Babylonians And Diophantus Quadratic Problems Babylonians And Diophantus

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

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? 25 - 16 = 9 units of area. Error = 9 sq. units

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

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

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

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.

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

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.

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.

Thanks for your attention and work !

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:

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:

OK, now we’re done!