Let us examine numbers as distances from some number.

If we choose as starting point the zero (0), we have the usual case.

0 = |0 – 0|

1 = |0 – 1|

2 = |0 – 2|

…

*n* = |0 – *n*|

Each *distance* can be divided into parts, let’s think the number space as the familiar set **R**. For example the distance from zero to two can be divided in *n* parts, we get distances 2 / *n* between zero and two.

The point from which we started isn’t a distance but — like said — a point, that can’t be divided into smaller parts. Philosophically speaking this point is now in words of Democritus “atomos”, the smallest unit.

In the book “The way to geometry” it is said: “The magnitude of a point is zero.”

When we founded the numbers as distances in relation to zero, the zero point is indivisible; because in question is a point not a distance, dividing zero (a point) by any number, we get zero. Zero doesn’t get smaller by dividing.

Of course zero can’t be divided by itself. In distance interpretation philosophically speaking indivisible would be divided by indivisible.

Now zero is different from other numbers so that it is a point, the other numbers are distances from this point.

As distance interpretation we could start the numbers also in relation for example to number eight (8).

Now eight is a point, we get the other numbers as distances to number eight.

0 = ||8 – 8| – 0||

1 = ||8 – 8| – 1||

2 = ||8 – 8| – 2||

…

n = ||8 – 8| – n||

Here from number eight is “created new zero”. First we determine the distance of number eight from itself, then the distance of examined number from the determined distance earlier..

We could get the numbers (distances) and the zero point also by denoting:

0 = |8 – (8 + 0)|

1 = |8 – (8 + 1)|

2 = |8 – (8 + 2)|

…

n = |8 – (8 + n)|

When we see the numbers as distances from some number, we notice, that the distance can be divided into smaller parts, but the point from which we started, we can’t divide into smaller parts.

In our example when we started from eight, we examine first the distance of eight from itself, |8 – 8| = 0, which is and remains as zero.

The distance can be even or uneven, but the point from which the distance is determined can’t be even or uneven, that is the whole point of this post.

If we want also the negative numbers as distance interpretation, we simply put the minus sign before the determined distance, for example -2 = -|0 – 2| and this is how we can get the entire set **R** so that the distances from zero point can be divided into smaller parts, but zero point is and remains zero and is not a distance but a point. In the words of Euclid that which has no part. In my own words: “Indivisible”.

In this way the specialty of zero can be seen better.

The title is a bit absurd, we could examine the distance of any number from itself..

In the previous post I examined the property of distance being even or uneven. Let’s go back to distance of zero from itself. Formally the distance of zero from itself is |0 – 0| = 0. 0 mod 2 = 0, why isn’t this distance even?

The distance doesn’t exist, instead of distance we have a point. Euclid’s definition to a point is: “A point is that which has no part.” In Euclid words the “distance” of zero from itself, |0 – 0| = 0 has no part. Something like this isn’t even or uneven.

In an e-book called “The Way To Geometry” it says, that magnitude of a point is zero.

Philosophically speaking a point is what Democritus called “atomos” applied to geometry.

In short, there isn’t distance of zero from itself. The meaning of distance |0 – 0| = 0 is geometrically a point, something that has no part.

The main point is, that I see zero as a neutral number. What does that mean?

- Zero isn’t even or uneven
- Zero isn’t negative or positive
- Zero doesn’t have numerical opposite

*The meaning of the video clip: “..achievements: Zero.”
*

*Probably zero is my greatest*

*achievement too…*🙂

Semantically zero is *none*, not nothing.

In multiplication zero overrides the neutral element property of one: 1 * *a *= *a*, but if *a *= 0, zero “zeroes” one. The number one doesn’t keep the identity of zero, but zero “zeroes” number 1’s neutral element property and keeps the identity of its own.

Let us examine the even and uneven property of zero from point of view of distances.

The distance of number 2 from zero is |0 – 2|= 2. 2 mod 2 = 0; 2 is even.

The distance of number 3 from zero is |0 – 3|= 3. 3 mod 2 = 1; 3 is uneven.

The distance of number 4 from zero is |0 – 4|= 4. 4 mod 2 = 0; 4 is even.

But what about our friend zero?

The distance of number zero from itself is |0 – 0| = 0. 0 mod 2 = 0, but now the meaning of zero is, that the distance doesn’t exist. There isn’t distance from zero to zero. The distance that doesn’t exist isn’t even or uneven. Therefore now zero can’t be considered even or uneven.

