1. Something Less Than Nothing? Negative Numbers By: Rebecca Krumrine and Kristina Yost

2. Introduction  Negative numbers were not generally accepted until a few hundred years ago.  Negative numbers first appeared when people began to solve equations.

3. Lets try a problem…  I am 18 years old and my sister is 11. When will I be exactly twice as old as my sister?  How would you react to that answer if you did not know about negative numbers?

4. Main Topics  Development of concepts of negative numbers in… China China Greece Greece India India Middle East Middle East Europe Europe

5. China 100BCE – 50BCE  In the “Nine Chapters of Mathematical Art” they used red rods as positive coefficients and black rods for negative coefficients to explain methods for finding area of figures.  The Nine Chapters also included rules for dealing with negative numbers.

6. Greece 570BCE – 300BCE  Greeks ignored negative numbers completely.  Aristotle made a distinction between numbers and magnitude, but gave no indications of the concept of negative numbers.  Euclid continued this distinction in his work Elements.

7. Greece 3 rd century CE  Diophantus did not deal with negative numbers but he was aware of rules for multiplying with the minus and solving equations.  In book V of his Arithmetica, he encounters the equation 4x+20 = 4 He believes that this problem is absurd, since to him 4x + 20 meant adding something to 20 to equal 4. He believes that this problem is absurd, since to him 4x + 20 meant adding something to 20 to equal 4.

8. India 7 th century CE  Brahmagupta recognized and worked with negative numbers. Positive numbers were possessions and negative numbers were debts Positive numbers were possessions and negative numbers were debts  Stated rules for adding, subtracting, multiplying, and dividing negative numbers in his work Correct Astronomical System of Brahma.  Expanded on Diophantus concepts of the quadratic equations (ax 2 + bx = c, bx + c = ax 2, ax 2 + c = bx) using negative numbers forming the general form of the quadratic equations.

9. India 12 th century CE  Bhaskara gives negative roots, but rejects the negative root since it was inappropriate in the context of the problem. “…For people have no clear understanding in the case of a negative quantity” “…For people have no clear understanding in the case of a negative quantity”

10. Middle East 9 th century CE  Arabs were familiar with negative numbers from the work of India mathematicians, but they rejected them. Muhammad Ibn Musa Al-Khqarizimi did not use negative numbers or negative coefficients in his two books. Muhammad Ibn Musa Al-Khqarizimi did not use negative numbers or negative coefficients in his two books.  Knew how to expand products such as (x – a)(x – b), but they only used this concept when the problems involved subtractions whose answers are positive.

11. Europe 16th Century  Negative numbers were still being ignored and considered as “fictitious solutions.”  Mathematicians of this time still resisted negative numbers and thought of them as “fictitious” or “absurd.”  Some of the mathematicians of this time were: Cardano from Italy Cardano from Italy Stifel from Germany Stifel from Germany Viete from France Viete from France

12. Europe 17th Century  Negative numbers started to become accepted.  Along with the acceptance, came the realization of problems with negative numbers. I.e. square roots of negatives I.e. square roots of negatives  Rene Descartes partially accepted these numbers.

13. Question:  When taking the square root of a negative number, we refer to the result as….?

14. IMAGINARY!!  Rene Descartes was the mathematician who called these results imaginary!

15. 17th century continued…  Many mathematicians who started accepting negatives didn’t know where they belonged in relation to positives. One math guy, John Wallis, thought that negatives were larger than infinity. One math guy, John Wallis, thought that negatives were larger than infinity.  Isaac Newton wrote a book in 1707 called Universal Arithmetick. In this book he states, “Quantities are either Affirmative or greater than nothing, or Negative, or less than nothing.”

16. Questions for thought…  How can a quantity of something be negative and less than nothing?  Can you have a negative quantity of books, food, clothing, or money?  It was hard for people to grasp the concept of negative numbers being debt.

17. Europe Middle 18th century  Negatives are officially accepted as real numbers!!  Euler was fine with negatives during the writing of his book Elements of Algebra.  Even though negative numbers were known and used, it was common for people to still ignore them as results to equation systems.  It was still common practice to ignore a negative results in any system of equations.

18. Europe 19th century  Negatives finally become important enough to not be ignored.  The works of Gauss, Galois, and Abel really had a big impact on equation systems with negative results.  Doubts of negative numbers finally disappear.

19. Summary  Although negative numbers were “discovered” in BCE, negative numbers were not completely accepted until the 1800’s.  Still, generally, mathematicians used negative numbers in computations, but did not understand the concept of them.

20. Timeline  4 th century BCE- Aristotle made a distinction between numbers and magnitude.  100 BCE- In the Nine Chapters of Mathematical Art, the Chinese used negative numbers in solving systems of equations.  3 rd century CE- Diophantus solved equations with negative numbers in Arithmetica, but then rejected the equation itself.  7 th century CE- Indians used negative numbers to represent debt.  9 th century CE – Arabs were familiar with negative numbers, but rejected them.  12 th century CE- Bhaskara (Indian) gives negative roots for quadratic equations, but rejects the roots because people do not approve of negative roots.

21. Timeline continued…  16 th Century CE- European Mathematicians thought of negative numbers as “fictitious” or “absurd.”  17 th Century CE- Rene Descartes claims the result of negative square roots as “imaginary.”  18 th Century CE- Negatives start to become accepted in Europe even though they are still commonly ignored.  19 th Century CE- Doubts of negative numbers finally disappear and negatives are known now as real numbers.

22. References  Berlinghoff, William P., and Fernando Q. Gouvea. Math through the Ages A Gentle History for Teachers and Others. 1st ed. Farmington, Maine: Oxton House Publishers, 2002.  Katz, Victor J.. A History of Mathematics. New York: Pearson/Addison Wesley, 2004.  Negative and non-negative numbers." Wikipedia. 2006. 7 Sep 2006.  "Number." Wikipedia. 2006. 7 Sep 2006.  Smith, Martha K.. "History of Negative Numbers." University of Texas at Austin. 19 Feb 2001. 9 Sep 2006.