Section 4.3 Cyclic Groups of Complex Numbers
To add two complex numbers and we just add the corresponding real and imaginary parts:
Remembering that we multiply complex numbers just like polynomials. The product of and is
Every nonzero complex number has a multiplicative inverse; that is, there exists a such that If then
The complex conjugate of a complex number is defined to be The absolute value or modulus of is
Example 4.3.1. Complex Number Operations.
There are several ways of graphically representing complex numbers. We can represent a complex number as an ordered pair on the plane where is the (or real) coordinate and is the (or imaginary) coordinate. This is called the rectangular or Cartesian representation. The rectangular representations of and are depicted in Figure 4.3.2.
Nonzero complex numbers can also be represented using polar coordinates. To specify any nonzero point on the plane, it suffices to give an angle from the positive axis in the counterclockwise direction and a distance from the origin, as in Figure 4.3.3. We can see that
Hence,
and
We sometimes abbreviate as To assure that the representation of is well-defined, we also require that If the measurement is in radians, then
Example 4.3.4. Complex Numbers in Polar Form.
Conversely, if we are given a rectangular representation of a complex number, it is often useful to know the number’s polar representation. If then
and
so
The polar representation of a complex number makes it easy to find products and powers of complex numbers. The proof of the following proposition is straightforward and is left as an exercise.
Proposition 4.3.5.
Theorem 4.3.7. DeMoivre.
Proof.
We will use induction on (see Section 2.1). For the theorem is trivial. Assume that the theorem is true for all such that Then
Example 4.3.8. Powers of Complex Numbers.
Suppose that and we wish to compute Rather than computing directly, it is much easier to switch to polar coordinates and calculate using DeMoivre’s Theorem:
The multiplicative group of the complex numbers, possesses some interesting subgroups. Whereas and have no interesting subgroups of finite order, has many. We first consider the circle group,
The following proposition is a direct result of Proposition 4.3.5.
Proposition 4.3.9.
The circle group is a subgroup of
Although the circle group has infinite order, it has many interesting finite subgroups. Suppose that Then is a subgroup of the circle group. Also, and are exactly those complex numbers that satisfy the equation The complex numbers satisfying the equation are called the th roots of unity.
Theorem 4.3.10.
If then the th roots of unity are
where Furthermore, the th roots of unity form a cyclic subgroup of of order
Proof.
By DeMoivre’s Theorem,
The ’s are distinct since the numbers are all distinct and are greater than or equal to 0 but less than We will leave the proof that the th roots of unity form a cyclic subgroup of as an exercise.
Example 4.3.11. Roots of Unity.
The 8th roots of unity can be represented as eight equally spaced points on the unit circle (Figure 4.3.12). The primitive 8th roots of unity are
We interrupt this exposition to repeat the previous diagram, wrapped as different figure with a different caption. The TikZ code to produce these diagrams lives in an external file,
tikz/cyclic-roots-unity.tex
, which is pure text, freed from any need to format for XML processing. So, in particular, there is no need to escape ampersands and angle brackets, nor is there employment of the CDATA
mechanism. But the real value is that there is just one version to edit, and any changes will be reflected in both copies.
You have attempted 1 of 1 activities on this page.