $\text{\hspace{0.17em}}1+5i\text{\hspace{0.17em}}$ in the complex plane.
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 $\text{\hspace{0.17em}}\left|z\right|.\text{\hspace{0.17em}}$ It measures the distance from the origin to a point in the plane. For example, the graph of $\text{\hspace{0.17em}}z=2+4i,\text{\hspace{0.17em}}$ in [link] , shows $\text{\hspace{0.17em}}\left|z\right|.$
Given $\text{\hspace{0.17em}}z=x+yi,\text{\hspace{0.17em}}$ a complex number, the absolute value of $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ is defined as
It is the distance from the origin to the point $\text{\hspace{0.17em}}\left(x,y\right).$
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, $\text{\hspace{0.17em}}\left(0,\text{}0\right).$
Find the absolute value of $\text{\hspace{0.17em}}z=\sqrt{5}-i.$
Using the formula, we have
See [link] .
Find the absolute value of the complex number $\text{\hspace{0.17em}}z=12-5i.$
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Given $\text{\hspace{0.17em}}z=3-4i,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}\left|z\right|.$
Using the formula, we have
The absolute value $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ is 5. See [link] .
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