# Can you explain this Babylonian math problem?November 30, 2019 4:02 PM   Subscribe

The link in the below explanation shows a math problem from thousands of years ago, in which someone appeared to have made a mistake. Can you explain the question and the mistake?

There is a number "16;40" which is being squared, to yield "4,37;46,40." I gather that Babylonians used a base 60 (ish) system... but what do the commas mean? What do the semicolons mean? What number is the scribe attempting to calculate?
posted by lewedswiver to Science & Nature (3 answers total) 4 users marked this as a favorite

My guess would be that commas separate "digits" and the semicolon is a decimal point. 16;40 is 16+40/60=16+2/3. The square of that is 277+7/9=4*60+37+46/60+40/3600 which is 4,37;46,40.
posted by eruonna at 4:18 PM on November 30 [3 favorites]

Here's Wikipedia's article on base 60, including how to decode that numbering system:
In the 1930s, Otto Neugebauer introduced a modern notational system for Babylonian and Hellenistic numbers that substitutes modern decimal notation from 0 to 59 in each position, while using a semicolon (;) to separate the integral and fractional portions of the number and using a comma (,) to separate the positions within each portion.
posted by WCityMike at 4:23 PM on November 30 [3 favorites]

a = 0;6,40 which is 6/60 + 40/3600 = 400/3600 = 1/9, so 1/a = 9
b = 4,37;40 which is 4*60 + 37 + 40/60 = 277 2/3

The formula is 1/a * 1/(1+b/a).

Do b/a first. Bear with me, we're gonna convert both to 1/3600s.
b = (4*21600 + 37*3600 + 40*60) = 999600/3600
a = 6*60 + 40 = 400/3600
b/a = (999600/3600) / (400/3600)
which reduces to 999600/400 which is 2499

Babylonian dude expressed that correctly as 41,39. Check: 41*60 = 2460, 2460 + 39 = 2499.

Now what should be the easiest part: add 1 to that. Should get 41,40 which is 2500.

But, oops, he added 1 to the wrong part, giving 42,39 which is 2559.

The formula should go 1/a * 1/2500. Remember 1/a is 9, so this is 9/2500. That means b+a is 2500/9. Instead the dude got 2559/9.

What a dumbass, you say. But there was no decimal point in Babylonian mathematics. (No equivalent of Neugebauer's colon.) If you see 41,39 it might be 41*60 + 39 or it might be 41*3600 + 39*60 or it might be 41/60 + 39/3600, etc. So it's an easy mistake to make.

Fun fact: you think a 60-based system is crazy? Think about hours, minutes, seconds. Or degrees of the circle / minutes/ seconds. Yep, that comes from Babylonia.
posted by zompist at 8:37 PM on November 30 [9 favorites]

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