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

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