Class 1/18/12

Salut mes amants des maths!

Monday was a holiday, but today you get another blog entry! Today's lecture doesn't quite match up with the readings assigned on the tentative schedule so we'll be looking more into sections 2.1 & 2.2 in Friday's blog entry. 

Today's class dealt with a lot of the material covered in previous sections of chapter 1. Specifically we talked about false position as it was used by both the Egyptian and Babylonian civilizations.

In the case of the Egyptains--who used it to deal with linear equation and proportional reasoning--the Rhind Papyrus illustrates that the method of false position consists of assuming a convenient but probably incorrect answer and then adjusting it by proportionality. For example, consider the following problem: 

Problem 26 of the Rhind Papyrus asked to find a quanitty such that when it is added to 1/4 of itself the result is 15. The scribe's solution was as follows: "Assume [the answer is] 4. Then 1 [+ 1/4] of 4 is 5. .  . Mulitply 5 so as to get 15. The answer is 3. Multiply 3 by 4. The answer is 12."

This interesting procedure used by scribe is used in several other problems similar to this, so one can consider the method an algorithm for the solution of a linear equation of this type. Unfortunately, there is no evidence pointing to the discovery of said algorithm nor information about why it works, but it is obvious that Egyptian scribes were well aware of the idea of proportionality of two quantities.

For the Babylonians, evidence shows that Babylonian scribes may have also used false position to deal with linear systems of equations. Although, presumably, the Babylonians solved complex single linear equations by false positions, the few problems that thought to show this do not reveal their exact methods; the answer is sometimes simply given as in shown in a problem from tablet YBC 4652 in which the scribe presents the answer to the problem, but doesn't give detailed steps as to how it was found. 


In addition to false position, lecture also covered the Egyptians approximation of pi which was [(8^2) (4)]/81  = 3.16049. They used this value to calculate the area of a circle with a given diameter. Also, the methods Babylonian scribes used to approximate the value of the square root of a natural number N. Evidence shows that scribes were able to derive the equation (a/2) + (N/2a) where a is the side of the square in the geometric representation of 2ac + c^2 = b = N - a^2. 

That's all for today, mes amis. Until next time!
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