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number.md
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@ -41,60 +41,86 @@ Here are some [fun](fun.md) facts about numbers:
*Number lines and some notable numbers -- the horizontal line is real line, the vertical is imaginary line that adds another dimension and reveals complex numbers. Further on we can see quaternion lines projected, hinting on the existence of yet higher dimensional numbers (which however cannot properly be displayed using mere two dimensions here).*
The following is a table demonstrating just one way of how you can play around with numbers -- here we just examine whole positive numbers (like number theorists would) up to 50, count each one's total number of divisors (excluding 1 and itself, 0 here means the number is [prime](prime.md) except for 1, if the number is highest in the series so far the number is called "highly composite"), unique divisors (excluding itself), sum of total and unique divisors (if the number equal sum of unique divisors, it is said to be a "perfect number"), average "dividing spread" (distance of each tested potential divisor's remainder after division from half of this tested potential divisor, kind of "amount of not dividing the number") in percents, maximum dividing spread and normalized range between smallest and biggest divisor expressed in percents (-1 if there are none). You can make quite interesting graphs from similar data and discover cool and interesting patterns.
The following is a table demonstrating just one way of how you can play around with numbers -- of course, we have generated it with a program, so we also practice [programming](programming.md) a bit ;) Here we just examine whole positive numbers (like number theorists would) up to 50 and take a look at some of their attributes -- we count each one's total number of divisors (excluding 1 and itself, 0 here means the number is [prime](prime.md) except for 1, if the number is highest in the series so far the number is called "highly composite"), unique divisors (excluding itself), minimum divisor (excluding 1 except for 1), maximum divisor (excluding itself except for 1), sum of total and unique divisors (if the number equal sum of unique divisors, it is said to be a "perfect number"), average "dividing spread" (distance of each tested potential divisor's remainder after division from half of this tested potential divisor, kind of "amount of not dividing the number") in percents, maximum dividing spread and normalized range between smallest and biggest divisor expressed in percents (-1 if there are none). You can make quite interesting graphs from similar data and discover cool and interesting patterns.
| number | divisors |divisors uniq.|divisor sum|uniq. div. sum|avg. div. spread (%)|max div. spread (%)|div. range (%)|
| -------- | -------- | ------------ | --------- | ------------ | ------------------ | ----------------- | ------------ |
| 1 | 0 | 1 | 0 | 1 | 0 | 0 | -1 |
| 2 | 0 | 1 | 0 | 1 | 0 | 0 | -1 |
