Towards An Ideal Numeral System
Jamie Stroud
- 3736y15w (oct) (ICS): Version 1.1: I reformatted this article and I fixed some typos.
- 3735y20-23w (oct) (ICS): Version 1
Table of contents
Introduction
America uses a combination of numeral systems, primarily base-10 (decimal) and base-12 (duodecimal). It'd be simpler to use a single numeral system. Furthermore, I'll demonstrate in this article that base-8 (octal) is the ideal numeral system. For clarity's sake, when I use numbers in this article they'll be in decimal, unless otherwise noted, so as not to cause confusion.
Different Numeral Systems In Common Usage
- Examples of octal in common usage:
- *Octal is less often used explicitly but rather more often implied by the desire to divide numbers in perfect halves.
- 16 ounces in a pound.
- 112 pounds in a long hundredweight. (Historically 112 was chosen for this unit rather than 100 since it divides in halves better. Even though 64 or 128 would've worked even better, they probably seemed too far of a stretch.)
- Practically all volume units are evidence of octal with the exception of imperial fluid ounces.
- We have 8 fingers excluding thumbs. We also have 8 spaces between our fingers including thumbs. (This is merely evidence of how base-8 numeral systems could have originated in different societies, and consider that the fingers might be counted with the thumbs which is why you'd exclude them in the first example.)
- Examples of decimal in common usage:
- Base-10 is the most commonly used numeral system.
- The meter has hierarchies in multiples of 10.
- 100 pounds in a short hundredweight.
- The gram has hierarchies in multiples of 10.
- The liter has hierarchies in multiples of 10.
- Fractions of seconds have hierarchies in multiples of 10.
- Celsius defines its 0° as the freezing point of water and it's 100° as the boiling point of water (100 being 10 x 10).
- We have 10 fingers including thumbs. (This is merely evidence of how base-10 numeral systems could have originated in different societies.)
- Examples of duodecimal in common usage:
- 12 inches in a foot.
- 12 troy ounces in a troy pound.
- Grocers deal in dozens (12) and grosses (144 which is 12 x 12) of products.
- 12 months in a year.
- 12 astronomical signs.
- 24 hours in a day which is generally split in 2 parts, 12 hours a.m. (ante meridiem, before noon) and 12 hours p.m. (post meridiem, after noon).
- Twelve is the largest single-morpheme named number (notice that you have "teen" on thirteen, fourteen, etcetera, making them nonsingle-morphemes).
- We have 12 finger bones on each hand (excluding the thumb). (This is merely evidence of how base-12 numeral systems could have originated in different societies, and consider that the finger bones might be counted with the thumb which is why you'd exclude it.)
- People say "dozens" but it's rare to unheard of for people to say "tens".
Evidence For Octal
There's no evidence to suggest that decimal offers anything of particularly special use, nor does nature seem to be especially fond of decimal apart from humans having 10 fingers (including thumbs). Duodecimal is said by many to be especially useful since it can be divided by both 2 and 3 without involving fractions, and this is something neither decimal nor octal is capable of. However, why should 3 be thought to be so special, so important, to be divided by without involving fractions? The world is more likely to function by pairs and repeating phenomena (e.g. particles/antiparticles, attraction/repulsion, diecious species, any properties with polar opposites, fractals, etcetera), and so in the end it's more important to work well with pairs and repeating phenomena. With all that being said, I'll note how nature and society seems most inclined towards and would make the most use of octal over all other numeral systems.
- 8 is divisible in halves all the way down to 1 without involving fractions.
- The most common meter in music is 4/4. This also relates to dance, in that it's most commonly played in double-time, resulting in the familiar, "one, two, three, four, five, six, seven, eight".
- Angles in athletics, and directions in general, are most often dealt with in either halves, quarters, or eighths. For example, north, north-east, east, south-east, south, south-west, west, and north-west, or up, up-right, right, down-right, down, down-left, left, and up-left.
- Based on a root number of 0 or 1, 8 is a Fibonacci number, which relates to golden ratios, which is a fundamental aesthetic found frequently in nature (e.g. snail shells).
- All fundamentally derived formulas (formulas derived from fundamental/base units) are either linear or quadratic. This suggests that the laws of nature only function as whole multiples (which is how 8 is essentially able to function).
- 8 points comprises the smallest perfect cube, excluding 1.
- Jason Padgett is an important case study because he has an *abnormal brain which causes him to 1) see fractals in everything, and 2) see the Pythagorean Theorem everywhere. (*Abnormal doesn't necessarily mean bad.) The Pythagorean Theorem is a formula for solving the hypotenuse of a triangle. It's formula is a^{2} + b^{2} = c^{2}. Recognize that any simple polygon can be comprised of triangles. A fractal can be defined or described as a shape that repeats itself symmetrically, that has any given part with the same characteristics as the whole, or that is the same from near as from far. Jason Padgett drew what he refers to as a Planck Lattice and states that it's his interpretation of the structure of space-time. (Padgett, 2014, p. 88) Notice that this structure is essentially a fractal of a perfect cube, and that the smallest perfect cube is comprised of 8 points (excluding 1), thereby suggesting that the universe functions by the number 8 at its most fundamental level.
