Problem 94 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The powers of a complex number \(z\), namely \(z^{1}, z^{2}, z^{3},\) and \(z^{4}\) when plotted in the complex plane form a counter-clockwise pattern along the circle's circumference. This rotation effect is a salient property of complex numbers.

Step 1: Calculation

The interesting feature about complex numbers is that their powers produce a rotation effect in the complex plane. To demonstrate this, we first need to calculate the powers of the given complex number. We have \(z = \frac{1}{2}(1+\sqrt{3}i)\). Then, the powers of \(z\) are computed as follows: \(z^{1} = z = \frac{1}{2}(1+\sqrt{3}i)\), \(z^{2} = \left(\frac{1}{2}(1+\sqrt{3}i)\right)^{2} = -1\), \(z^{3} = -1*\frac{1}{2}(1+\sqrt{3}i) = -\frac{1}{2}(1+\sqrt{3}i)\), and finally \(z^{4} = \left(-\frac{1}{2}(1+\sqrt{3}i)\right)^{2} = -1*-1 = 1\).

Step 2: Graphing

Once we have the results, we can plot these numbers in the complex plane. Recalling that the horizontal axis (real axis) represents the real part and the vertical axis (imaginary axis) represents the imaginary part, we can proceed with the plotting. For \(z^{1}\), we plot the point (\(\frac{1}{2}, \frac{\sqrt{3}}{2}\)). For \(z^{2}\), we plot the point \((-1, 0)\). For \(z^{3}\), we plot the point \(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\). Finally, for \(z^{4}\), we plot the point \((1, 0)\). Note that the points are arranged in a counter-clockwise pattern.

Step 3: Pattern observation

Observe that the powers of \(z\) form a pattern in the complex plane. Starting from \(z\), the points generated by the subsequent powers lie along a circle's circumference and go in a counter-clockwise direction (due to the multiplication by \(i\), which causes a 90 degrees rotation). This is a general property of complex number powers and gives an important insight into their behavior.