March 28
Plan:
- Questions (There should be some from the homework. I may postpone answering them until I've read the homework.)
- Starting Chapter 5 - solving equations. Introduction/outline, then Euclid's solution to quadratics.
Discussion:
The discussion page can be found here, for observations and remarks.
Today in Class:
We started Chapter 5 which is concerned with solving equations.
Topics for this chapter include: solving linear equations, quadratic equations, cubic and finally fourth degree
Solving Equations with one unknown: ax + b = c
- al-Khwarizmi defined two actions al-jabr, the movement of unknowns, and al-muqabala, the movement of constants in equations.
- The babylonians also knew how to solve this equation
Solving Quadratic Equations (Refer to example on 205 of the textbook)
- First done numerically by the Babylonians (pg 206 in the textbook shows the babylonian way to solve)
- Computations were done in base 60 (this may have been because of its relation to calendar year of 365 days)
- The algorithm that they used involved these equations involved the product and sum of two unknowns:
$xy=a\\x+y=b\\w=\sqrt{({\frac{a}{2})}^2 - b}\\x={( \frac{a}{2})^2}+w\\y={( \frac{a}{2})^2} -w$
These are the components of the quadratic formula