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.

Complex numbers and functions, Quaternions and Riemann Surfaces

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.

I was planning to write a long post and add some visuals on the subject for some time. But after watching the available videos in internet, I have decided that they don’t require any additional explanation and do a much better job than I could do. Therefore I will only give a brief and give links to those amazing videos.

(These videos are comparatively long for today’s video consuming standards so they can be pretty quickly boring for you. Watch them only if you are interested with this subject and want to learn what the concepts in the title really mean. Try to take your time and enjoy while watching. If you will put a timer in your brain to finish, you will never do.)

First let’s start with this awesome explanation about what the imaginary and complex numbers actually are and what they are useful for:

Once you understand the concept of imaginary and complex numbers, inevitably you will ask what about more dimensions than 2? How can I deal with rotations in higher dimensions? Apparently, this question gave trouble to great mathematician William Rowan Hamilton and after 10 years of looking for the solution for rotation in 3 dimensions, he finally has figured out that the answer actually lies in 4 dimensions instead of 3 and he came up with the idea of Quaternions. Here is another amazing video from the great Youtube channel of Numberphile:

If you will get deep enough understanding of what the complex numbers and quaternions are and continue to go deeper, you may figure out that each quadratic equation in even in their basic forms like y=x²+1 must have complex roots on the imaginary numbers axis. This can show you a way to imagine how can a function with a 2nd order have 3 dimensional visualization.

Even better, there are videos which garnish their explanation with very cool 3D visualizations like the one below. Ok, this is not a single video but a series of 13 great videos. What they show with that colored 3d quadratic equation in first video that you can see below is very cool, indeed. But what really has blown my mind is what they have did with that Phyton code while they were explaining the complex functions starting from video 10. If you will be patient enough and give yourself to what is being explained from the beginning, I promise that you will be rewarded. It will be as enlightening as the moment when Morpheus was explaining Neo what the matrix is. Here they are; Welch Labs presents, Imaginary Numbers are Real:

I forgot to say that starting from video 12, they explain what a Riemann surface is and how they are useful for our understanding. Especially, if you are into physics and Relativity, this is a must see for you. Those videos have finally gave me some clues about how to visualize the Riemann Surfaces.

That’s all for now. If I come to know more cool videos in future, I will keep updating this post with them.

Pythagorean Theorem and Cross Product of the Vectors in Higher Dimensions

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.

In my previous post about the Pythagorean Theorem, I have tried to give a different look at the subject. In summary; it was about applying the theorem in 2 dimensions on a coordinate system instead of a triangle and to show the geometrical evidence of it using one higher dimensional axis as in below sketch.

That gave us a new concept which is similar to the cross product of the vectors. In this post, I will pursue the same idea in higher dimensions than 2 and look for new perspectives.

To deal with a 3 dimensional object, I have set up my playground in a 3 dimensional coordinate system as in below sketch.

There is nothing unusual about it. I can set the lengths of x, y, z and carry on to apply Pythagorean Theorem to find the result and try to imagine its correspondence square in 4th dimension, which I already did in my previous post.

But I have noticed that it doesn’t take me anywhere. Maybe it is helpful to find insights about imagining the 4th dimension but while playing within this playground, I have noticed something else. First, I have connected x, y, z points as in below sketch and created some triangles.

You can notice that they resemble a tethrahedron but a face of it is missing. You can see that face marked as d which is made of x, y, z points as below.

There is something very similar with our first Pythagorean Theorem application with the 1d lines. But instead of lines, this time 3 pieces of triangles are resulting in a 4th one (d). I have wondered if Pythagorean Theorem was also applicable to the areas of these triangles. I have created a simple excel sheet where I can enter different values for each edge length and can easily see the results for different values quickly. That exercise has showed me that that it works as in below equation.

a² + b² + c² = d²

That’s strange. Nobody has taught us that the Pythagorean Theorem is also applicable to the the areas of the triangles in such an arrangement. Probably that was because they didn’t find it useful or thought that the 1d application of it is all we needed. In any case, there is definitely something to pursue.

After noticing that, I have tried to apply the geometrical proof of 1d application into this one and I have quickly failed. That was most probably because taking the square of a 2 dimensional object requires to imagine a 4 dimensional one which we are not quite capable of.

After that, I have wondered, if the same logic can be taken into further steps. For example, can we apply it to volumes? But there is one problem with that. Since our scenario about 2d surfaces above had to be set in a 3d coordinate system, to apply it to the volumes, we need to set a scenario in 4d coordinate system, similarly. The problem is, we are not capable of visualizing a 4d coordinate system effectively. To overcome this problem, we need to push the limits of our imagination a bit. In below sketch, you see an illustration of a 4d coordinate system with 4 axes which each are perpendicular to the others.

