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Complex Numbers: The Imaginary Number

The Imaginary Number

Numbers that have real and imaginary parts are complex numbers. They can be expressed in the form  where i is  where . Real numbers are actually a special case of complex numbers. Take the above expression. If b is zero then there is no imaginary part and the number is real. As with all special cases the original case has properties that are very much different for example, complex conjugates: Let      (where      denotes that z is a complex number) then the complex conjugate of      is

.  This is also true for numbers such as  , here the complex conjugate is   .  The complex conjugate is generally signified by either the asterisk  or the bar above the number . Since real numbers have no complex part the complex conjugate of a real number is itself i.e.   .

Before moving forward let's consider a complex number of the form . This number can be rewritten by multiplying by one as follows



Complex numbers can be viewed as vectors in the complex plane as shown below:

Math and physics use  as the symbol for  but many other fields including engineering use  instead. One reason for this may be that engineers use the letter i to represent current and so the next letter j became the symbol for the imaginary number.

Among some of the most useful equations to use with imaginary numbers are the following formulas:





Notice that if we plug in a+bi then we see that the sine component of the imaginary number is b and the cosine part of is a.

While we didn't talk about it, De Movier's Theorem can be very useful, and for information on De Movier's Theorem (and for more on complex numbers) Dr. Schneider at Portland State University wrote a very article here.

Derivatives of Complex Functions

While complex analysis can easily become very difficult, here we shall look through it at a beginner's level.

Hopefully, you've had some experienced some calculus. If not, then this section may be a little difficult to understand.

Now a function is differentiable at a point, c, if  exists. Also, the function must be holomorphic (or an analytic) in a specified region (holomorphic implies that the function is infinitely differentiable).

This brings us to one of the interesting facts discovered by Cauchy and Riemann: If a function  is complex differentiable, then its real and imaginary parts satisfy the Cauchy-Riemann equations:



and



Where , , , and 

Well, with that let's practice this!

Find the real and imaginary part of the following expressions:

1. 



Real: -88

Imaginary: 166

2.      



Real: 

Imaginary: 

Plot the values of  for 

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Prove the magnitude of a complex number, z, is equal to 

Let , . Then .

Prove that  and that 





Prove De Moivre's Theorem. In other words, prove that if  is the magnitude of z and  is the angle between the real axis and the line that goes from (0,0) to z in an Argand Diagram, then 

Let . Then 

Use Euler's formula to prove that 

If we were to first look at the exponential form of these two equaions we would see that . Now, . For this to be true, the real part on the left has to equal the real part of the equation on the right. So, we see the following identities:

1. 
2. 

Use Euler's formula to write  and  in terms of exponentials





Focus on Math: Frieze Groups

Frieze groups are a mathematical concept used to explain patterns or symmetries arising in various forms of art including wallpaper and architecture. The term frieze refers to the band of decorative designs around a ceiling or another piece of work on a building. Frieze groups are very similar to the frieze form of architecture since both have a finite length (typically wide enough for one pattern sequence) and an 'infinite' width (naturally a frieze on building but the principle is still valid right up to the end).

There are seven types of frieze groups each with their own patterns. Each pattern depends on the symmetry of the group, whether it be translational, mirror, or reflection symmetry. A visual of each group may be viewed here.

Frieze groups are the 'one-dimensional' counterpart to wallpaper groups which have reoccurring patterns in two-dimensions. Frieze groups, as previously mentioned, only have seven different possible patterns. Similarly, the structure of atomic lattices (see lattice groups for more details) have reoccurring patterns which have properties that vary according to the structure of the lattice. Frieze groups are the simple one-dimensional case of repeated designs which appear all throughout history and their larger dimensional counterparts appear everywhere.