Complex numbers: time to turn on the beacon
Math is motivated by analogy and the desire to solve new problems, among other things. We'll see that this is the pattern that leads from real numbers to complex numbers and further to matrix algebra.
Complex numbers are not that complex
Problem. For any real number the square is non-negative. Therefore the equation doesn't have solutions in the set of real numbers.
Idea. Expand the set of real numbers in such a way that a) the square root exists in the new set of numbers and b) the properties of real numbers involving addition/subtraction and multiplication/division are preserved.
Solution to the problem. Step 1. Formally introduce the imaginary unit This implies, in particular, that and that everywhere is encountered it should be replaced by
Step 2. Formally introduce complex numbers as linear combinations of the real unit and imaginary unit with real Manipulate them using the properties of real numbers involving addition/subtraction and multiplication/division. For example,
Step 3. A complex number can be interpreted as a vector on the plane because summation of complex numbers
corresponds to summation of vectors. In particular, use the norm of the vector as the absolute value of
These formalities plus the geometric interpretation is all one needs to know about the set of complex numbers
Definition 1. The number is called a conjugate of the number
With this definition, we have from (1) and (2)
Besides, it's easy to see that if and only if This is the way to identify real numbers in the set of complex numbers:
(4) a number is real if and only if
Exercise 1. Express the ratio of two complex numbers in the form
The set of complex numbers corresponds to the whole plane and the set of real numbers corresponds to the axis. Similarly to we can define as the set of all vectors with components
Definition 2. The scalar product in is defined by
Exercise 2. Check that it has all properties of the scalar product in except that instead of one has where
Let us write
Here the numbers satisfy Therefore there exists an angle such that (see Figure 1, sorry about the notation discrepancy). This implies
Euler established a wonderful formula
For example, with we have an interesting relationship between three most important numbers in mathematics: Combining (5) and (6) we get the polar representation:
The additive form is better for addition/subtraction and the polar (multiplicative) form is better for multiplication/division. In particular, by (7)