To solve the above ancient Math problem, lets do some translation first!
“ The first field gave 4 gur for each bur and the second field gave 3 gur for each bur. The first field gave 8,20 more than the second. The some of the field area is 30,0. What is the area of each field?
* “bur” and “sar” are units of field area.
1 bur= 1800 sar and 1 sar is about 36 square meters.
* “gur” and “ Sila” are units of grain volume.
1 gur= 300 Sila and 1 sila is about 1 liter.
As Babylonian numbers were written in sexagesimal we have some calculations first to convert 8,2 and 30 to base 10.
8,20( base 60)= 500 ( base 10) sila= 5/3 gur
30 ( base 60)= 1800 ( base 10) sar= 1 bur
Solving this word problem, using our modern math, looks very easy!
if x= area of first field and y= area of second field, we can have:
X+ Y= 1
4X- 3X= 5/3
and then the area of first field is X= 2/3 bur= 1200 sar
And the area of second field is Y=1/3 bur= 600 sar
We can also calculate the yield of each field:
The first field produced 4X grain , which is 4*2/3 gur or 800 sar grain
The second field produced 3Y grain, which is 3*1/3 gur or 300 sar grain.
But how the Babylonian solved this problem without using our modern algebra🧐
What would be this problem like if it was not just about content, but they were trying to get better at doing something? What would be solving a problem like, if it weren’t just about routine, if it required thinking with what they already knew and pushing further?
How we can get the whole game?
I have tried to think like them. Let’s see what information they had about the field, the yield,....
1-They knew that they have two fields with different productivity( maybe base on experience)
2- First field gave 500 sila more grain.
3- They know in field 1, each 1800 sar gave 4*300 sila( meaning 2/3 of field was 100% productive)
4- in field2, each 1800 sar gave 3*300 sila ( meaning 1/2 of field was 100% productive)
5- the sum of two area is 1800 sar
So they knew if two fields were equal, they would have: (2/3)*900= 600 sila from first field and (1/2)*900 from second field.
In this case the difference would be 600-450=150 not 500 sila!
So 500-150=350 sila must be divided into (2/3 + 1/2)=7/6 ( the productive parts of fields)
It means 350/(7/6)=300 sar
Then
The area of first field is 900+300=1200 sar
The area of second field is 900-300=600 sar
Until Islam’s Golden Age, the civilization that inherited Babylonian mathematics practiced algebra in procedural methods
Is our modern method better? Yes, it has the advantage of contribution of every great civilization over the last 4000 years
Resource:
Robert Coolman’s Blog, Thing Are Interesting: Ancient Babylonian Mathematics
Good interpretive work, Roya!
ReplyDelete