Treebeard's Homepage : Stumpers

# Treebeard's Stumper Answer1 May 98

Metric Time

The clock is ticking as we head towards The Dunn Middle School Science Fair next Friday. Our way of measuring time goes back to ancient Babylon, but 12, 24, and 60 are curious factors. A metric alternative was proposed during the French Revolution: 10 days per metric week, two periods of 10 metric hours per day, 100 metric minutes per hour, and 100 metric seconds per minute. What would a metric clock look like? Are the metric intervals longer or shorter than normal? Does our normal system have any benefit besides familiarity?

With a 20 metric hour day, the hours would be 12 minutes longer, and the 100 metric minutes and seconds would be a bit shorter, 43.2 and 0.432 seconds respectively. Usable. I sketched a 10 hour metric clock and found the problem. 10 doesn't divide into quarters or thirds like 12 does. 12 and 60 have more factors than 10 and 100, which makes them especially useful for dividing intervals. It's interesting that 5,280 (feet per mile) has 48 different factors, while 1,000 only has 16. Is this design or coincidence? Bonus question: Science Fair starts at 5:00 p.m. on Friday. What time is that on a metric clock?

Note: Graybear answered (among other things) that

It's by design since larger measurements were extrapolated from smaller ones. 5:00 standard time would convert to 4:16:66.7 in metric time. What other benefits of standard time were thought of? If we went to metric time, we would probably stop using quarter-hours in favor of two-tenth-hours.

I asked Graybear to elaborate on the claim that "larger measurements were extrapolated from smaller ones" and he answered:

There are a lot of steps between a foot and a mile which give more factors. 3 feet = 1 yard, 2 yards = 1 fathom, 11 fathoms = 1 chain, 10 chains = 1 furlong, 8 furlongs = 1 mile. 3x2x11x10x8 = 5280 = 2x2x2x2x2x3x5x11 or 8 prime factors. Even if you compare it with the Roman 'metric' mile of 5000 feet (more comparable in value to 5280 than 1000), you only have 7 prime factors (2x2x2x5x5x5x5). But, as Paul Harvey says, the rest of the story is how many ways those prime factors can be uniquely combined. Using 5280 as an example, all of the factors can be expressed in the form: 2^a * 3^b * 5^c * 11^d where a,b,c,d are integers, and a is between 0 and 5 (6 possible values), b is either 0 or 1, as is c and d. The total number of factors is therefore 6*2*2*2=48. The number of factors for 5000 is 4*5=20
Thanks Graybear. Yes, our system has character!

Back to Stumper