Complex Numbers

1. i1 = i, i2= -1, i3= -i, i4= 1, i5= i, i6= -1 and so on.

2. Euler's constant e, when raised to the power of a complex number, gives a point in the complex plane according to Euler's Identity, e^(iθ

) = cos(θ

) + i*sin(θ

).

Whenθ

is replacedby pi, we get:

e^(iπ)

= cos(π

) + i*sin(π

)

= -1 + 0

= -1

In other words, e^(iπ)

+ 1 = 0.

3. Many intriguing fractals have been discovered in the complex plane, the most famous of which is the Mandelbrot Set. This is formed by an iterative process of a point in the plane. It produces a beautiful image, nicknamed by many as the "Thumbprint of God". (Google Images is your friend here - check it out.)

4. De Moivre's Theorem holds true in the realm of the imaginary, stating that

(r cisθ

)n= rncis nθ

.

This is a fascinating theorm in itself, but it also has hidden implications.

For example, if r = 1 (the point was 1 unit away from the origin), and you were to plot this point with a small angle above the positive real axis, then raising this number to an increasing power would result in that point making a full revolution about the origin and coming right back to where it started.

This is because rnstays the same (1^(anything) = 1), but the angle is slowly increasing as the power does. Therefore, as the angle increases, a circle is plotted around the origin with radius 1.

There are many other fascinations around complex numbers, but this covers the basics.

3 x 5 = 15

2 x 2 x 3 x 3 = 36

The GCF is 3.

X = 1.31356+0.612045*i

Steps to solve, take the natural log of both sides:

ln(X^(3-5i)) = ln(23-14i).

(3-5i)*ln(X) = ln(23-14i). Convert 23-14i to exponential form: A*e^(iΘ) {A = 26.926 and Θ = -0.54679 radians}

(3-5i)*ln(X) = ln(A*e^(iΘ))= ln(A) + iΘ = ln(26.926) - 0.54679i.

divide by (3-5i): ln(X) = (ln(A) + iΘ) / (3-5i) = (3.2931 - 0.54679i)/(3-5i)

So we have ln(X) = 0.370978 + 0.436033i, then:

e^(ln(X)) = e^(0.370978 + 0.436033i) --> X = 1.31356+0.612045*i