Clarification
Many people have a difficult time understanding what it means to use a different numeral system and/or how to process numbers in a different numeral system. I'll explain how numeral systems work for clarity's sake.
Numeral systems are based on multiples of 10, with different numeral systems having different number values/quantities for "10". After 10, the cycle essentially repeats. Notice how similar "teen" and "ty" sound to "ten". You can tell they're a modification of "ten" based on how they function. For example, look at "forty", it's the fourth multiple of 10. 4 x 10 = for (FOUR) x ty (TEN). You can notice this for almost all numbers, twen(ty)one (Two x Ten + One), twen(ty)two (Two x Ten + Two), etcetera.
Eleven through nineteen (11-19) is somewhat of a special case from the rest of the numbers in English. This is because they essentially function backwards from the rest of the numbers in regards to how they're named. For example, fourteen could hypothetically be teenfour or tyfour and it'd be more logical. Most languages do this actually, and so English's way of naming numbers is more complicated than most.
With all that being explained, an octal numeral system would function something like this:
- 0, 1, 2, 3, 4, 5, 6, 7
- 10, 11, 12, 13, 14, 15, 16, 17
- 20, 21, 22, 23, 24, 25, 26, 27
- 30, 31, 32, 33, 34, 35, 36, 37
- 40, 41, 42, 43, 44, 45, 46, 47
- 50, 51, 52, 53, 54, 55, 56, 57
- 60, 61, 62, 63, 64, 65, 66, 67
- 70, 71, 72, 73, 74, 75, 76, 77
- 100, 101, 102, 103, 104, 105, 106, 107
- etc.
- zero, one, two, three, four, five, six, seven
- eight, eight(+)one, eight(+)two, eight(+)three, eight(+)four, eight(+)five, eight(+)six, eight(+)seven
- two(x)eight, two(x)eight(+)one, two(x)eight(+)two, two(x)eight(+)three, two(x)eight(+)four, two(x)eight(+)five, two(x)eight(+)six, two(x)eight(+)seven
- three(x)eight, three(x)eight(+)one, three(x)eight(+)two, three(x)eight(+)three, three(x)eight(+)four, three(x)eight(+)five, three(x)eight(+)six, three(x)eight(+)seven
- four(x)eight, four(x)eight(+)one, four(x)eight(+)two, four(x)eight(+)three, four(x)eight(+)four, four(x)eight(+)five, four(x)eight(+)six, four(x)eight(+)seven
- five(x)eight, five(x)eight(+)one, five(x)eight(+)two, five(x)eight(+)three, five(x)eight(+)four, five(x)eight(+)five, five(x)eight(+)six, five(x)eight(+)seven
- six(x)eight, six(x)eight(+)one, six(x)eight(+)two, six(x)eight(+)three, six(x)eight(+)four, six(x)eight(+)five, six(x)eight(+)six, six(x)eight(+)seven
- seven(x)eight, seven(x)eight(+)one, seven(x)eight(+)two, seven(x)eight(+)three, seven(x)eight(+)four, seven(x)eight(+)five, seven(x)eight(+)six, seven(x)eight(+)seven
- hundred, hunred(+)one, hundred(+)two, hundred(+)three, hundred(+)four, hundred(+)five, hundred(+)six, hundred(+)seven
- etc.
*Saying one-hundred is redundant. In octal, you wouldn't say one-eight (10), one-eight-one (11), one-eight-two (12), etcetera. Or in decimal, you wouldn't say one-ten (10), one-eleven (11), one-twelve (12), etcetera.
*Zero and seven are awkward in that they're 2 syllables while the rest of the single digits are one syllable. For example, they could be called "zee" and "ven", respectively.
*In practice, people use numeral systems with the end of the base number functioning as the end of a sequence, but logically the end of a base number actually functions as the beginning of a new sequence or new "row".
As you can see, the last digit of each respective number always lines up in the latter example whereas they're one off in the former example. This has implications when it comes to creating symmetrical arrays.
In Closing
If social network companies like Facebook and Twitter implemented codes to automatically put octal numbers in parenthesis whenever someone posts with decimal numbers, it'll get people used to seeing the numbers of that numeral system. Through repetition of frequently seeing octal numbers next to decimal numbers, they'll be able to more easily adapt to it. To avoid confusion for those who want to progress quickly to octal but don't want to cause confusion or risk misinterpretation by others, the clarification can be put in parenthesis. For example, 100 in octal could be written as 100 (oct), whereas 100 in decimal could be written as 100 (dec).
References
- Padgett, J. (2014). Struck by genius: How a brain injury made me a mathematical marvel. New York, New York: Houghton Mifflin Harcourt Publishing Company.