# 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?

See here: https://twitter.com/tamizdatum/status/1200619910809952256

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?

See here: https://twitter.com/tamizdatum/status/1200619910809952256

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?

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

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]

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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posted by eruonna at 4:18 PM on November 30 [3 favorites]