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For everyone who is intrigued by numers in nature, fractals, and co-incidental numbers.
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Jun. 16th, 2005 @ 08:40 pm Hex Pi?
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Ven Ra
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Date:June 17th, 2005 06:53 am (UTC)

On bases...and our minds....

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Actually, the only reason that we see base ten "easy" is because we use it most of the time, most of the day. We have been trained to use base ten and base six in real life that we see these two bases as possibly the only easiest bases that exist. In mathematical reality, that is not true. As long as you are willing to learn, every base is easy.
Date:October 23rd, 2006 06:55 pm (UTC)

Re: On bases...and our minds....

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It's true that people find decimal "easy" simply because that's what they're used to and what their language reflects: There is nothing inherently easier in something like decimal "6 x 7 = 42" over hexadecimal "6 x 7 = 2A", except for the fact that we all spent years memorizing the decimal multiplication tables in school (so that they now seem "natural"), while almost no-one knows or has even heard about the hexadecimal ones; and we have words in our languages to read decimal "42" as "forty-two", while we lack words to appropriately read the hex number "2A" (in fact, we even lack a complete set of digits to properly write hex, so we have to rely on letters as a substitute). But I wouldn't say every base is as easy; some bases can certainly facilitate or hinder some tasks or purposes.

For example, 1/3 expresses a "simpler" or "more basic" proportion than 1/5, but decimal makes working with fifths easy (0.2, 0.4, 0.6, 0.8) and working with thirds "uneasy" (those "pesky" unending 0.3333333333... and 0.666666666...).

Binary arithmetic is the easiest there is, by far. In fact, it's addition and multiplication tables consist solely of the trivial rows (those of 0 and 1), which are usually left out of the tables in other bases because they are so obvious. However, binary numbers quickly get uncomfortably lengthy, and it's easy to misread numbers such as 111011000101011101010001 (decimal 15,488,849), unless you pay good attention and insert lots of commas to help in reading (1110,1100,0101,0111,0101,0001 or 111,011,000,101,011,101,010,001 which is like reading binary "hexadecimally" or "octally").

Hexadecimal, for its part, makes it easy and compact to work with binary fractions (those with only powers of two as their denominator). For example, three sixty-fourths (3/(2^6)) is 0.046875 in decimal, but just 0.0C in hexadecimal. However, all of the other rational numbers display recurring digits in hex. Although, if you don't mind having to deal with infinite strings of recurring digits (which is impossible in practice, so this means dealing with rounded approximations, i.e. with inaccuracies), hex makes it easy to remember many of their periods, because they are usually short (typically, 1 or 3 digits long). For example, 1/3 = 0.555555555..., 1/5 = 0.333333333..., 1/7 = 0.249249249..., 1/9 = 0.1C71C71C7....

Sexagesimal allows for compact representations of large and tiny numbers (for example, sexagesimal 4:3:2:1.0:9 corresponds to decimal 874,921.0025), but its addition and multiplication tables are enormous, which means in practice it's not very easy to work with this base (in fact, the ancient Mesopotamians didn't even try to memorize the multiplication tables).

On the other hand, the duodecimal multiplication chart is larger than the decimal one, but because of twelve's versatility (it has many factors, which means it "fits" in many of the patterns created by other numbers), the rows show a larger proportion of easy-to-see patterns that help memorization (e.g., the table of 3 goes like: 0,3,6,9, 10,13,16,19, 20,23,26,29, 30..., and the table of 4 goes like: 0,4,8, 10,14,18, 20,24,28, 30,34,38, 40...). So duodecimal is easier than decimal for multiplication and division.

If we talk about counting on fingers, senary (and not decimal) is the most efficient (using both hands you can count up to senary 55, i.e. decimal 35 -- BTW, yes, you can count up to decimal 1023 using fingers to codify binary digits, but that's not really an easy way of counting on fingers), and it also makes it easy to identify potential primes (all of them, except for 2 and 3, are adjacent to a senary "round" number ending in 0; i.e., they end in 1 or 5). However, due to the relatively small size of its base, senary representations are not as compact as decimal, duodecimal or hexadecimal ones.