Ancient Babylon and Quadratics


Any schoolkid taking more than algebra 1 will probably tell you that this form:

Image

is the algebraic expression of a Quadratic Equation.  Quadratic equations are important for a lot of reasons, but the most real-life example is measuring land ownership, which has been with us since civilization existed.  The Latin word for “square” (a handy form for measuring land) is quadratus, which is where “quadratic” comes from.

Solving for x is probably something most post-algebraic people can do with a little boning up on theory:

ImageThat’s modern notation.  Naturally.  You can find this in a textbook, or wikipedia, or pretty much anywhere.  However, most people don’t think like algebraic notation.  I know I don’t.  The Babylonians certainly didn’t.  What’s cool– blog-postingly cool… is that they had a rule of thumb method for handling quadratics which pretty much did the same thing as algebraic notation.

ImageThis contains the solution to the following problem:

The area of a square, added to the side of the square comes to 0.75.  What is the side of the square?

The tablet actually works out a solution step by step, using the Babylonian rule of thumb for quadratics.  Like so:

  1. “I have added the area and the side of my square. 0.75″
  2. “You write down 1, the coefficient”
  3. “You break half of 1. 0.5″
  4. “You multiply 0.5 and 0.5.  0.25.”
  5. “You add 0.25 and 0.75.  1″
  6. “This is the square of 1″
  7. “Subtract 0.5, which you multiplied”
  8. “0.5 is the side of the Square”

That’s pretty impressive.  Using the mathematics they had on hand, and logic, they figured out something pretty danged important.  This is roughly arriving at this formula:

to solve 

Not bad for no computers, no google, no internet!

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