1. Geometry and Complex Arithmetic

Introduction to Complex Numbers

The general cubic equation can be expressed in the form:

and Bombelli considered , which yields

Euler’s Formula

Consider the power series of :

So just use substitution for in and compare with , then we can prove

and we also can get two formulas and .

Problems

Deduction of the Cubic Root Formula

Use translation because of the point of inflection. We can transfer the question to solve

Use Lagrange pre-resolution, which is a translation

then we have

and then the question transfers to finding the and to meet that equation, and we assume is true and use to form an equation since then

Use the substitution , then we simplify the cubic equation to a quadratic equation, and finally we get

What’s more we also can use equation and the root will be in the form of

Geometric Proof

Show geometrically that if then

We can decompose the into . Because of observation, we know the angle of is twice that of , then we can prove it.

The Constant-Argument Locus

Explain geometrically why the locus of such that

is an arc of a certain circle passing through the fixed points and .

In words this means that the angle between the vectors and is constant.

Equilateral Triangle Condition

By using pictures, find the locus of for each of the following equations

We can deduce

and also

and then

finally we can deduce

i.e. the triangle is equilateral.

Summing Cosines of Odd Multiples

Let . Show that

Method 1. We can consider

then

Method 2 (Geometric series). Consider the series

which can be written as

we have

Use the equation

to rewrite the numerator and the denominator, we can get the same result.