base thirteen numbers

  • about ten years ago, i wondered if a culture might have had six fingers on one hand and seven fingers on the other, if they might have developed a number system based on thirteen (instead of ten). i wondered if a system based on thirteen might have similar ability to use numbers, and that once a culture adapted to base thirteen and invented or discovered it's mathemagic that they would be convinced that they have the universe's best easiest most productive way to work/play with numbers.

    i had time, energy and curiosity, so i worked out multiplication tables and then used them to work out the division tables. i was fascinated by the patterns i found in base thirteen,but it does seem like a crappy system for numbers. but, again, if i had a whole culture who lived for lifetimes using it, maybe it could be better.

    there are some similarities to base ten. instead of the magic of nine (3x3) there is the magic of twelve (4x3), which is not as obviously balanced. anyway, there it is. i have the tables written down somewhere in my storage.

    i woke up this morning with an urge to share this.
    with the base thirteen multiplication table, i used d, e, f, similar to hexadecimal (base 16) system.  the diagonal line goes thru the squared numbers, and show the mirror effect of the commutative property.  i added four more numbers on the top right just to see if any patterns were unique with extra digits.  i temporarily decided to name the numbers like this...   d is dec, e is ell, f is eff, 10 is tthen, 11 is onetheen, 12 is twotheen, thirtheen, etc, 20 is twenthy, 30 is thirthy, etc, 100 is thundred.  i haven't gotten beyond that.  because i'm fascinated with toroidal based math, i added the mod-equivalent numbers above the two digit numbers to begin to check for patterns.

    the base thirteen division page starts with 1/2.  long division of 1/2 shows 6x2=f (which is twelve) remainder of 1 (which indicates a repeating fraction).  i chose to represent zeroes with qs, because it helped me remember that it represents a digit base13.  1/3, 2/3, 1/4, 3/4 all repeated similarly, as did 1/6, 5/6, 1/f, 5/f, 7/f, e/f.

    fifths and elevenths behaved similar to how sevenths do in base ten.  

    with fifths, to get 1/5, the long division is shown. then, like in base ten, to multiply x2 from the right 2x5=d, 2xd=7 (20, which is 13 remainder 7, carry the 1), 2x7=2 (14 + the carried 1=15, which is 13 remainder 2, carry the 1), 2x2=5 (4+the carried 1=5)
    1/5= .2 7 d 5  
    2/5= .5 2 7 d
    3/5= .7 d 5 2
    4/5= .d 5 2 7
    5/5= .f f f f = 1

    with elevenths, its the same thing, i did the long division on the page, then i noticed the same pattern of shifting numbers (the same way 1/7 magically does in base 10) i multiplied by 2 to get 2/11, then below, added 1/11 to 2/11 to get 3/11 to double check the pattern.
    if you start adding from the right side, 7+1=8, 3+7=d, 8+3=e, d+8= 5 (18, which is 13 remainder 5, carry the 1), e+d=9 (21 + the carried 1=22, which is 13 remainder 9, carry the 1), 5+e=4 (16 + the carried 1=17, which is 13 remainder 4, carry the 1), 9+5=2 (14 + the carried 1=15, which is 13 remainder 2, carry the 1), 4+9=1 (13 +the carried 1=14, which is 13 remainder 1, carry the 1), 2+4=7 (6+ the carried 1=7) and of course 1+2=3.  
    1/e = .1 2 4 9 5 e d 8 3 7
    2/e = .2 4 9 5 e d 8 3 7 1
    3/e = .3 7 1 2 4 9 5 e d 8
    e/e= .f f f f f f f f f f = 1

    with sevenths, eighths, nineths, tenths there are hybrid patterns.

    this is as far as i got.  it seemed there headaches ahead trying to get any deeper into it.  ...but maybe not.