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Chapter 7: Problem 93

Represent the powers \(z, z^{2}, z^{3},\) and \(z^{4}\) graphically. Describe the pattern. $$z=\frac{\sqrt{2}}{2}(1+i)$$

Short Answer

Expert verified
The pattern observed when graphically representing the powers of the complex number \(z = \frac{\sqrt{2}}{2}(1+i)\) is one of increasing distance from the origin and a counter-clockwise rotation of \(\frac{\pi}{4}\) radians for each increasing power. This directly relates to DeMoivre's theorem for complex numbers.

Step by step solution

01

Convert to Polar Form

The first step involves transforming the given complex number from rectanglar to polar form. The polar form of a complex number is given by \(r(\cos(\theta)+i\sin(\theta))\). In this case, \(z=\frac{\sqrt{2}}{2}(1+i)\) can be rewritten as \(z=\sqrt{2}\left(\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\right)= \sqrt{2}\left(\cos\left(\frac{\pi}{4}\right)+i\sin\left(\frac{\pi}{4}\right)\right)\) which has a magnitude of \(\sqrt{2}\) and an angle of \(\frac{\pi}{4}\) radians.
02

Calculate Powers of z

Now we need to calculate the complex number powers \(z, z^{2}, z^{3},\) and \(z^{4}\) using DeMoivre's theorem, which in general form states that \((r(\cos(\theta)+i\sin(\theta)))^n=r^n(\cos(n\theta)+i\sin(n\theta))\). Thus we find: \(z = \sqrt{2}(\cos(\frac{\pi}{4}) + i\sin(\frac{\pi}{4})), z^2 = 2(\cos(\frac{\pi}{2}) + i\sin(\frac{\pi}{2})), z^3 = 2\sqrt{2}(\cos(\frac{3\pi}{4}) + i\sin(\frac{3\pi}{4})), z^4 = 4(\cos(\pi) + i\sin(\pi))\).
03

Plot Graphically

Each of these values can then be plotted in a complex plane. A complex plane is a geometric representation of the complex numbers established by the horizontal (real) axis and the vertical (imaginary) axis, and each of these values represent a point in this plane. It's important to remember to convert the polar form back to rectangular form before plotting on the cartesian plane.
04

Describe the Pattern

The pattern formed when plotting these points shows that as you raise the complex number to higher powers, each point is further from the origin and rotated counter-clockwise by additional \(\frac{\pi}{4}\) radians. This directly corresponds to DeMoivre's theorem, wherein the magnitude is raised to the power, and the angle is multiplied by the power when calculating the powers of a complex number.

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