History of the Quadratic Equation Sketch 10

1. History of the Quadratic EquationSketch 10 By: Stephanie Lawrence & Jamie Storm

2. Introduction • Around 2000 BC Egyptian, Chinese, and Babylonian engineers acquired a problem. • When given a specific area, they were unable to calculate the length of the sides of certain shapes. • Without these lengths, they were unable to design a floor plan for their customers.

3. Preview • Egyptian way of finding area • Babylonian and Chinese method • Pythagoras’ and Euclid’s contribution • Brahmagupta’s Contribution • Al-Khwarzimi’s Contribution

4. Egyptian’s Contribution • Their Problem • They had no equation • Tables • Solution!!!

5. Babylonian and Chinese Contribution • Started a method known as completing the square and used it to solve basic problems involving area. • Babylonians had the base 60 system while the Chinese used an abacus. These systems enabled them to double check their results.

6. Pythagoras’ and Euclid’s Contribution • In search of a more general method • Pythagoras hated the idea of irrational numbers • 268 years later Euclid proves him wrong

7. Euclid’s Contribution C D B A K L E F G • Using strictly a geometric approach. -If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half.

8. Brahmagupta’s Contribution • Indian/Hindu mathematician • Gives an almost modern solution of the quadratic equation, allowing negatives • Brahmagupta’s formula: s=a+b+c+d 2 s=semiperimeter Proof

9. Al-Khwarizmi’s Contribution • An Arab Mathematician • Lived in Baghdad; a generalist who wrote books on mathematics • He considered single squares and used the following formula: • This part of the quadratic formula was brought to Europe by the Jewish mathematician Abraham bar Hiyya (Savasaorda), who then wrote a book containing the complete solution to the quadratic equation in 1145 called Liber Embadorum

10. Squares equal to roots • Squares equal to numbers • Roots equal to numbers • Squares and roots equal to numbers • Squares and numbers equal to roots • Roots and numbers equal to squares Al-Khwarizmi’s Contribution • Gave a classification of the different types of quadratics which include: • His book Hisab al-jabr w-al-musqagalah (Science of the Reunion and the Opposition) starts out with a discussion of the quadratic equation.

11. The Discussion • Ex: One square and ten roots of the same are equal to thirty-nine dirhems. (i.e. What must be the square that when increased by ten of its own roots, amounts to thirty-nine?)

12. Can you Show this Geometrically? • We draw a square with side x and add a 10 by x rectangle. -The area is 39 • To determine x cut the number of roots in half • Move one of these halves to the bottom of the square (total area is still 39) • What is the area of the missing square? Total area? -Missing square: 25 Total area: 64 • So what is the length of one of the sides of this bigger square? -Answer: √ 64=8 • Therefore how can we conclude that x=3? Answer: Since the side of the big square is x+5, we can conclude that x=3

13. Answer: Back to The Discussion • X is the unknown; the problem translates to x2+10x=39 • Proof: You halve the number of roots, which in the present instance yields five. This you multiply by itself; the product is twenty-five. Add this to thirty-nine; the sum is sixty-four. Now take the root of this, which is eight, and subtract from it half the number of the roots which is five; the remainder is three. This is the root of the square which you sought for; the square itself is nine. ?? Does this remind you of anything ??

14. Try One • One square and 6 roots of the same are equal to 135 dirhems. (i.e. What must be the square which, when increased by 6 of its own roots amounts to 135?) • Answer: The square is 81

15. Extra Information • Methods and justifications became more sophisticated over time • From the 9th Century to the 16th Century, almost all algebra books started their discussions of quadratic equations with Al-Khwarizmi’s example • In the 17th Century European mathematicians began representing numbers with letters • Finally Thomas Harriot and Rene Descartes realized that it is much easier to write all equations as something = 0

16. Today • In 17th Century Rene Descartes published La Geometrie, which developed into modern mathematics • General equation: ax2+bx+c=0 • Written:

17. Timeline • 1500BC Egyptians made a table. • 580 BC Pythagoras hates irrational numbers. • 400 BC Babylonians solved quadratic equations. • 300 BC Euclid developed a geometrical approach and proved that irrational numbers exist. • 598-665AD Brahmagupta took the Babylonian method that allowed the use of negative numbers. • 800AD Al-Khwarizmi removed the negative and wrote a book Hisab al-jabr w-al- musqagalah (Science of the Reunion and the Opposition) • 1145AD Abraham bar Hiyya Ha-Nasi (Savasaorda)wrote the book Liber embadorum –contained the complete solution to the quadratic equation. • 1637AD Rene Descartes published La Geometrie containing the quadratic formula we know today.

18. References • Artmann, Benno (1999). Euclid: The Creation of Mathematics. New York, NY: Springer-Verlag. • Fishbein, Kala, & Brooks, Tammy. “Brahmagupta's Formula.” The University of Georgia. 16 September 2006 <http://jwilson.coe.uga.edu/EMT725/Class/Brooks/ Brahmagupta/Brahmagupta.htm>. • Katz, Victor J. (2004). A History of Mathematics. New York, NY: Pearson Addison Wesley. • Lawrence, Dr. Dnezana. “Math is Good for You!” 17 September 2006 <http://www.mathsisgoodforyou.com/index.htm>. • Merlinghoff, W, & Fernando, G (2002). Math Through the Ages A Gentle History for Teachers and Others.Farminton, ME: Oxton House Publishers. 105-108. • O'Conner, J. J., & E. F. Robertson. "History topic: Quadratic, cubic, and quartic equations." Quadratic etc equations. Feb. 1997. 4 Sept.2006<http://www-groups.dcs.st-and.ac.uk/~history/PrintHT/Quadratic_etc_equatins.html>. • "The History Behind the Quadratic Formula." BBC homepage. 13 Oct. 2004. 11 Sept. 2006<http://www.bbc.co.uk/dna/h2g2/A2982576>.