Complex Numbers
It is complex, but not imaginary…
complex numbers
Complex numbers are a brand new group of numbers and involve something unique called the “imaginary number”. This term already sounds not serious and something out of blue. But understandably, a tool to solve newly developed problems has to be innovative as well.
In this section, we want to present to you with some basic ideas about complex numbers to bring you closer to this new section of mathematics.
A little post-it at the start notes down some most important information.
Cheat sheet
complex plane and imaginary number i
We can represent all real numbers by the real axis(1D). To complement the real numbers(with unit 1), we want to introduce another axis, the so-called imaginary axis, with unit i (the imaginary number). The imaginary number has the following property.
With these two axes, we have spanned a 2D plane(the complex plane). And any point on this plane, which is a linear combination of “1” and “i”, is a complex number.
You see here: the complex plane is really similar to the cartesian coordinate system with x and y axes.
- real axis with unit 1 and imaginary axis with unit i.
- We can specify a unique point in the complex plane by a set of numbers. Ex. (3,4) in the complex plane means 3+4i.
- x axis with unit 1(x) and y axis with unit i(y).
- We can specify a unique point in the x-y plane by a set of numbers. Ex. (3,4) in the x-y plane means 3 units of x and 4 units of y.
Since we know now that a complex number is composited by two parts, one of unit 1 and the other of unit i, we can take a closer look at the definition of a complex number.
z = a+bi
- real number a(real part), how many units 1
Re(z) = a
for a = 0, we call the complex number pure imaginary.
- real number b(imaginary part), how many units i
Im(z) = b
for b = 0, the complex number has no imaginary part. This is our old friend, a real number.
We can represent all real numbers by the real axis(1D). To complement the real numbers(with unit 1), we want to introduce another axis, the so-called imaginary axis, with unit i (the imaginary number).
With these two axes, we have spanned a 2D plane(the complex plane). And any point on this plane, which is a linear combination of “1” and “i”, is a complex number.
addition & subtraction
In the x-y plane, we can also use a set of numbers to represent a vector and perform various calculations. In the complex plane, the complex numbers can be added/subtracted in the same ways as vectors.
To understand what exactly is happening, we’ll use from now on x1,y1 to denote the real part and the imaginary part of the complex number z1.
And the same for subtraction.
multiplication
We define the multiplication of two complex numbers as follows:
We can simply derive this:
division
For division it holds:
To cancel out the imaginary part in the denominator, we made use of the complex conjugate of a complex number. For a given complex number, it is obtained by changing the sign of the imaginary part.
- When a complex number is added to its own complex conjugate, the result is a real number.
- When a complex number is multiplied by its own complex conjugate, the result is a real number.
complex numbers and rotation
After some definitions and properties, it really didn’t bring us any closer to the intuition. It is still confusing and not encouraging at all. But probably you can call the same feeling when you first met the family of negative numbers, say, ten years ago. Back then, in the good old days, we upgraded our number system. They gave answer to a new problem: what is a like to have something less than nothing?
So today’s new problem that the family of complex number answers is how to take the square root of something less than nothing. Let’s consider the following question:
And if we consider that with help of a 1D number axis, the answer is to span a perpendicular axis and rotate each time by 90 degrees. Each rotation corresponds to multiplication with i.
i corresponds to a 90-degree rotation. We can imagine the family of different complex numbers corresponds to different rotation angles. Since we observed some equalities complex numbers and vectors share, we can use complex numbers to rotate vectors. For example, we want to rotate (2,1) by 45 degrees.