In order to test that if we can apply Pythagorean Theorem to the volumes, first we need to define what we are looking for.

• 1d application of Pythagorean Theorem, was giving us the length of the hypotenuse of a right triangle by taking squares of the lengths of 2 short edges.
• In surface application of it, we have shown that if we take the sum of squares of the areas of 3 triangles in a 3 dimensional environment, we can get the square for the area of their interconnecting triangle.
• Now, using this principle, we can claim that; in a 4 dimensional coordinate system, taking the sum of the squares of the 4 tetrahedrons should give us the square of the volume of the 5th interconnecting volume of the 5-cell.

Since it is difficult to express this idea by words, I will try to explain it by sketches. But before that, since talking about higher dimensional objects’ volumes and surfaces is a very tricky subject, we need to declare our common language.

The difficulty is; while it is very clear when you talk about the length and area of a 2 dimensional object like triangle, it quickly becomes confusing after 3 dimensional objects. Because, for example; a 4 dimensional object has a surface area, surface volume and as an additional concept; a hyper volume. This is where things get mixed. In order to avoid that, people have invented clever ways to name those concepts. One of these ways is, instead of giving individual names to each deeper! volume of higher dimensional objects, we can call them simply as face and give them face numbers as shown in Wikipedia page of the Simple.

So according to this table, whenever I say;

Area of a 1 face, it will mean the length of a line,

Area of a 2 face will mean the surface area of a triangle,

Area of a 3 face will mean the volume of a tetrahedron,

Area of a 4 face will mean the hyper volume of 5-cell and so on.

And whenever I use them in a calculatio,n I will use them in that fashion: |xyzw…| number of letters will indicate the number of faces.

Deal? Ok let’s back to our subject. Here is our 4 dimensional object, a 5-cell:

Looks confusing, right? That’s expected since it is a 4d object which our minds are not familiar with. As I said, you have to push your imagination to its limits in order to visualize it. If you cannot, no problem. We can use our previous tetrahedron example as an analogue to our 5-cell in order to understand what are we dealing with.

Surface of 3d tetrahedron is covered with 2d triangles, Similarly, a 4 dimensional 5-cell is also covered with 3d tetrahedrons. So, the a, b and c triangles of below tetrahedron are analogue to the volumes of our 5-cell:

So, the a, b and c triangles in below sketch,

Are analogue to the below volumes of our 5-cell:

And the interconnecting d triangle ( |xyz| ) in below sketch,

Is analogue to interconnecting 5th |xyzw| volume below:

With this volume, the number of volumes in a 5-cell becomes 5 in total, which gives its name.

So, what we need to do is to verify if below equation is correct or not:

|xyzw|² = |oxyz|² + |oxyw|² + |oxzw|² + |oyzw|²

So, if we calculate those volumes for below values using the determinant of (A . ( B x C)) / 6 formula:

x: 2

y: 3

z: 4

w: 5

|oxyz| = 4

|oxyw| = 5

|oxzw| = 6,6666…

|oyzw| = 10

|xyzw| = 13,6178…

Which proves our formula by;

13,6178…² = 4² + 5² + 6,6666…² + 10²

I guess that, it gives us enough evidence for our case. I don’t know if this has been previously declared anywhere before. If not, I would like to call it Pythagorean Simplex Conjecture and declare it as;

In a n dimensional simplex; square of n-1 faced hypotenuse area is equal to sum of square of all other n-1 dimensional faces’ area.

I don’t have enough mathematical skills to write it as an equation. If you are interested and have an idea about how to express it, you are welcome to the comments section.

P.S. After a careful read at Pythagorean theorem page in Wikipedia I have come to know that this has been previously thought as you can see in below picture. That means; a) there is no need to name it as a new conjecture, b) I have discovered an already thought concept once again. What can I say, it was a good exercise.

And from the page of De Gua's theorem:

So it is possible to write it as a general equation like that.

P.S.2 As I understand, De Gua has declared this only for 3 dimensional tetrahedron and didn’t generalize it for higher dimensions. I don’t know if anybody else did it before but I could find this paper in internet which is generalizing this idea to all dimensions and it is dated 2002. It seems like I wasn’t 300 years late but just about 14. Not that bad.

Pythagorean Theorem and Cross Product of the Vectors

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.

In previous post, I have mentioned that I will explain the motion based relation of dimensions in my next post. I believe that this is the most counter intuitive idea which I came up with long time ago , but since then I couldn’t figure out a way to explain it in a meaningful way. Recently, while I was trying to understand the Pythagorean Theorem in depth, I have noticed that all I needed was to apply Cross product to the vectors in my model.