| 3 | 0 | 1 | 0 | 1 | 0 | 0 | -1 |
| 4 | 2 | 2 | 4 | 3 | 33 | 100 | 0 |
| 5 | 0 | 1 | 0 | 1 | 16 | 50 | -1 |
| 6 | 2 | 3 | 5 | 6 | 43 | 100 | 16 |
| 7 | 0 | 1 | 0 | 1 | 24 | 66 | -1 |
| 8 | 4 | 3 | 10 | 7 | 44 | 100 | 25 |
| 9 | 2 | 2 | 6 | 4 | 36 | 100 | 0 |
| 10 | 2 | 3 | 7 | 8 | 40 | 100 | 30 |
| 11 | 0 | 1 | 0 | 1 | 34 | 80 | -1 |
| 12 | 5 | 5 | 17 | 16 | 53 | 100 | 33 |
| 13 | 0 | 1 | 0 | 1 | 35 | 83 | -1 |
| 14 | 2 | 3 | 9 | 10 | 43 | 100 | 35 |
| 15 | 2 | 3 | 8 | 9 | 44 | 100 | 13 |
| 16 | 7 | 4 | 24 | 15 | 49 | 100 | 37 |
| 17 | 0 | 1 | 0 | 1 | 38 | 87 | -1 |
| 18 | 5 | 5 | 23 | 21 | 47 | 100 | 38 |
| 19 | 0 | 1 | 0 | 1 | 42 | 88 | -1 |
| 20 | 5 | 5 | 23 | 22 | 51 | 100 | 40 |
| 21 | 2 | 3 | 10 | 11 | 45 | 100 | 19 |
| 22 | 2 | 3 | 13 | 14 | 43 | 100 | 40 |
| 23 | 0 | 1 | 0 | 1 | 42 | 90 | -1 |
| 24 | 8 | 7 | 39 | 36 | 55 | 100 | 41 |
| 25 | 2 | 2 | 10 | 6 | 45 | 100 | 0 |
| 26 | 2 | 3 | 15 | 16 | 45 | 100 | 42 |
| 27 | 4 | 3 | 18 | 13 | 44 | 100 | 22 |
| 28 | 5 | 5 | 29 | 28 | 49 | 100 | 42 |
| 29 | 0 | 1 | 0 | 1 | 45 | 92 | -1 |
| 30 | 6 | 7 | 41 | 42 | 52 | 100 | 43 |
| 31 | 0 | 1 | 0 | 1 | 45 | 93 | -1 |
| 32 | 9 | 5 | 42 | 31 | 48 | 100 | 43 |
| 33 | 2 | 3 | 14 | 15 | 45 | 100 | 24 |
| 34 | 2 | 3 | 19 | 20 | 47 | 100 | 44 |
| 35 | 2 | 3 | 12 | 13 | 48 | 100 | 5 |
| 36 | 10 | 8 | 65 | 55 | 54 | 100 | 44 |
| 37 | 0 | 1 | 0 | 1 | 45 | 94 | -1 |
| 38 | 2 | 3 | 21 | 22 | 45 | 100 | 44 |
| 39 | 2 | 3 | 16 | 17 | 46 | 100 | 25 |
| 40 | 8 | 7 | 53 | 50 | 51 | 100 | 45 |
| 41 | 0 | 1 | 0 | 1 | 47 | 95 | -1 |
| 42 | 6 | 7 | 53 | 54 | 51 | 100 | 45 |
| 43 | 0 | 1 | 0 | 1 | 46 | 95 | -1 |
| 44 | 5 | 5 | 41 | 40 | 49 | 100 | 45 |
| 45 | 5 | 5 | 35 | 33 | 47 | 100 | 26 |
| 46 | 2 | 3 | 25 | 26 | 47 | 100 | 45 |
| 47 | 0 | 1 | 0 | 1 | 47 | 95 | -1 |
| 48 | 12 | 9 | 85 | 76 | 53 | 100 | 45 |
| 49 | 2 | 2 | 14 | 8 | 48 | 100 | 0 |
| 50 | 5 | 5 | 47 | 43 | 49 | 100 | 46 |
{ Be warned the following is just me making some quick unoriginal antiresearch, I may mess something up, it's just to show the process of playing around with numbers. ~drummyfish }
| number | divisors |divisors uniq.|min. div.|max. div.|divisor sum|uniq. div. sum|avg. div. spread (%)|max div. spread (%)|div. range (%)|
| -------- | -------- | ------------ | ------- | ------- | --------- | ------------ | ------------------ | ----------------- | ------------ |
| 1 | 0 | 1 | 1| 1| 0 | 1 | 0 | 0 | -1 |
| 2 | 0 | 1 | 2| 1| 0 | 1 | 0 | 0 | -1 |
| 3 | 0 | 1 | 3| 1| 0 | 1 | 0 | 0 | -1 |
| 4 | 2 | 2 | 2| 2| 4 | 3 | 33 | 100 | 0 |
| 5 | 0 | 1 | 5| 1| 0 | 1 | 16 | 50 | -1 |
| 6 | 2 | 3 | 2| 3| 5 | 6 | 43 | 100 | 16 |
| 7 | 0 | 1 | 7| 1| 0 | 1 | 24 | 66 | -1 |
| 8 | 4 | 3 | 2| 4| 10 | 7 | 44 | 100 | 25 |
| 9 | 2 | 2 | 3| 3| 6 | 4 | 36 | 100 | 0 |
| 10 | 2 | 3 | 2| 5| 7 | 8 | 40 | 100 | 30 |
| 11 | 0 | 1 | 11| 1| 0 | 1 | 34 | 80 | -1 |
| 12 | 5 | 5 | 2| 6| 17 | 16 | 53 | 100 | 33 |
| 13 | 0 | 1 | 13| 1| 0 | 1 | 35 | 83 | -1 |
| 14 | 2 | 3 | 2| 7| 9 | 10 | 43 | 100 | 35 |
| 15 | 2 | 3 | 3| 5| 8 | 9 | 44 | 100 | 13 |
| 16 | 7 | 4 | 2| 8| 24 | 15 | 49 | 100 | 37 |
