History of the Quadratic Equation Sketch 10 By: Stephanie Lawrence & Jamie Storm

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.

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

Egyptian’s Contribution Their Problem They had no equation Tables Solution!!! Egyptians had lofts in which they would store bales of papyrus. They wanted to find the best fitting design for the shape and amount of papyrus they owned. Since the Egyptians did not have an equation to solve this problem, the area of all possible sides and shapes of rectangles, squares, and T-shapes were calculated. Egyptian scribes recorded this information in a master look-up table. Engineers used this table to find the answers to their customer’s problems.

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. There are no indications that these people used a specific mathematical procedure to find out the solutions, so probably some educated guessing was involved.

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 Geometry and trigonometry provided the means by which the problem of finding the perimeter for a particular area was made possible. Now a general method could be used, rather than a table or making an educated guess. Pythagoras realized that “the ratios between the area of a square and the respective length of the side - the square root - were not always an integer, but he refused to allow for proportions other than rational” ones. In his book “The Elements”, Euclid expanded upon Pythagoras idea and discovered that proportion is not always rational. Thus proving the existence of irrational numbers.

Euclid’s Contribution 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. A B C D K F For let a straight line AB be cut into equal segments at C and into unequal segments at D. 2. I say that the rectangle contained by AD, DB together with the square on CD is equal to the square on CB. 3. The proposition is true because rectangle AL and DF are of equal content. 4. Later in the 17th century, mathematicians used propositions II-5 and II-6 as geometric justifications of the standard algebraic solutions of quadratic equations. G E L

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

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: 1.Some claim that Al-Khwarizmi is the father of Algebra while others say it was Diphoantus. Diphoantus was a Greek mathematician who lived during the third century AD in Alexandria, Egypt. The main difference between the two mathematicians is that Al-Khwarizmi's algebra was more involved with what we today call algebraic manipulation, and Diophantus' algebra was more involved with the theory of numbers. 2.Al-Khwarizmi was born in what is now Uzbekistan but lived in Baghdad where he studied in an academy of science called “The House of Wisdom”. 3.His work with the quadratic equation was made famous by the Jewish mathematician Abraham bar Hivva who is recognized by his Latinized name Savasaorda. Savasaorda brought Al-Khwarizmi’s work to Europe publishing it in his book Liber Embadorum. 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

Al-Khwarizmi’s Contribution Gave a classification of the different types of quadratics which include: 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 It is important to note that Al-Khwarizmi rejected negative solutions since they were not yet accepted by mathematicians in Arabia. The Arabs did not know about the advances of the Hindu mathematicians, who allowed for negative solutions (Brahmagupta). So the Arab mathematicians of the time had neither negative quantities nor abbreviations for their unknowns. Al-Khwarizmi gave a classification to different types of quadratic equations. However, he only gave numerical examples of each. Squares equal to roots. x2 = x Squares equal to numbers. x2 = 4 Roots equal to numbers. x = 2 Squares and roots equal to numbers, e.g. x2 + 10x = 39. Squares and numbers equal to roots, e.g. x2 + 21 = 10x. Roots and numbers equal to squares, e.g. 3x + 4 = x2. 3. The title of his book is Hisab al-jabr w-al-musqabalah, which means "Science of the Reunion and the Opposition" or in more modern terms "science of the transposition and cancellation". 4. Al-Khwarizmi gives the rule for solving each type of equation, essentially the familiar quadratic formula given for a numerical example in each case, and then a proof for each example which is a geometrical completing the square. His book Hisab al-jabr w-al-musqagalah (Science of the Reunion and the Opposition) starts out with a discussion of the quadratic equation.

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?)

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

?? Does this remind you of anything ?? Back to The Discussion X is the unknown; the problem translates to x2+10x=39 Answer: 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. 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 ??

Try One Answer: The square is 81 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

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

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

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.

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>.