Concept Byte: Powers of Complex Numbers
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How can you relate the distances with the exponent of each number?
It will lie on the imaginary-axis at point (0,32).
In Example 1, (2i)^n was computed for n=0, 1, 2, and 3.
| n | (2i)^n |
|---|---|
| 0 | 1 |
| 1 | 2i |
| 2 | -4 |
| 3 | -8i |
Additionally, these points where plotted on the complex plane.
Our mission is to predict the location of (2i)^5. To do this, we will make a table of values where we point out some important facts from the graph above.
| n | (2i)^n | Location | Distance to the Origin |
|---|---|---|---|
| 0 | 1 | Real-axis | 1 |
| 1 | 2i | Imaginary-axis | 2 |
| 2 | -4 | Real-axis | 4 |
| 3 | -8i | Imaginary-axis | 8 |
The table above leads us to say that (2i)^4 will be on the real-axis, so (2i)^5 will lie on the imaginary-axis. Next, let's rewrite the numbers in the last column.
| n | (2i)^n | Location | Distance to the Origin |
|---|---|---|---|
| 0 | 1 | Real-axis | 2^0 |
| 1 | 2i | Imaginary-axis | 2^1 |
| 2 | -4 | Real-axis | 2^2 |
| 3 | -8i | Imaginary-axis | 2^3 |
| 4 | (2i)^4 | Real-axis | ? |
| 5 | (2i)^5 | Imaginary-axis | ? |
With this last table we can say that the distance from (2i)^4 to the origin is 2^4=16. Since it lies on the real-axis, its coordinates are (16,0). Consequently, the distance from (2i)^5 to the origin is 2^5=32, and because it lies on the imaginary-axis its coordinates are (0,32).
| n | (2i)^n | Location | Distance to the Origin |
|---|---|---|---|
| 0 | 1 | Real-axis | 2^0 |
| 1 | 2i | Imaginary-axis | 2^1 |
| 2 | -4 | Real-axis | 2^2 |
| 3 | -8i | Imaginary-axis | 2^3 |
| 4 | 16 | Real-axis | 16 |
| 5 | 32i | Imaginary-axis | 32 |
