With those cool thumbs of ours, we find ways to count to 12 (or 16) with one hand. Just move the thumb from fingertip to fingertip and then from joint to joint and most of us can keep track of numbers 1-12 (or 1-16) without having any math skills. Hence numbers that revolve around 12 and 16: a dozen numbers before we get into the teens (or 16 (a pint of?) numbers for the french before they start adding tens: "dix sept (10 7), dix huit(10 8)" etc).Most interesting to me is the Babylonian base 60 number system (from whom the Greeks got many mathematical ideas). I think if you are counting 12's on one hand, a natural extension may be to count five sets of 12's on your other hand - giving you 60 as a natural break point (seventy in french is soixante-dix (60 10)).
From this we get 60 seconds in a minute, 60 minutes in an hour and (most interestingly) 60 degrees in an equilateral triangle angle (6 of these inscribed in a circle gives you the 360 degrees in a circle!)
Now we come to the point I have in mind in class when my students start complaining about radians. They complain about radians being unnatural and degrees being simpler. But I take the opposite tack and say we should abolish the historical baggage of the base 60 system and convert completely to the beauty of the radian.
The radian is a geometric ratio of two lengths: the arclength of a circle divided by its radius gives us the angle in radians. Much confusion reigns because degrees and radians are seen as non-units and as interchangeable to boot. Not so! Throw away the degree and adopt the radian as a true unit I say! Angles need units too! Rotation is as important as translation! Fight the power!
I didn't know degrees and radians weren't really units! What are they, then, if they aren't considered "units"?!?!
ReplyDeleteI think it's cool to find meaning behind numbers twelve and sixteen.
But I don't like that diagram of the hand. I think it's kind of creepy because thumbs don't actually move in such a full circle!
They are ratios (meters/meters = no units). I suppose the argument is that an angle doesn't have definite size until you specify a radius. It's very confusing - think of the difference between hertz (cycle/second) and angular speed (radians/second) - technically they are both (1/second) - same units but not the same!
ReplyDeleteI'll try to get a better pic. (probably one taken by you and unattributed ;) )
I'm so glad you posted this cause I see my mom doing math counting off on one hand sometimes with that posable thumb like you mentioned! I always thought, "what exactly is she counting and how is she doing it!" I never asked her but now I see.
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