Something Less Than Nothing Negative Numbers By Rebecca

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

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

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

Main Topics Ø Development of concepts of negative numbers in… l l l China Greece India Middle East Europe

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

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

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

India 7 th century CE Ø Brahmagupta recognized and worked with negative numbers. l 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. Ø 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. Ø

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

Middle East 9 th century CE Ø Arabs were familiar with negative numbers from the work of India mathematicians, but they rejected them. l Ø 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.

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

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

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

IMAGINARY!! Ø Rene Descartes was the mathematician who called these results imaginary!

17 th century continued… Ø Many mathematicians who started accepting negatives didn’t know where they belonged in relation to positives. l One math guy, John Wallis, thought that negatives were larger than infinity. Ø 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. ”

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

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

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

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

Timeline Ø Ø Ø 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.

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

References Ø Ø Ø Berlinghoff, William P. , and Fernando Q. Gouvea. Math through the Ages A Gentle History for Teachers and Others. 1 st 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 <http: //en. wikipedia. org/wiki/Negative_numbers>. "Number. " Wikipedia. 2006. 7 Sep 2006 <http: //en. wikipedia. org/wiki/Number>. Smith, Martha K. . "History of Negative Numbers. " University of Texas at Austin. 19 Feb 2001. 9 Sep 2006 <http: //www. ma. utexas. edu/users/mks/326 K/Negnos. html>.