I came up last sleepless night with a fractal of my own. There are two versions of it. The first version has the following rules:

- Build a circle with radius of
*r* - Draw inside the previous circle a circle of radius of
*r*/ 2 - Draw circles below, above and sides of the previous circle, the radius of these smaller circles being
*r*/ 4

Choose for a new circle the circle that’s radius is *r / *2. Repeat the previous steps *ad infinitum.*

Unfortunately on WordPress.com I can’t add the JavaScript-code where you could change the amount of iterations, so let’s just settle for pictures.

This kind of fractal might be invented already, but I haven’t seen one ever before.

Version 2 of this fractal:

The previous has a bit more complicated rules.

I made two color version of the previous too:

The second one:

Last night I was a bit playing with these fractal images in JavaScript, below is one more picture:

Perhaps this fractal of mine has at least some value of curiosity.

I try to find the time sometime in September to come back to this topic.

The general form of a equation of second degree is

*ax*^{2 }+ *bx* + *c *=* *0

In this post we’ll deduce the general solution formula to that.

The constant *c *just shifts the curve vertically in the *xy*-plane.

Let’s start by determining the slope of the general case, which is given by the derivative of the general case: 2*ax* + *b*.

The zero point of the derivative gives the symmetry point of the general case. Let’s examine parabola *x*^{2 } – 2*x *and its derivative 2*x* – 2:

The zero point of the derivative is at *x *= 1. We notice that the symmetry point of the parabola lies at this *x-*position, the roots of the parabola are at *x = *0 and *x = *2.

In general the symmetry point of equation of second degree lies at the zero point of its derivative.

Now we are ready to deduce the general solution formula.

Let us solve the zero point of the derivative: *x *= –*b* / 2*a.*

Next we put this in the general equation, and we get:

*ax*^{2 }– *b*^{2 }/ 2*a+* *c = *0

By solving the *x*, we get

The formula isn’t ready yet, but we know how the derivative is related to this and how it behaves.

The zero point of the derivative behaves symmetrically to the previous, so all we have to do is subtract the solved *x *from the symmetry point and add the solved *x *to the symmetry point. We get:

That’s it! We have general solution formula of the equation of the second degree.

* * *

I hope I haven’t chosen any funny words in this article. I don’t have my Finnish-Swedish-English math dictionary with me right now… 🙂

I have to correct some things I’ve written about the empty set.

Let’s start. Let A = {0}. Now A \ {0} = {} = Ø.

That is: We get the empty set. Let’s go beyond that.

I’ve written in this blog earlier ”emptyness can be created, ‘nothing’ can’t be created; nothing is from which the creation begins”.

At the beginning of this post, we created emptyness, we got the empty set as a result. By the means of the set theory we can’t get rid of the empty set. It’s as empty space as we can get. Also, in the poetic thought of mine I’ve written, that emptyness can be created, ‘nothing’ can’t be created.

So, we can’t get to this ‘nothing’, we can just see, that it is a ”state” that is ”before” the empty set, for the sake of perfectness. 🙂

A symbol to this ”state” could be

The order: ‘nothing’ → emptyness → something.

This new “collection of nothing” can’t be considered really a set; it actually doesn’t exist, but still, it is there — at least it was… Perhaps it will be… Somewhere…

In the previous post I wrote, that as far as I can see at the moment, semantically zero is *none*, not nothing. I’ll clarify that a bit. Let’s ask a question: How many? None.

The opposite to “none”, might be everything of some finite amount (in the sense of the set theory). But as I wrote in the previous post, zero might be considered neutral in the way, that zero doesn’t have an opposite.

And I can’t really say, that the opposite to none is everything of some finite amount. If zero is none and the opposite to none is everything of some finite amount, every positive integer could be considered as an opposite to zero, that doesn’t make sense.

Or does it, if we have special cases, where something in some collection or set is some finite amount of elements, for example we have 7 elements in a set. Then, if we have all the elements from the set, we have 7 elements, all of them, instead of having none of the elements from the set.

What about dividing by zero, particularly 0/0? If there is zero amount of something, there isn’t this something at all. So 0/0 could be phrased as “none isn’t divided at all”.

Though I’m using too many negatives in one sentence. Perhaps now it’s better phrased: “None is divided.” Now there’s only one negative in one sentence and it’s better English and we don’t divide anything, particularly not zero. Word “none” prohibits the division.

In case *a*/0, *a ≠ *0, perhaps we could say: “Something is not divided.”

Therefore particularly zero can’t be divided by zero. 🙂