Where to find the Complex Roots of Quadratic Equations in Our Lives?

Below post is a back-up of my posts in my Quora blog. I take a copy of them to here in case if something goes wrong in there.

After this title, I won’t be surprised to hear someone saying; “We were expecting that you will go nuts one day but didn’t expect it that fast”. I definitely see your point but my time has not come yet. In this post, I will try to convince you that I am still sane and why my title makes sense (well, kind of).

In my previous post, I gave links to some videos. A part of it was about the imaginary and complex numbers. While digesting the rotation related meaning of complex numbers, I have noticed one important and overlooked issue about the complex roots of quadratic equations. But before going into that, let’s first quickly went through what is a quadratic equation.

Quadratic equations are the ones which has the forms of ax²+bx+c. The most important part here is our equation to have a variable which has the highest power of 2. Therefore, b and c can be equal to 0 but a cannot. So, let’s have a look at its simplest form with

a=1, b=0, c=0: y = 1x²+0x + 0

Which we can simplify as:

y = x²

So, it has the shape of a parabola. If we play with rest of the equation and change the value of a, b and c, we simply change the scale or location of our parabola. In summary, while changing the “a” value, our parabola becomes thinner or wider, changing the “b” and “c” values pans the location of parabola on our x-y coordinate plane.. If you think a little bit about that, you may easily notice that each and every parabola is actually the same in their essence. This is not something I claim on my own:

Great mathematician and lecturer Matt Parker explains in this video, why There is only One True Parabola.

Alternatively you can also watch the same idea in this video.

So, what all these things have to do with the complex numbers? One thing they have taught us in high school about the equations is that they always have the same number of roots as with the highest power of variable in them. In this case, a quadratic equation must always have 2 roots since x² has the highest power. This rule becomes a problem with the equations in the form of y=ax² + c because for y=0, its roots must be equal to x=± square root of (-c/a). Since there is “–c” term inside our square root, that means our roots are complex numbers. More interestingly, when you try to geometrically explain them, those roots must lie on an imaginary x axis which is perpendicular to both x and y real axes. Something like this:

In this picture, the red parabola is showing the real part of our y=x²+2 quadratic equation while the black parabola is the imaginary part of it. Now we have a much different picture than our first single parabola. Everything is suddenly in 3d. There is even more to that. The first video in that video series which I have given at the end of my previous post has a very cool 3d effect to show all of this structure and it is pretty amazing. You can watch it starting at 1:40 of this video in action or have a look at below image. :

The reason of this 3D effect is our 1 dimensional x axis has turned into a 2D plane (just as he pulled it out from the surface of the paper in the video) as soon as we added imaginary x axis in addition to real x axis. And these things have happened because our parabola was above the x axis and had complex roots on imaginary axis since it wasn’t intersecting the real x axis. But wait a moment. We have seen the evidence of all the parabolas being the same at the beginning of this post. If so, what is so special about the parabolas which don’t intersect with the real x axis?

The answer is; absolutely nothing. Intersection with x axis is only a tool for us to determine the location and scale of the parabola. As previously said, there is only one true parabola. Therefore, doesn’t matter if it intersects the real x axis or not, all parabolas as a result of quadratic equations have their imaginary part independent from their scale and location on the coordinate system. Even if they intersect the real x axis, they have the imaginary part. They look something like the dashed black parabola seen at the bottom of this picture:

So, what am I trying to reach with all those things? I am not going to claim any relation with these quadratic equations and any physical structure that we meet in our real life like those ones:

Equations are about varying values, so they have a dynamic structure. Therefore, I won’t claim that in our real life, there are imaginary parts of physical objects which somehow took the form of a parabola. Because they are static and they are what they are, no variables. But there are other concepts in our real life which has a dynamic structure just like a quadratic equation and in the form of a parabola. For example; displacement function of an accelerating object. When we are accelerating, our displacement curve is actually a parabola and can be written as ∆s= 1/2.a.t²+Vo.t + So. ∆s is being our total displacement, “a” is the acceleration, “Vo” is starting velocity, “So” is starting point and “t” is time. “t” is our variable here and replaces the x in our previous quadratic equations. In this form, it is clear that displacement function of an accelerating object is in quadratic form and has a parabolic structure. It is not a physical structure in this case, so we can’t claim the existence of something physical in an imaginary axis out of nowhere. But I believe that there is something worth to investigate and it also has something to do with my theory about the true nature of dimensions. I have not much in my hand other than this idea, yet. If I will come up with something, I will have a new post about it and inform you here.

P.S. This observation is not limited with the quadratic equations and the equations which have more than power of 2 also have more than 2 complex roots, accordingly. I think, they also correspond to higher time derivatives of acceleration like jerk, jounce, etc. Even more interestingly, this logic also works with the powers with fractions of 1 in the form of x^(1/n). So as long as the power of x is not 1 or 0, imaginary part exists. But not to make things more complex, I didn’t mention about this. I just wanted to add this note to let you know that I am aware of it.