I know there is a lot of arguments towards a better base, but what about all of the bases equally as useless as b10. How would we rate bases that only differed in their odd prime factors.

Base 6 your great, but it's much too late

Base 14 may not be so bad, the number 7 makes me glad

For base 26 and base 22, how I long to be with you.

Base 34, your such a bore

But give me 36 and I'll ask for more

Odd-number bases (especially those based on primes) are not very useful. Base 12 has the advantage of being divisble by the first four integers, and Base 60 by the first five; moreover, base 60 is more congruent with base 12 than with base 10. Forf this reason, I see no need to change circular/angle measure if we adopt dozenal for everything else.

I disagree. Base 60 works well with dozenal fractions in decimal notation (counting by 5's), or decimal fractions in dozenal notation (counting by 6's). Not so well with dozenal fractions in dozenal notation (counting by 5's, but the base is coprime to 5).

In particular, if we kept degrees as-is and just changed the base, then the commonly-used angles of 30° and 90° become 26° and 76°, forcing you to deal with an extra significant digit.

Ah! Beat me to it. But yes, I agree with Dan here. Since we use a clock based on a circle all the time, and that clock has twelve hours on it, that seems to make sense for daily use.

Scientists, engineers, and mathematicians would probably like something based on radians, and the dozenal system also offers a convenient means of dealing with that. The TGM manual offers a good system for this on page 19 et seq.

something about that old 6 on ten relationship in base sixty. It's nice and smooth. 2 by 3 and 2 by 5. With the 5-12 relationship you've got the meeting of two quite disagreeable characters. 5 wants little to do with twelve (after all they are relatively prime. a fifth in base twelve is quite a measure worse than a decimal third). The usage of sixty in my business leads me to notice, at 4:40 pm (16:40) that it is exactly 1000 minutes from midnite. Someone ran a distance (5k?) in 28:48 and yes, it was exactly the cube of twelve seconds. My wife thinks it's odd that I ever put 36 or 48 seconds in the microwave. Hey, they're exact fifths of a minute! but 6 on 10, even though base twelve is keen, I'd avoid sub-grouping things in dozens when using base sixty, it seems so eccentric, when 2*3 and 2*5 seem so well-balanced... like 10 on 12 is in base 120.

I agree that a dozenal fifth is worse than a decimal third ( 1/5 = 0.2497.... while 1/3 = 0.3....), but a third is arguably more common than a fifth and a fifth is probably only as frequent as it is in a decimal system because it is only one of two fractions that allow for a simple numeric, non-repeating representation. If we had a system that allowed for thirds, the use of fifths would probably be far less frequent.

I don't consider your point a valid argument against a dozenal in favour of a sexagesimal radix. Furthermore, the sheer number of symbols and the size of addition and multiplication tables would would weigh heavily against such a radix. While I recognize your points about the factors of 2, 3 and 5, they are not strong enough to support such a large radix in my view. Others may have other opinions.

A dozenal fifth is a true monstrosity, no doubt. But I don't think anyone here will deny that the extremely frequent fractions half, third, and fourth, as well as their halves (sixth and eighth), are much more important to get right than the fifth, which is arguably (as Ruthe observed) only as common as it is thanks to it being the only non-half easy fraction in decimal, and is still run across relatively rarely even then. All of these important fractions (half, third, quarter, sixth, and eighth) are handled more easily in dozenal than in decimal.

Sexagesimal does, of course, include an easy fifth; but the multiplication of symbols makes it impractical as a base. I think Ruthe is dead right here.

The number of symbols isn't that much of a problem, because sexagesimal is notated a mixed-base 6×:A system. The Babylonians built their digits out of only 2 symbols (3 if you count a placeholder for zero): One for :A (resembling Y) and one for 1 (resembling <). And of course, our standard notation for time uses the ten digits 0-9 to make the digits 00-59.

If you meant "multiplication" as in ×, then base-sixty does get difficult, as the multiplication table approach is impractical.

My friends, I guess it's time to share the secret...need only 104 unique values to multiply in sexagesimal. Compare this to hexadecimal, 136 unique values; dozenal 78 unique values, decimal 55 unique values. I never even use a multiplication table as we know it. Pretty busy right now but will share more later. wrote a paper about it in 2007, shared it with dozenalists at the DSA annual meeting. will post a link but will paraphrase it tomorrow or Tuesday. Method is good to around base 120, for what I call the mid-scale of bases, between the high end of "human scale" at 16-20 and around 120. ciao

While obviously we currently use only ten symbols for 00-59, that's really dodging the issue. We use ten symbols and not sixty because we count in base ten, not in base sixty. I can see a system in which we use what are basically sub-bases, in which symbols for higher numbers are made up of symbols for lower numbers, so that fewer unique symbols would be needed. But you're still manipulating sixty individual units of numbers, even if those units are made up of smaller units themselves.

Unless I'm missing something, you need a number of symbols equal to the value of your base in order to use placeholder notation. Reducing the number of unique symbols by making them up of other symbols just dodges the issue.

Now, icarus, when you refer to 104 unique "values" to multiply in sexagesimal, what do you mean? Am looking forward to you sharing more when you get the time.