When somebody talks about the vectors, an average person usually becomes dull. In our daily lives, we usually don’t use vectors and scalar quantities like speed, distance, weight are enough for all of us. Therefore, explaining something using vectors can be meaningless to the most. But what I have noticed while playing with the Pythagorean Theorem, even its geometrical explanation with a different interpretation can be useful to visualize and understand what the cross product means. Let’s see how;

We have been thought in the school that the Pythagorean Theorem is a useful way to find the longest edge of a perpendicular triangle and its geometric explanation is usually given as in the below sketch:

There are even other geometrical convincing proofs like the below one. You can see more inPythagorean theorem Wikipedia page.

Simple, isn’t it? That means, sum of square of the short edges’ lengths on a perpendicular triangle is equal to the square of the long one’s length. With this formula, once you know the length of any 2 sides of a perpendicular triangle, you can find the 3rd one. Very useful for the geometry problems.

Interestingly, this theorem is also applicable in higher dimensions. You can find the derivation of it in 3 dimensions below:

As per our previous proof, it is clear that the red line’s length can be calculated from the x and y coordinates of that triangle. After calculating that length, by using the z coordinate we can also calculate the length of the brown line which is located in a 3 dimensional coordinate system. It is obvious that the same logic can be even taken further into the higher dimensions. What a useful idea.

But now our picture has changed from our starting one in 2 dimensions. In 2 dimensions, we were using it on a triangle but now we have started to use it in a coordinate system. What if, we step back and use the coordinate system logic in our 2D example again. Take a look at the below sketch to understand what I mean:

Now, everything seems a bit different even though our formula still applies. The perpendicular triangle is no longer there. You can only have it by translating the blue “a” line to the location of dashed red line. Then the triangle is visible. But, without having that triangle, how can we apply the first geometrical proof that we have seen? You see 2 attempts for that below:

It is not as nice and neat as we did before, isn’t it. Our green square is messing with the squares of the other sides and it losses the true meaning of our proof. I wish there was an extra space that we could locate our green square.

Then I saw it. There was an available extra space; 3rd spatial axis! If I place our square along that axis, it would not mess with any other squares. That kind of thinking literally gave a new depth to my look at the problem. Check out the below sketch to see what I mean:

Seeing it this way, have changed my look at the Pythagorean Theorem completely. Before, the whole theory was about finding the hypotenuse but now it seems like the real result of it lies on the z axis and only its geometrical proof which is the c² sized area has its one side on the x-y plane and this could be interpreted as an equivalent to the hypotenuse after translating a or b to the dashed locations.

After seeing that, I had the pleasure of discovering a paradigm shifting concept for a while. That was before I find out its relation to the cross product in the Wikipedia page of Pythagorean Theorem. Then I have started to remember our classes in the high school where they were teaching us about the vectors and how to take their products. Then all these ideas started to make sense (by leaving me wondering; why they don’t teach us this stuff like that at school).

I wanted to take this one step further and try to see how it could be applied to 3 dimensions. I have figured out the directions for the squares of the lengths along x, y, z as in below sketch. They are shown with blue, yellow and green shaded squares and not to mess the big sketch, I have given their intersection view next to it separately.

(I know, some of them doesn’t look like a square at all, but this is the best I could draw after a half day of struggle, so please assume that green, blue and yellow shades are squares.)

Now, there is one important thing is missing compare to our previous 2D version. As the theory says, there should be a d² sized square which its area is equal to the sums of a²+b²+c². The problem is, there is no available space to draw that into. In our previous 2D example, we also couldn’t find any available space and had to use the axis in one higher dimension. Likewise, we should also draw our d² square onto the 4th spatial axis but unfortunately we don’t have one.

Time direction could be a good one to draw that onto, since it is also known as 4th dimension but instead of calling it as 4th dimension, physicists today name our universe as space-time, meaning it has 3 spatial dimensions + 1 temporal time dimension. That means, time dimension cannot be treated like a spatial one. Although, in general relativity, it is possible to convert it into a spatial dimension by multiplying it with a constant called “c”. This constant is speed of light. Coincidentally, this is exactly the same thing to do for calculating the distance we have covered on time axis, if we were travelling in that direction with speed of light as in the universe model which I currently try to work out. Things start to fit perfectly and its my main reason of writing this post about the Pythagorean Theorem. I will use it in my future posts for further explanations.

Even if you are not interested in new universe models, I hope this post could give you some new look on a very well known theorem.