Here's a good example I just came across of why it's useful to be able to do approximate multiplication in your head (perhaps even while intoxicated): http://news.nationalpost.com/full-comment/robyn-urback-sorry...
Is there any point to the 12 times table?
161–170 of 198 posts
Re: Is there any point to the 12 times table?
#162Only 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…
What is the advantage of using degrees (e.g. "30 degrees") versus radians (π/6)?
Regardless, your observations seem to come down to "12 is highly composite and we use highly composite numbers for time"—valid, but interesting?
Re: Is there any point to the 12 times table?
#163Jon McLoone really seems to be approaching this topic from a place of frustration—rather than from an overwhelming sense of beauty. I think both mathies and visual designers can agree that 12 is a very beautiful number. In fact the ONLY single thing that makes sense about the imperial system is the division of feet into 12 inches, something mirrored by picas and points in typography: https://en.wikipedia.org/wiki/Pica_(typography)
This beautiful number is evenly divisible by 1, 2, 3, 4, 6, and itself—a very high percentage (50%!) of the positive integers leading up to and including itself. (To be sure 60 is also a beautiful number but its ratio of factors to non-factors is of course lesser.) I can see you gearing up to argue that 6 is more beautiful, and I get that. I really do. But the ability to evenly divide into quarters is just so ... pleasant :)
Does the ability to divide evenly in several ways serve as a soapbox from which to argue that it is mathematically important to memorize the first 12 numbers of which 12 is a factor? I suppose not. But could we simply do 12 the honor of learning these two extra columns of the multiplication table because it is beautiful? (And really you’re just learning ONE extra column because 11 is such a simpleton. Silly 11.)
Of course dividing things into 12 units can SOUND pleasant as well. Western music is founded on dividing an octave into 12 units—though the exact method of division can take many forms. Here’s just a sample: https://github.com/stewdio/beep.js/blob/master/beep/Beep.Not...
So I love 12. Join me in loving 12. Together we can praise 12.
Full disclosure, I have been known to previously adore the number 12. On December 12, 2012 I proposed this “Cleaner Grid of Time” beginning with the months in a year: http://stewd.io/b/2012/12/12/twelve
Re: Is there any point to the 12 times table?
#164Earlier quoted context omitted.
I'd assume if we used base 12 for everything we'd also use it for writing numbers too. 1 2 3 4 5 6 7 8 9 a b so 10 would be 12
You have to be clear on the difference between the number and its representation. In English we represent numbers in base 10 unless otherwise specified. Hence you say things like, The base 12 number 10 is 12. I did not otherwise specify, so I represented my numbers in base 10 everywhere even though I was talking about base 12.
If we weren't afraid to confuse our Greek readers, we could just use greek letters: ω (=0) α β γ δ ε ζ η θ ι κ λ, with ' to indicate a number: 15 would be written 'αγ.
Re: Is there any point to the 12 times table?
#165Twelve 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…
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…
> You have to multiply by 3.6, can you easily do that in your head?
Multiply by 36 (12*3) then divide by 10?
Re: Is there any point to the 12 times table?
#166Earlier 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…
> You have to multiply by 3.6, can you easily do that in your head? Multiply by 36 (12*3) then divide by 10?
I'd multiply by 6 twice then divide by 10.
But it isn't the "move the decimal point" of other metric unit conversions.
Re: Is there any point to the 12 times table?
#167Earlier 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…
Rome had 8 day weeks.. actually the history of the Roman calendar is fascinating:
https://en.wikipedia.org/wiki/Roman_calendar#Nundinal_cycle
"Because the term of office of elected Roman magistrates was defined in terms of a Roman calendar year, a Pontifex Maximus would have reason to lengthen a year in which he or his allies were in power, or shorten a year in which his political opponents held office. For example, Julius Caesar made the year of his third consulship in 46 BC 445 days long."
Re: Is there any point to the 12 times table?
#168Earlier quoted context omitted.
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…
Those seem like minor use cases for disagreement. I'm not sure there should be an education policy or tradition just because our current unit of time is divisible by 12, or just because circles can be represented with the arc degree, especially when a lot of students go on to use radians, even in non-metric countries. And if there's some domain-specific application, like in carpentry, then let those people use their…
carpentry isn't a closed domain. It is a wide range of skills. If you're in a country using inches and feet you would benefit from knowing the 12 times tables, more than you need 8x or 7x actually you'll have practical benefits from 12x. having comfort in fractions of a fourths, eighths and 16ths is also nice for applying measurements. It's for everyone to put things to use in their life.
Re: Is there any point to the 12 times table?
#169Twelve 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…
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…
> (You have to multiply by 3.6, can you easily do that in your head?)
I think so, by multiplying by 12 and then 3.
Re: Is there any point to the 12 times table?
#170Earlier quoted context omitted.
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…
What is the advantage of using degrees (e.g. "30 degrees") versus radians (π/6)? Regardless, your observations seem to come down to "12 is highly composite and we use highly composite numbers for time"—valid, but interesting?
Radians are superior IMO. Using degrees for measurement is archaic compared to radians, but I imagine whole arabic-numeral numbers are more familiar than PI symbols in the learning process at the point in life when most kids are first learning geometry something related to angles. Understanding PI symbols for the first time could possibly come alongside an algebra curriculum, but I'm not a K-12 educator so if I had an opinion, it wouldn't be worth any weight.
Meanwhile, your response got me to look up whether there's a metric equivalent to degrees or radians. Apparently that exists and it's a measure called Gradians. There are 400 Gradians in a circle, and 400 is not divisible by 12. I guess that idea went the way of the French Republican Calendar.