| 17 | 0 | 1 | 17| 1| 0 | 1 | 38 | 87 | -1 |
| 18 | 5 | 5 | 2| 9| 23 | 21 | 47 | 100 | 38 |
| 19 | 0 | 1 | 19| 1| 0 | 1 | 42 | 88 | -1 |
| 20 | 5 | 5 | 2| 10| 23 | 22 | 51 | 100 | 40 |
| 21 | 2 | 3 | 3| 7| 10 | 11 | 45 | 100 | 19 |
| 22 | 2 | 3 | 2| 11| 13 | 14 | 43 | 100 | 40 |
| 23 | 0 | 1 | 23| 1| 0 | 1 | 42 | 90 | -1 |
| 24 | 8 | 7 | 2| 12| 39 | 36 | 55 | 100 | 41 |
| 25 | 2 | 2 | 5| 5| 10 | 6 | 45 | 100 | 0 |
| 26 | 2 | 3 | 2| 13| 15 | 16 | 45 | 100 | 42 |
| 27 | 4 | 3 | 3| 9| 18 | 13 | 44 | 100 | 22 |
| 28 | 5 | 5 | 2| 14| 29 | 28 | 49 | 100 | 42 |
| 29 | 0 | 1 | 29| 1| 0 | 1 | 45 | 92 | -1 |
| 30 | 6 | 7 | 2| 15| 41 | 42 | 52 | 100 | 43 |
| 31 | 0 | 1 | 31| 1| 0 | 1 | 45 | 93 | -1 |
| 32 | 9 | 5 | 2| 16| 42 | 31 | 48 | 100 | 43 |
| 33 | 2 | 3 | 3| 11| 14 | 15 | 45 | 100 | 24 |
| 34 | 2 | 3 | 2| 17| 19 | 20 | 47 | 100 | 44 |
| 35 | 2 | 3 | 5| 7| 12 | 13 | 48 | 100 | 5 |
| 36 | 10 | 8 | 2| 18| 65 | 55 | 54 | 100 | 44 |
| 37 | 0 | 1 | 37| 1| 0 | 1 | 45 | 94 | -1 |
| 38 | 2 | 3 | 2| 19| 21 | 22 | 45 | 100 | 44 |
| 39 | 2 | 3 | 3| 13| 16 | 17 | 46 | 100 | 25 |
| 40 | 8 | 7 | 2| 20| 53 | 50 | 51 | 100 | 45 |
| 41 | 0 | 1 | 41| 1| 0 | 1 | 47 | 95 | -1 |
| 42 | 6 | 7 | 2| 21| 53 | 54 | 51 | 100 | 45 |
| 43 | 0 | 1 | 43| 1| 0 | 1 | 46 | 95 | -1 |
| 44 | 5 | 5 | 2| 22| 41 | 40 | 49 | 100 | 45 |
| 45 | 5 | 5 | 3| 15| 35 | 33 | 47 | 100 | 26 |
| 46 | 2 | 3 | 2| 23| 25 | 26 | 47 | 100 | 45 |
| 47 | 0 | 1 | 47| 1| 0 | 1 | 47 | 95 | -1 |
| 48 | 12 | 9 | 2| 24| 85 | 76 | 53 | 100 | 45 |
| 49 | 2 | 2 | 7| 7| 14 | 8 | 48 | 100 | 0 |
| 50 | 5 | 5 | 2| 25| 47 | 43 | 49 | 100 | 46 |
Now we may start working with the [data](data.md), let's for example notice we can make a nice [tree](tree.md) of the numbers by assigning each number as its parent its greatest divisor:
```
1
|
.---.------------.------------.-----'--.-----.---.--.--.--.--.--.--.--.--.
| | | | | | | | | | | | | | |
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 <--- primes
| | | | | | | | |
| .'---. .----+----. .---.-'-.--. .-'-. .-'-. | | |
| | | | | | | | | | | | | | | | |
4 6 9 10 15 25 14 21 35 49 22 33 26 39 34 38 46
| | | | | | | | |
| | .-'-. | .-'-. | | | |
| | | | | | | | | | |
8 12 18 27 20 30 45 50 28 42 44
| | | |
16 24 36 40
| |
32 48
```
Here patterns start to show, for example the level one of the tree are all prime numbers. Also in this tree we can nicely find the [greatest common divisor](gcd.md) of two numbers as their closest common ancestor. Also if we go from low numbers to high numbers (1, 2, 3, ...) we see we go kind of in a zig-zag direction around the bottom-right diagonal -- what if we make a program that plots this path? Will we see something [interesting](interesting.md)? We could use this tree to encode numbers in an alternative way too, by indicating path to the number, for example *45 = {2,1,1}*. Would this be good for anything? If we write numbers like this, will some operations maybe become easier to perform? You can just keep diving down rabbit holes like this.
## Numbers In Math