Complex numbers are among the most important in mathematics, so it’s worth spending some time to get to know them. Actually, complex numbers have developed from what were originally called “imaginary numbers” based on trying to solve the simple quadratic equation:

x^2 + 1 = 0

Of course, one gets:

x^2 = -1

but when one solves for x, he obtains:

x = (-1) ^1/2 or the square root of minus 1. This is defined and referred to as i.

Thus: i×i= -1

i× i×i= -1i

i× i×i×i = 1

and so on.

The importance of complex numbers, before we get started, is relevant to everything from the quantum wave function to oscillating-alternating circuits and wave forms. For example, the representation of monochromatic plane waves in free space:

E = E(o) exp[i(wt – kz)]

Or the voltage at the end of a transmission line:

V(x=0) = A exp iwt[1 + K]

Where K is the voltage reflection coefficient.

Anyway, an important diagram for use with analyzing the properties of complex numbers is the Argand diagram – such as shown in Fig.1. Here, several vectors are shown. The key point is that any complex number may be written: x + iy, so that we see the imaginary axis will be associated with the y-axis, and the real numbers with the x-axis.

In this first instalment we will focus on the vector C. Later, we will look at vectors A and B and then see a way to combine all the vectors in the complex plane.

As can be seen on inspection, the vector C may be written in the form:

C = 4 + 3i

A critical part if finding the angle shown – referred to as the argument. We can see from the diagram that the angle may be found using:

arctan (y/x) = arctan(3/4) = 36.8 deg

Thus, theta = 36.8 degrees is the argument

Now, any complex number (x + iy) may be written in polar form:

x + iy = r(cos (theta) + isin(theta))

to get r:

r = [x^2 + y^2]^1/2 = [4^2 + 3^2]^1/2 = [25]^1/2 = 5

Therefore we may write:

(x + iy) = 5(cos (36.8) + isin(36.8))

Note there is also the abbreviated function (based on the combo sine and cosine):

cis (theta) = cos (theta) + isin(theta)

so we can finally write:

C = r cis(theta) = 5 cis (36.8)

More fun to come!

x^2 + 1 = 0

Of course, one gets:

x^2 = -1

but when one solves for x, he obtains:

x = (-1) ^1/2 or the square root of minus 1. This is defined and referred to as i.

Thus: i×i= -1

i× i×i= -1i

i× i×i×i = 1

and so on.

The importance of complex numbers, before we get started, is relevant to everything from the quantum wave function to oscillating-alternating circuits and wave forms. For example, the representation of monochromatic plane waves in free space:

E = E(o) exp[i(wt – kz)]

Or the voltage at the end of a transmission line:

V(x=0) = A exp iwt[1 + K]

Where K is the voltage reflection coefficient.

Anyway, an important diagram for use with analyzing the properties of complex numbers is the Argand diagram – such as shown in Fig.1. Here, several vectors are shown. The key point is that any complex number may be written: x + iy, so that we see the imaginary axis will be associated with the y-axis, and the real numbers with the x-axis.

In this first instalment we will focus on the vector C. Later, we will look at vectors A and B and then see a way to combine all the vectors in the complex plane.

As can be seen on inspection, the vector C may be written in the form:

C = 4 + 3i

A critical part if finding the angle shown – referred to as the argument. We can see from the diagram that the angle may be found using:

arctan (y/x) = arctan(3/4) = 36.8 deg

Thus, theta = 36.8 degrees is the argument

Now, any complex number (x + iy) may be written in polar form:

x + iy = r(cos (theta) + isin(theta))

to get r:

r = [x^2 + y^2]^1/2 = [4^2 + 3^2]^1/2 = [25]^1/2 = 5

Therefore we may write:

(x + iy) = 5(cos (36.8) + isin(36.8))

Note there is also the abbreviated function (based on the combo sine and cosine):

cis (theta) = cos (theta) + isin(theta)

so we can finally write:

C = r cis(theta) = 5 cis (36.8)

More fun to come!

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