Imagine the Imaginary Numbers

Quick Introduction

Actually, you need to know that Mathematicians don’t like this terminology ("Imaginary Numbers"). They are not imaginary. 

The second important thing to know is that these numbers were invented (or discovered) to complete our numbers system. Some mathematical problems didn’t have a solution before the "Imaginary Numbers". For example, the square root of negative numbers.

This video will give you a nice imagination and 3D representation of an imaginary number function.

Application in Chemical Engineering

Through more than 10 years of industrial experience (Fertilizers, Refining, consultancy, petrochemicals), through B.Sc. in chemical engineering, and through M.Sc. in Mathematical Modeling of processes, I cannot recall that I used the "Imaginary Numbers". Except for one time.

The only one time I saw "Imaginary Numbers", I was developing my Peng-Robinson fluid package tool, this tool is an Excel add-in which calculates different physical properties of hydrocarbon mixtures. One of these properties is the Compressibility Factor (Z). Z is calculated by a 3rd-degree equation, when you solve for Z you should have 3 solutions (roots), here you may see the "Imaginary Numbers", some of the roots may be imaginary and at this point, you will neglect them. So, you don’t really need them. :D

However, at least for me, "Imaginary Numbers” still an interesting topic and they might be useful for chemical engineers who would like to know more about Process Control, they are essential for understanding different topics like Laplace transformation.

The need for "Imaginary Numbers”

Before we start and to be on the same ground, we will agree that the solution (root) of an equation is the intersection with the X-axis this is what we call the graphical solution, also there should be a number of solutions equal to the degree of the polynomial, for example, 2nd-degree polynomial should have 2 solutions. 

Now, to imagine the "Imaginary Numbers”, let’s consider this simple equation (x^2 + 1 = 0) which can be graphically represented as below. It is clear that there is no intersection with the x-axis, consequently no solution. Simply this means that there is no number (real number) that can make this equation equal to zero, this is at least from the real numbers point of view.

So, while expecting to see 2 solutions as this function is 2nd-degree polynomial, we get no solution! Here is where things start to be interesting, here is where we will use "Imaginary Numbers”.

The graphical representation of "Imaginary Numbers” functions

In the above graphical representation, we have used one dimension for the input x, and one dimension for the output y. However, when we start using "Imaginary Numbers”, we will need 2 dimensions for the input x (x real, x imaginary), these 2 variables will represent the 2 part of the input Imaginary Number x real + i * x imaginary. And on the other hand, the output also will need 2 dimensions (y real, y imaginary). This creates the need for 4 dimensions to illustrate this function. In a 3D world, it is hard to imagine how 4D will look like. therefore, the two methods below can help in this imagination. 

Some software can produce a 3D graph using the available 3 dimensions and then uses color to represent the 4th dimension. For example, by replacing the real input by imaginary input, the given function above x2+1 = 0 was represented by MATLAB as the below figure, where input x real on the x-axis, input x imaginary on the y-axis, output y real on the z-axis, and output y imaginary is represented by color. 

 The other method, which I prefer, is using Excel. Excel can’t create figures like the one above, it can’t assign colors based on the values of the 4th dimension, it rather assigns the colors based on the value of the 3rd dimension. To overcome this, 2 figures will be produced. The 1st figure can represent 2 input dimensions with the 3rd dimension representing the output real part. The 2nd figure will represent 2 input dimensions with the 3rd dimension representing the output imaginary part.

Mathematics Software also can produce 2 curves as Excel, below is the output of Mathematica. 

The graphical solution of "Imaginary Numbers” functions

As we agreed, the solution is the intersection with the x-axis, in other words where the z-axis equals 0. So here we need both outputs real and imaginary parts equal to zero. 

For the imaginary part, the figure below, we see that the value of the output will equal zero if any of the input parts, real or imaginary, equals 0. This can be represented by the blue and green lines. 

By applying these two lines on the real part output figure below, we will have the two lines which represent the 0 value of the imaginary output part. So the solution will be the intersection of these lines (green and blue) with the lines representing 0 value of the real output part which is the red line. Note that there should be another identical red line on the other side of the curve but is not shown below. 

We will notice no intersection between the green and red lines, and only 2 intersection points between blue and red lines (one shown above and another identical one on the other side of the curve). These 2 points can be shown on the Side View curve below. These 2 points (0 + 1*i and 0 - 1*i) represent the 2 solutions of this function as both output parts equal 0. This matches our expectations to have 2 solutions for this polynomial. 

Reference

This work was inspired by a very nice youtube playlist which discusses the imaginary numbers and their history.