The Quadratic Quandary So Many Methods, So Little Time

1. The Quadratic QuandarySo Many Methods, So Little Time Kelly Jackson Camden County College CCLR Spring 2011

2. Major General Stanley… • Gilbert and Sullivan's operetta the Pirates of Penzance • "The Major General's Song" "I am the very model of a modern Major-General, I've information vegetable, animal, and mineral, I know the kings of England, and I quote the fights historical, From Marathon to Waterloo, in order categorical; I'm very well acquainted too with matters mathematical, I understand equations, both the simple and quadratical, About binomial theorem I'm teeming with a lot o' news-- With many cheerful facts about the square of the hypotenuse."

3. Does he have a brain? • After receiving his brains from the wizard in the 1939 film The Wizard of Oz, the Scarecrow recites the following: • "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side."

4. What is it? • The term "quadratic" comes from quadratus, which is the Latin word for "square". Area = x2 x x

5. What is it? • In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is: • where x represents a variable, and a, b, and c, constants, with a ≠ 0.

6. Babylonians • Babylonians (about 400 BC) were the first to solve quadratic equations? • This is an over simplification • Babylonians had no notion of 'equation'. • Solved problems which, in our terminology, would give rise to a quadratic equation. • The method is essentially one of completing the square. • Problems had answers which were positive, since the usual answer was a length.

7. Plimpton 322

8. Egypt The first known solution of a quadratic equation is the one given in the Berlin papyrus from the Middle Kingdom (ca. 2160-1700 BC) in Egypt.

9. Greeks • Pythagoras, 500BC • Knew that the ratio of the area of a square to its side length (its square root) was not always a whole number. • He would not accept that it could be irrational

10. Greeks • In about 300 BC Euclid developed a geometrical approach. • Finding a length which in our notation was the root of a quadratic equation. • Euclid had no notion of equation, coefficients etc. but worked with purely geometrical quantities. • He did allow for irrational solutions.

11. Greeks Diophantus, 3rd Century BCE • Diophantine equations are polynomial equations with integer coefficients to which only integer solutions are sought. • Most of the problems in Arithmetica lead to quadratic equations. • 3 different types of quadratic equations: • ax2 + bx = c • ax2 = bx + c • ax2 + c = bx • Diophantus was always satisfied with a rational solution and did not require a whole number which means he accepted fractions as solutions to his problems. • Diophantus considered negative or irrational square root solutions "useless", "meaningless", and even "absurd".

12. India • Hindu mathematicians took the Babylonian methods further. • Brahmagupta (598-665 AD) gives an, almost modern, method which admits negative quantities. • He also used abbreviations for the unknown

13. Arabs • The Arabs did not know about the advances of the Hindus so they had neither negative quantities nor abbreviations for their unknowns. • However al-Khwarizmi (800 BCE) gave a classification of different types of quadratics • no zero or negatives. • six chapters each devoted to a different type of equation (five to quadratics) • Equations made up of three types of quantities • roots, squares of roots and numbers • x, x2 and numbers.

14. Five Types of Quadratics • Al-Khwārizmī

15. Al-Khwārizmī One square, and ten roots of the same, are equal to thirty-nine dirhems. That is to say, what must be the square which, when increased by ten of its own roots, amounts to thirty-nine.

16. Al-Khwārizmī • The solution is this: 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 half the number of roots, which is five; the remainder is three. This is the root of the square which you sought for; the square itself is nine.

17. Al-Khwārizmī • In our symbols:

18. Al-Khwārizmī • His rule: • The Quadratic Formula (with a=1):

19. Al-KhwārizmīWhy Does it Work?

20. Al-KhwārizmīWhy Does it Work?

21. Al-KhwārizmīWhy Does it Work?

22. Al-KhwārizmīWhy Does it Work?

23. Al-KhwārizmīWhy Does it Work?

24. Al-KhwārizmīWhy Does it Work?

25. Europe • Jewish Mathematician Abraham bar Hiyya Ha-Nasi of Spain • Latin name Savasorda, • book Liber embadorum published in 1145 • first book published in Europe to give the complete solution of the quadratic equation

26. Europe • 1545 GirolamoCardano compiled the works related to the quadratic equations • Al-Khwarismi's solution with the Euclidean geometry. • He allows for the existence of complex, or imaginary numbers - that is, roots of negative numbers. • Amateur-mathematician François Viète, in France. • End of the 16th Century • Mathematical notation and symbolism • In 1637, when René Descartes • published La Géométrie, quadratic formula adopted the form we know today.