Imaginary Numbers
- Invented to solve problems where an equation has no real roots, e.g.,
. The idea of declaring the existence of a quantity such that allows us to express the solution as:
The set represented by
Powers of i
If
Therefore, we have the sequence:
Complex Numbers
A complex number is just the sum of a real and an imaginary number:
Operations on Complex Numbers
Given two complex numbers:
Addition and Subtraction
Product
Given the complex number:
Norm (Modulus or Absolute Value)
Complex Conjugate
The product of two complex numbers where the only difference between them is the sign of the imaginary part is:
This quantity
Inverse
Multiplying the numerator and denominator with the conjugate of
Square Root of
We’re trying to find a complex number
Assuming that
Therefore:
Equating real and imaginary parts:
Therefore,
Finally, the value of
The value of
Matrix Representation of a Complex Number
The matrix
which can be written as:
Where
The matrix representation of
To find the matrix representation of
Squaring the following matrix gives the matrix above. Then the value of
Finally, the value of
The Complex Plane
The powers of
We can see that the positions of
e.g.,
A complex number is rotated
by multiplying it by .
Let’s graph the roots of
We can see that
Let’s multiply the complex number
Multiplying
Which is exactly what we find if we multiply
A complex number is rotated
by multiplying it by .
A complex number is rotated
by multiplying it by .
Polar Representation
Instead of using coordinates in the complex plane, we can represent a polar number with the length of the vector from the origin to the complex coordinate and the angle between the complex vector and the positive real axis:
The horizontal component of
Euler provided the identity:
Which allows us to represent any complex number as:
Given two polar numbers:
Their product is:
Which effectively rotated the complex number
A rotor is a complex number that rotates another complex number by an angle
(through multiplication) and has the form:
Rotating a complex number
Which in matrix form is:
Note that because of the way the complex product is defined, the multiplication between two complex numbers commutes: