Greetings,

Firstly I got to admit this is not a very philosophical topic or connected to ponies in any form. I am trying lately to get my mind to think a bit more mathematical. I simply do this because I am going to take a computer based test next month to see if I could become a professional pilot. Usually this means I toss numbers pointlessly around until I get bored, but this evening those numbers suddenly decided to make sense, at least to me. I was looking at a rather basic problem, the conversion of metric units in its single steps. Meaning converting a centimetre into decimetres, square decimetres into square metres, and cubic millimetres into cubic centimetres and so on. Now, most people would say that this problem is no problem at all. Most people, namely those not as mathematically incompetent as me, would say that this is a very simple process of multiplication/division by the factors 10/100/1000.

One millimetre times ten is a centimetre, a centimetre times ten is a decimetre, a decimetre times ten is a metre BUT a metre times one thousand makes a kilometre.

If we're dealing with surfaces, a smart person would have lectured me that, because a square's surface is calculated by its base multiplied by its base, or its base squared, you also have to multiply two times by ten between every step, meaning taking it times one hundred.

One square millimetre times one hundred is a square centimetre, one square centimetre times one hundred is a square decimetre, one square decimetre times one hundred is a square metre, a square metre times one hundred is a hectare and a hectare times one hundred is a square kilometre.

Taking the third dimension into play, a smart person obviously would have continued, may will require you to multiply everything by one thousand, simply because the volume of a cube widely known is calculated by multiplying its base with its base with its high which has the same value as its base, simply making it base cubed. So, according to the logic applied above, we need to multiply the value three times by ten when making a step of one unit, and ten times ten times ten, shortly ten cubed, is one thousand.

One cubic millimetre times one thousand is one cubic centimetre, one cubic centimetre times one thousand is a cubic decimetre (formally known as 'litre'), a cubic decimetre times one thousand is a cubic metre, a cubic metre times one thousand is a cubic decametre, a cubic decametre times one thousand is a cubic hectometre and a cubic hectometre times one thousand then is a cubic kilometre.

After hours of throwing numbers around I finally realized that this indeed is a rather easy process. And I thought, my, why is there no formula making this clear?

//This is where the problem actually starts

So I began to try myself in the dark arts of mathematical witchcraft and came up with the following: if one ignores the infamous jump from a metre to a kilometre, all steps are of the exact same size. And since there is basically nothing else but multiplication/division by 10 happening, why shouldn't we always work with always multiplying by ten to the power of something? It would cast the irritation away that I always had when working with square or even cubic units. So I thought about what this something should be.

For starters, I'd offer to set that something equal to the dimension one is working in. Is it only a distance, then it's one-dimensional, setting that something = 1. This looks right to me, since ten to the power of one is still ten, and multiplying by ten is what you do as shown above. Except this infamous metre to kilometre, what I in consequence have to shoot unless something in between gets this right again.

Also for surfaces this seems right, for surfaces are two-dimensional, setting this something = 2. Now, ten to the power of two is one hundred. As seen above, multiplying by hundred is exactly what you do when dealing with surfaces.

And even for dealing with whole rooms this seems correct to me. A room is three-dimensional, setting this something = 3. Now ten to the power of three is one-thousand. As seen above this is, again, exactly what you have to multiply stuff with. Excellent!

So, if we give something a name now, let's say D for Dimension, we would have something like that:

Old Value Size * 1 * 10^D = New Value Size

But wait. Ultimately, this will make the value larger. But when converting into a larger scale, the value has to become smaller, as the amount of stuff in existence may is now measured differently but doesn't get more or less. If we take ten to the power of this negative something, meaning:

Old Value Size * 1 *10^-D = New Value Size

In this case our value always will get smaller, but never bigger, thus two different equations would be required, depending on what action is desired. Also, there is the problem that, even if you don't change the measure unit at all, making a ghost calculation so to speak, the value size still would change by at least ten, as there may not is a zero-dimensional unit. Thus, I saw the need of something being added.

To find this something, I first looked at another problem that comes with the equations above: you may only jump one unit at a time. This wouldn't make anything easier except of telling you what to do how often. So I created a new something. This something, again, would need to affect this ten, because we only want to change the since of the value but not the numeric... worth? Anyways, I wanted this new something to enable me to jump a variable amount of steps and in both directions at that. I decided to call it Steps, given its function, and multiply the old something with it.

Old Value Size * 1 * 10^D*S = New Value Size

With that I had a value that affected the equation in the way I wanted. It changed ten without ever possibly causing it to become something else than a multiple of ten. The problem with the ghost division seemed solved. If we give the new something the direct value of the steps you do, the problem of taking no steps becomes solved. Because: ten to the power of zero is one. Witchcraft.

And yet, completely satisfied I still was not. If I now go one step upwards and put all the values into the equation then I get a raised value even though the new unit would require a lower value size in order to keep the... mass equal. The minus seemed to serve me well.

Old Value Size * 1 * 10^D*(-S) = New Value Size

With that I then seemed satisfied and still am. If no step is taken, it still becomes zero, and if the step is positive the value size now is lowered, because obviously ten to the power of minus 2 equals one divided by ten to the power of two, what then is 0,01. And that then is the same solution as to divide through one hundred, what would be the proper thing to to if one would go backwards, as the opposing action to multiplication is division. Thus, ten divided by ten equals ten multiplied by one divided by ten: 10 : 10 = 10 * 1/10 = 1.

It furthermore allows you to take any desired step (all except from going to kilometres, what I defy with this equation), as taking three steps backwards from metres to millimetres would require you to multiply by ten three times, what you now would do in the equation, as you would do ten to the power of the dimension (1) times the steps (3):

27m * 1 * 10^1*(-3) = 27m * 10^-3 = 0,027 mm

And that is right. I tried multiple of such examples and they all seemed right.

TL;DR: My Question

My question now is, did I do this properly (would be a first) and, in last consequence, is this equation right and, ultimately, what can I do against this damned metres to kilometres?

Does the equation Old Value Size * 1 * 10^D*(-S) = New Value Size describe the relation between the jumps in measuring units inside of metric units inside the single dimensions properly? D would equal the dimension one's working in, D={1,2,3}, while S equals the steps taken, S={Z}.

Carpe noctem,
--Chaodiurn