Professor Simonson History of Math Ingenuity

Fall 1997 MA 149

Write your solutions to these problems as a group. They will be discussed in detail during class. If you cannot solve a problem then report on your group discussion and explain where you got stuck. You will always have the opportunity of rewriting an assignment after class discussion. The assignment is due Wednesday after Columbus day.

1. Quadratic Pictures

Use a picture to prove that (a+b)² = a² + 2ab + b²

Use a picture to prove that the square root of X(X-A) is approximately X-A/2.

2. Babylonian Quadratic Methods

A babylonian manuscript from around 1800 B.C.E., says the following: The difference between two quantities is 7. The product of the two is 60. To find the quantitites, divide 7 by 2 and square the result. Add this to 60 and take the square root. This gives 8 and 1/2. Then add 3 and 1/2 to get 12; and subtract 3 and 1/2 to get 5. The numbers 5 and 12 is the solution.

What "formula" or method is this implying for solving quadratic equations?

How does it compare to the formula you know from high school?

3. Chinese Square Root Methods

The Chinese method for calculating square roots was widely taught in the US school system in the last 50 years. Here is an example: To find the square root of 25,281:

Let the square root be equal to 100a + 10b + c. Then set a to 1 and square 100 to get 10,000. Then subtract 10,000 from 25,281 leaving 15,281. (Note a=2 would give 40,000 which is too big).

100a 10b c

100a

10b

c

Then choose b=5, and calculate 50*100+50*100+50*50 = 12,500 and subtract this from 15,281 leaving 2781. (Note b=6 would give 15,600 which is too big). Then, set c=9, and calculate 9*(100 + 50) + 9*(100+50) + 9*9 = 2781 and subtract from 2781 leaving 0. The square root then is exactly 159.

Using a square with columns labeled 100a, 10b, c and d/10, use the Chinese procedure to calculate the square root of 41,000 to the nearest tenth.

4. More Mysteries from India

In the 12th century, Bhaskara in India, was posing questions like: Does 2x²+1= y² have any integer solutions greater than zero? Does 61x²+1= y² have any integer solutions? How are these questions similar/different from standard quadratic equations? Can you find solutions for either of the two problems above?