The Fundamental Building Blocks
Kesar was beginning to calm down. He and his legion were on the way to Velnias I, and he could not afford to allow his frustration with the slow progress of his research to cause him to enter combat, on a world that will likely be the most difficult they have yet faced, unfocused. To help his calm, he picked up his trusty data-slate, and started working through the basics of what he hoped would eventually become a means of understanding reality. Something he was sure the ancients mastered tens of millennia ago. The basics of mathematics. How numbers worked, how they operated, and the basic understanding of why the did so.
He started with just the numbers themselves. What would mean what? The way many planets were named, and the way the Legions were named was simply an incredibly archaic system that had massive issues, namely the lack of the concept of nothing. There was also the issue of what base to use for these numbers. The Mechanicus favored a base 2 number system, only two different characters. Something simple that their beloved Machine Spirits seemed to understand well. However, this was not a system meant for use by the Mechanicus only, but eventually, for everyone. Then he thought of using ten, the decimal system as he had heard it called. Something used by many an administrator for ease of operations. However, he then thought of runes, and the number five. A number that apparently had effects that mitigated warp corruption, though he knew not why... at least not yet.
Having decided to at the very least to start with five, he wrote out the characters to be used. 0,1,2,3,4. Every value he could think of, could be written out as some combination of those 10 numbers. He decided to name this system Quinary Numbers. Next he had to figure out the appropriate methods to manipulate those numbers. How to signal what manipulations were occuring, and decide why specific methods should be used to get specific results. This... was surprisingly easy to consider, at least at first. Addition and Subtraction were the easiest. Able to be described in terms even a child could learn to comprehend. If you have a rifle, and were given another. How many rifles did you have? In this case, addition would be used. Adding the initial 1 rifle, to the new 1 rifle. Giving 2 rifles. So 1+1=2. An important aspect of this is that adding 0 to any number would still give that number. As well, it did not matter what order you added the numbers in, they'd give the same result either way.
Subtraction was only slightly more difficult. Requiring that concept of nothingness that was missing in the numerals used by the Legion naming sense, though he supposed there would be nothing to name so no 0 was necessary. Still, an issue. Back to the topic at hand however he thought, shaking out his hand as if cleaning away the previous thoughts. If you had a rifle, and then a rifle was taken away from you, how many rifles did you have? Well, you started with 1 rifle. Then 1 rifle was removed. That leaves you with 0 rifles. Still simple enough to comprehend. Here 0 again played an odd role. Here, subtracting 0 from a number game the original number, but it was very important to keep which number is being subtracted and which doing the subtraction clear, as a putting it in the wrong spot could lead to issues. Here also was the first major complication, what did you do with a 1-2? Something to be considered later he supposed.
Then, came adding multiple sets of numbers at once. If you had no rifles, and someone handed you a single rifle 4 times, how many rifles did you have? Here was a slightly different problem. You started with nothing, so 0 rifles. Then you were given a rifle. +1. Then another, +1 again, and again, and again. For a total of 0+1+1+1+1, which was just a mess to write out, and could be understood quicker by adding those numbers together. For 0+1x4. Giving 0+4, or 4. This same process could be used with any combination of numbers. Though, the idea of multiplying by 0, or multiplying 0 by any other number was an odd one. Because regardless of how it was written, it always gave 0. An interesting fact based on how the numbers meaning was decided. For to add nothing any amount of times would still be a total of nothing, and adding any number no times, would also lead to nothing. However multiplying by 1 kept the number the same, in a similar manner to adding zero. In this case for a different reason for a single set of a number was simply that number. If you one armful of 3 rifles, you had 3 rifles. Simple enough.
Division is where things began to get complicated. If multiplying by 0 was interesting, dividing by it was an absolute nightmare. It didn't seem to have any possible meaning, and yet was a thing that could theoretically be done based on his rules so far. Instead, he simply wrote into at the top of this file in his data-slate, where he had been writing the other basic rules he'd noticed going through these different operations, that anything divided by zero simply didn't have an answer, and moved on. Hoping it wouldn't come back to bite him later. Beyond that incredibly frustrating issue, he found that dividing by 1 acted in a similar way as multiplying by 1. Giving the same number one started with. Though the reasoning was slightly different. Instead of having 1 set of things, and that being why multiplying by 1 gave the same number, in division, dividing by 1 was a matter of figuring out how many sets of a number made up another. So with a number such as 4, dividing by 1 was the same as asking what single number by itself was 4. Which, seemed almost like an identity of the number itself. Though, he also supposed it could instead be asking, how many times must you add 1 together to get 4. In which case you'd need to do so 4 times. He may, he realized, have to consider that difference and whether or not it was important another time.
These were relatively simple and basic operations regarding numbers only up to 4. But how did these operations react to numbers from 10 or higher, and how would he go about teaching the process? Well first an example. Something simple to start. 10-4. Here, it is similar to saying 4+1-4, and so describing it that way might work? It does give the expected answer of 1. So what about another example. 20-3. Well 4+2-3 still seems accurate, though writing it out this way seemed excessive. Perhaps there was a better way to go about it that he could figure out sometime? Still this seemed to work. Then addition instead? 13+4. Well, that seemed simple enough. 22. To be certain though, double check the work. 22-4. So, using the same kind of method 4+2-4...+4 again. Did not give 13. It seems he'd made a mistake before. 20-3 was not 4+2-3, and he should have double checked. It was instead 10+4+1-3, or 12. 12+3=20. There we go! Much better. Alright and for multiplication. 10*10. Easy enough here, just add an extra 0 at the end of the 10 for 100. Then what about 11*2? 22. Alright then. What about division? 4/10.... well. Huh. How would he solve this? How many times could you add 10, before you got 4? Well, less than 1 time. But... how to notate that? Just leave it in that form? Or maybe another way. Something like... 0.4 perhaps? With .1 being 1/10, or... a fifth? yes that would work, and 4/10 would be four fifths? Yes. That seems to make sense. Well, that seems simple enough... but then, how would he handle adding and subtracting those fractions of the numbers? Or would using the quinary places be better? Perhaps it would depend on the individual task? Something to look into. So a simpler problem instead. 20/2. That would simply be 10. 10*2 would give 20. How many sets of 2 would it take to make 20? 10. How much is 10 things of 2? 20. Yes, this seemed to work just fine.
Kesar went through more problems, just verifying that everything he had come up with worked. He hoped to teach these principles to his Legion in time, but for now, simply finding order in things brought him calm, as he prepared to lead his 330200 sons at Velnias I, all the while he ran through the names of the Lost and Fallen in the back of his mind, knowing that by the end of the campaign there, it was near inevitable that the list of both would grow.