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## Algebra (all content)

### Unit 16: Lesson 10

Multiplying & dividing complex numbers in polar form

# Visualizing complex number powers

Learn how powers of complex numbers behave when you look at their graphical effect on the complex plane.

## Connection between $i^2 = -1$i, squared, equals, minus, 1 and where $i$i lives

We began our study of complex numbers by inventing a number i that satisfies i, squared, equals, minus, 1, and later visualized it by placing it outside the number line, one unit above 0. With the visualizations offered in the last article, we can now see why that point in space is such a natural home for a number whose square is minus, 1.
You see, multiplication by i gives a 90, degrees rotation about the origin:
You can think about this either because i has absolute value 1 and angle 90, degrees, or because this rotation is the only way to move the grid around (fixing 0) which places 1 on the spot where i started off.
So what happens if we multiply everything in the plane by i twice?
It is the same as a 180, degrees rotation about the origin, which is multiplication by minus, 1. This of course makes sense, because multiplying by i twice is the same as multiplying by i, squared, which should be minus, 1.
It is interesting to think about how if we had tried to place i somewhere else while still maintaining its characteristic quality that i, squared, equals, minus, 1, we could not have had such a clean visualization for complex multiplication.

## Powers of complex numbers

Let's play around some more with repeatedly multiplying by some complex number.

### Example 1: $(1 + i\sqrt{3})^3$left parenthesis, 1, plus, i, square root of, 3, end square root, right parenthesis, cubed

Take the number z, equals, 1, plus, i, square root of, 3, end square root, which has absolute value square root of, 1, squared, plus, left parenthesis, square root of, 3, end square root, right parenthesis, squared, end square root, equals, 2, and angle 60, degrees. What happens if we multiply everything on the plane by z three times in a row?
Everything is stretched by a factor of 2 three times, and so is ultimately stretched by a factor of 2, cubed, equals, 8. Likewise everything is rotated by 60, degrees three times in a row, so is ultimately rotated by 180, degrees. Hence, at the end it's the same as multiplying by minus, 8, so left parenthesis, 1, plus, i, square root of, 3, end square root, right parenthesis, cubed, equals, minus, 8.
We can also see this using algebra as follows:
\begin{aligned} &\phantom{=}\left(2(\cos(60^\circ) + i\sin(60^\circ)) \right)^3\\\\ &= 2^3 ( \cos(60^\circ + 60^\circ + 60^\circ) + i\sin(60^\circ + 60^\circ + 60^\circ) \\\\\ &= 8(\cos(180^\circ) + i\sin(180^\circ)) \\\\ &= -8 \end{aligned}

### Example 2: $(1 + i)^8$left parenthesis, 1, plus, i, right parenthesis, start superscript, 8, end superscript

Next, suppose we multiply everything on the plane by left parenthesis, 1, plus, i, right parenthesis eight successive times:
Since the magnitude of 1, plus, i is
vertical bar, 1, plus, i, vertical bar, equals, square root of, 1, squared, plus, 1, squared, end square root, equals, square root of, 2, end square root,
everything is stretched by a factor of square root of, 2, end square root eight times, and hence is ultimately stretched by a factor of left parenthesis, square root of, 2, end square root, right parenthesis, start superscript, 8, end superscript, equals, 2, start superscript, 4, end superscript, equals, 16.
Since the angle of left parenthesis, 1, plus, i, right parenthesis is 45, degrees, everything is ultimately rotated by 8, dot, 45, degrees, equals, 360, degrees, so in total it's as if we didn't rotate at all. Therefore left parenthesis, 1, plus, i, right parenthesis, start superscript, 8, end superscript, equals, 16.
Alternatively, the way to see this with algebra is
\begin{aligned} &\phantom{=}(1 + i)^8 \\\\ &= \left(\sqrt{2}\cdot(\cos(45^\circ) + i \sin(45^\circ) \right)^8 \\ &= (\sqrt{2})^8 \cdot \left( \cos(\underbrace{45^\circ + \cdots + 45^\circ}_{\text{8 times}}) + i\sin(\underbrace{45^\circ + \cdots + 45^\circ}_{\text{8 times}}) \right) \\\\ &= 16 \left(\cos(360^\circ) + i\sin(360^\circ) \right) \\\\ &= 16 \end{aligned}

