131–140 of 198 posts
Most of the multiples of eleven are really, really easy, and probably just has simple pedagogical value. Multiples of 12 come up a lot because dozens are used a lot in real-world counting of length and time. Perhaps less in a metric country, but hours and days are still divisible by 12.
Earlier quoted context omitted.
Why? No matter what radix you pick, there's always going to be some rational-value you cannot accurately encode. Moving to binary just gives you a different distribution of "un-encodeable values". (I'd say a larger/worse set, but I'm not yet sure how to prove it.) In the long run, trinary gives you the best radix-economy, being closest to e: https://en.wikipedia.org/wiki/Radix_economy
Making a self-reply here, I think I know how to show binary is "worse" when it comes to un-representable rational numbers. I'm not a math major, so this is probably some incredibly obvious textbook stuff to somebody else, but... In base-N, you can accurately write any fraction of (1/y) provided that y can be expressed using the same prime-factors found in N. For example, 10 has the prime factors of 2 and 5, leading t…
In this fashion, primes p seem "better" as they increase, because for a given size limits on k and n, you get more 'numbers' per unit space on the real line. Composite numbers are 'even better', but there are a lot of pains in the ass with composite numbers, stemming from Z/nZ not being a field if n not prime.
Ultimately, though, your internal representation is binary, using anything not two produces other inefficiency. That's why IEEE 754 BCD is not base-10 but base-1000 (since 2^10 = 1024 which is the closest a power of two gets to a power of 10, for bit-wise representation efficiency). We don't have bi-quinary computers.
If you need 11-times or 12-times tables, you will calculate them enough many times that your cache will become "hot" and you remember them out of the box.
If you don't, then you don't have enough use for those multipliers.
Only in non-metric countries. Back in the British era of pounds, shillings, and pence, hardware had to be built to do arithmetic in that system. This resulted in one of the strangest, and most complex, purely mechanical computing devices ever built - the McClure Multiplying Punch [1], from Powers-Samas. This device came out in 1938. The comparable IBM machine was the IBM 602 Multiplying Punch, but IBM only did decima…
I disagree. 1. We have time. Days are 24 hours, a multiple of 12. 2. We still have time. Hours are 60 minutes, a multiple of 12. 3. Yet more time. Minutes are 60 seconds, a multiple of 12. 4. We have circles. There are 360 degrees in a circle, another multiple of 12. 5. The numbers 1, 2, 3, 4, 6 and 12 itself divide into 12 evenly. The next smallest number that has more factors than 12 is 24, which manages to be a mu…
Thanks to the seconds and minutes counting in 60s and hours and months (somewhat) counting in 12s knowing base 2, 3, 4, 6 and 12 is exactly what someone needs to work up and down.
Many baking items are also sold in units of 12 (E.G. eggs, 144 being that packet size that businesses buy in).
Finally there's the learning aspect. Learning the 12s is a lot like extending the factor tables beneath it in a way that makes sense to analog brains. It's actually not /that/ much more costly. 13 however is a whole new prime, which isn't that useful in daily life; all of the larger numbers are big enough to be handled as estimates or 'long math' precision numbers.
PS: If I could magically force all humans to learn, know, and use a new base (and that's the only option I had) I'd probably pick Octal. Hex (for the above reasons) is useful, and great for computer addressing, but less great for actual human-analog computation.
Earlier quoted context omitted.
You're right about 86400 and 1729. I know up to 2¹⁷=131072 and also the special cases 2²⁰=1048576 and 2²⁴=16777216 (the 7s in the middle make it easier to memorize, and it seems to come up a lot). I would really like to remember 2³²=4294967296 (I always just remember "4.2 billion") and should probably get around to that. I learned a lot of pi in middle school but even then I knew that it wouldn't be useful for anythi…
4.2 Billion is useful, it's the addressable memory in 32 bits.
Earlier quoted context omitted.
Every representation has its limits. :-) In physics it is uncommon to have exact measurements for much of anything except integers. In physics, measurement error is usually a bigger issue than roundoff errors from floating point, so the non-existence of an exact floating point representation of the stated measurement is a non-issue. In the real world, most cases that I'm aware of where we need exact base 10 calculati…
depends. If you're running a simulation in physics, e.g. n-body problem, you can choose your parameters to be exact to start off with, and rescale your units to be exact. Moreover, even if you start with measurement error, the error moves in generally a predictable fashion, so it tends to be O(1). A floating point roundoff error is a martingale, with O(sqrt(N)) - unless you "always round up" or "always round down" wh…
This is a complex topic, with ongoing research across multiple fields of study.
That said, floating point is a "good enough" default for a surprisingly wide variety of situations.
Twelve has more divisors than ten (1, 2, 3, 4, 6 & 12 vs 1, 2, 5 & 10). If we were really smart, we'd switch from base 10 to base 12: many more 'decimals' (really duodecimals) would be non-repeating. One can very quickly count by twelves on the joints of one's fingers, using the thumb as an index (0-143 is a much larger range than 0-10, and it's easier to hold one's hands in the shape necessary). If we were really sm…
I came here for this. I am disappointed it is only the third response. I suggest anyone reading this to also watch this video https://www.youtube.com/watch?v=U6xJfP7-HCc
Earlier quoted context omitted.
Easy divisibility is the reason why when we switched to metric, the one thing that didn't switch is time. Which means that, for example, converting from m/s to km/h is a mess. (You have to multiply by 3.6, can you easily do that in your head?) With a base 12 version of everything you would have 1/12 of a day being 2 hours, 1/144 of a day is 10 minutes, and 1/1728 of a day is 50 seconds. These units would give us both…
I am in the process of reading 'The Story of French' by Nadeau and Barlow. There's a chapter on how France was able to push their influence, during the renaissance, to get the world to switch to the metric system. Prior to developing the metric system, France had no standard for the pound; unlike England. This was the height of enlightenment in France, so the reformers wanted to replace the Gregorian Calendar with a…
Most famous of all is the Eighteenth Brumaire, when Napoleon overthrew revolutionary institution and became the de facto dictator: https://en.wikipedia.org/wiki/Coup_of_18_Brumaire
The date was made even more famous by one Karl Marx, who used it in the evocative title of his essay https://en.wikipedia.org/wiki/The_Eighteenth_Brumaire_of_Lou... (or maybe the essay got famous because the title was evocative, don't know).
Also quite famous was the https://en.wikipedia.org/wiki/Thermidorian_Reaction which saw Robespierre and the Jacobins overthrown.
For other slightly less known events, see https://en.wikipedia.org/wiki/Glossary_of_the_French_Revolut...
Only in non-metric countries. Back in the British era of pounds, shillings, and pence, hardware had to be built to do arithmetic in that system. This resulted in one of the strangest, and most complex, purely mechanical computing devices ever built - the McClure Multiplying Punch [1], from Powers-Samas. This device came out in 1938. The comparable IBM machine was the IBM 602 Multiplying Punch, but IBM only did decima…
Just imagine how strange today's world will look when people will switch to binary (and probably hex for writing).