1. Babylonians And Diophantus
Quadratic Problems
Babylonians
And
Diophantus

2. This type of problem appears on a Babylonian tablet ~1700BC
Write this down.

3. Like we were doing in false position, let’s make a guess:
5 x 5 = 25 sq. units, but we wanted
xy = 16
How far off are we?
= 9 units of area.
Error = 9 sq. units

5. Let’s try another: Our solutions for the sides of the rectangle
are the lengths 8 and 2.
Let’s try another:

6. Try this one on our own: In English, I am looking for two numbers
whose product is 45 and whose sum is 18.

7. What did you get? The two numbers whose product is 45
and whose sum is 18 are … 3 and 15.

8. Try this in your group. Diophantus (200 - 284)
“There are, however, many other types of problems considered by Diophantus.” (MACTUTOR Biography)
Try this in your group.

9. Did you get 9 and 1 for y and z ?
= = = = = = = = = = = = = =
Diophantus’ idea is to plan ahead a bit.
He introduces x to be the difference that we
would soon be adding and subtracting from 5
if we were to do it the Babylonian’s way.
So he replaces y by (5 + x) and z by (5 - x):
yz = 9 becomes (5 + x)(5 - x) = 9

10. yz = 9 becomes (5 + x)(5 - x) = 9 25 - x2 = 9 x2 = 16 So, X = 4 **
So we get y = 9 and z = 1
just as Diophantus tells us to do.

11. What’s been the point of the two talks on False Position?
To see and understand how early mathematicians solved equations;
To experience a style of doing algebra that is different from the way we have been taught; and
To wonder at how much early scribes and others really understood.

12. Thanks for your attention and work !

13. How does this system relate to “our” quadratic formula?
Let’s consider:
* Then our first guess is 1/2 b.
* Our error will be (1/2 b)2 - c.
* Take the square root of that:

14. So that square root is the amount that we must add and
subtract from 1/2b. This is what we get for solutions:
and
leads to
->
Our equation:

15. OK, now we’re done!