### Example 3: $z^5 = 1$z, start superscript, 5, end superscript, equals, 1

Now let's start asking the reverse question: Is there a number z such that after multiplying everything in the plane by z five successive times, things are back to where they started? In other words, can we solve the equation z, start superscript, 5, end superscript, equals, 1? One simple answer is z, equals, 1, but let's see if we can find any others.
First off, the magnitude of such a number would have to be 1, since if it were more than 1, the plane would keep stretching, and if it were less than 1, it would keep shrinking. Rotation is a different animal, though, since you can get back to where you started after repeating certain rotations. In particular, if you rotate start fraction, 1, divided by, 5, end fraction of the way around, like this
then doing this 5 successive times will bring you back to where you started.
The number which rotates the plane in this way is cosine, left parenthesis, 72, degrees, right parenthesis, plus, i, sine, left parenthesis, 72, degrees, right parenthesis, since start fraction, 360, degrees, divided by, 5, end fraction, equals, 72, degrees.
There are also other solutions, such as rotating start fraction, 2, divided by, 5, end fraction of the way around:
or start fraction, 1, divided by, 5, end fraction of the way around the other way:
In fact, beautifully, the numbers which solve the equation form a perfect pentagon on the unit circle:
Solutions to z, start superscript, 5, end superscript, equals, 1

### Example 4: $z^6 = -27$z, start superscript, 6, end superscript, equals, minus, 27

Looking at the equation z, start superscript, 6, end superscript, equals, minus, 27, it is asking us to find a complex number z such that multiplying by this number 6 successive times will stretch by a factor of 27, and rotate by 180, degrees, since the negative indicates 180, degrees rotation.
Something which will stretch by a factor of 27 after 6 applications must have magnitude root, start index, 6, end index, equals, square root of, 3, end square root, and one way to rotate which gives 180, degrees after 6 applications is to rotate by start fraction, 180, degrees, divided by, 6, end fraction, equals, 30, degrees. Therefore one number that solves this equation z, start superscript, 6, end superscript, equals, minus, 27 is
\begin{aligned} \sqrt{3}(\cos(30^\circ) + i\sin(30^\circ)) &= \sqrt{3}\left(\frac{\sqrt{3}}{2} + i \frac{1}{2} \right) \\ &= \frac{3}{2} + i \frac{\sqrt{3}}{2} \end{aligned}
However, there are also other answers! In fact, those answers form a perfect hexagon on the circle with radius square root of, 3, end square root:
Solutions to z, start superscript, 6, end superscript, equals, minus, 27
Can you see why?

## Solving $z^n= w$z, start superscript, n, end superscript, equals, w in general

Let's generalize the last two examples. If you are given values w and n, and asked to solve for z, as in the last example where n, equals, 6 and w, equals, minus, 27, you first find the polar representation of w:
w, equals, r, left parenthesis, cosine, left parenthesis, theta, right parenthesis, plus, i, sine, left parenthesis, theta, right parenthesis, right parenthesis
This means the angle of z must be start fraction, theta, divided by, n, end fraction, and its magnitude must be root, start index, n, end index, since this way multiplying by z a total of n successive times will in effect rotate by theta and scale by r, just as w does, so
z, equals, root, start index, n, end index, dot, left parenthesis, cosine, left parenthesis, start fraction, theta, divided by, n, end fraction, right parenthesis, plus, i, sine, left parenthesis, start fraction, theta, divided by, n, end fraction, right parenthesis, right parenthesis
To find the other solutions, we keep in mind that the angle theta could have been thought of as theta, plus, 2, pi, or theta, plus, 4, pi, or theta, plus, 2, k, pi for any integer k, since these are all really the same angle. The reason this matters is because it can affect the value of start fraction, theta, divided by, n, end fraction if we replace theta with theta, plus, 2, pi, k before dividing. Hence all the answers will be of the form
z, equals, root, start index, n, end index, dot, left parenthesis, cosine, left parenthesis, start fraction, theta, plus, 2, k, pi, divided by, n, end fraction, right parenthesis, plus, i, sine, left parenthesis, start fraction, theta, plus, 2, k, pi, divided by, n, end fraction, right parenthesis, right parenthesis
for some integer value of k. These values will be different as k ranges from 0 to n, minus, 1, but once k, equals, n we can note that the angle start fraction, theta, plus, 2, n, pi, divided by, n, end fraction, equals, start fraction, theta, divided by, n, end fraction, plus, 2, pi is really the same as start fraction, theta, divided by, n, end fraction, since they differ by one full rotation. Therefore one sees all the answers just by considering values of k ranging from 0 to n, minus, 1.