Something Less Than Something Less Than Nothing? Nothing?

Negative NumbersNegative Numbers

By: Rebecca Krumrine and By: Rebecca Krumrine and Kristina YostKristina Yost

IntroductionIntroduction

Negative numbers were not generally Negative numbers were not generally accepted until a few hundred years accepted until a few hundred years ago. ago.

Negative numbers first appeared Negative numbers first appeared when people began to solve when people began to solve equations. equations.

Lets try a problem…Lets try a problem…

I am 18 years old and my sister is 11. I am 18 years old and my sister is 11. When will I be exactly twice as old as When will I be exactly twice as old as my sister? my sister?

How would you react to that answer if How would you react to that answer if you did not know about negative you did not know about negative numbers?numbers?

Main TopicsMain Topics

Development of concepts of negative Development of concepts of negative numbers in…numbers in… ChinaChina GreeceGreece IndiaIndia Middle EastMiddle East EuropeEurope

China 100BCE – 50BCEChina 100BCE – 50BCE

In the “Nine Chapters of Mathematical Art” In the “Nine Chapters of Mathematical Art” they used red rods as positive coefficients they used red rods as positive coefficients and black rods for negative coefficients to and black rods for negative coefficients to explain methods for finding area of figures. explain methods for finding area of figures.

The Nine Chapters also included rules for The Nine Chapters also included rules for dealing with negative numbers. dealing with negative numbers.

Greece 570BCE – 300BCEGreece 570BCE – 300BCE

Greeks ignored negative numbers Greeks ignored negative numbers completely. completely.

Aristotle made a distinction between Aristotle made a distinction between numbers and magnitude, but gave no numbers and magnitude, but gave no indications of the concept of negative indications of the concept of negative numbers. numbers.

Euclid continued this distinction in his Euclid continued this distinction in his work work Elements.Elements.

Greece 3Greece 3rdrd century CE century CE

Diophantus did not deal with negative Diophantus did not deal with negative numbers but he was aware of rules for numbers but he was aware of rules for multiplying with the minus and solving multiplying with the minus and solving equations. equations.

In book V of his Arithmetica, he In book V of his Arithmetica, he encounters the equation 4x+20 = 4encounters the equation 4x+20 = 4 He believes that this problem is absurd, He believes that this problem is absurd,

since to him 4x + 20 meant adding since to him 4x + 20 meant adding something to 20 to equal 4. something to 20 to equal 4.

India 7India 7thth century CE century CE

Brahmagupta recognized and worked with Brahmagupta recognized and worked with negative numbers. negative numbers. Positive numbers were possessions and negative Positive numbers were possessions and negative

numbers were debts numbers were debts Stated rules for adding, subtracting, Stated rules for adding, subtracting,

multiplying, and dividing negative numbers in multiplying, and dividing negative numbers in his work his work Correct Astronomical System of Correct Astronomical System of Brahma.Brahma.

Expanded on Diophantus concepts of the Expanded on Diophantus concepts of the quadratic equations (axquadratic equations (ax22 + bx = c, bx + c = ax + bx = c, bx + c = ax22, , axax22 + c = bx) using negative numbers forming + c = bx) using negative numbers forming the general form of the quadratic equations. the general form of the quadratic equations.

India 12India 12thth century CE century CE

Bhaskara gives negative roots, but rejects Bhaskara gives negative roots, but rejects the negative root since it was the negative root since it was inappropriate in the context of the inappropriate in the context of the problem. problem. “…“…For people have no clear understanding in For people have no clear understanding in

the case of a negative quantity” the case of a negative quantity”

Middle East 9Middle East 9thth century CE century CE

Arabs were familiar with negative numbers Arabs were familiar with negative numbers from the work of India mathematicians, but from the work of India mathematicians, but they rejected them. they rejected them. Muhammad Ibn Musa Al-Khqarizimi did not use Muhammad Ibn Musa Al-Khqarizimi did not use

negative numbers or negative coefficients in his negative numbers or negative coefficients in his two books. two books.

Knew how to expand products such as Knew how to expand products such as

(x – a)(x – b), but they only used this concept (x – a)(x – b), but they only used this concept when the problems involved subtractions when the problems involved subtractions whose answers are positive. whose answers are positive.

Europe 16th CenturyEurope 16th Century

Negative numbers were still being ignored Negative numbers were still being ignored and considered as “fictitious solutions.”and considered as “fictitious solutions.”

Mathematicians of this time still resisted Mathematicians of this time still resisted negative numbers and thought of them as negative numbers and thought of them as “fictitious” or “absurd.”“fictitious” or “absurd.”

Some of the mathematicians of this time Some of the mathematicians of this time were: were: Cardano from Italy Cardano from Italy Stifel from Germany Stifel from Germany Viete from France Viete from France

Europe 17th CenturyEurope 17th Century

Negative numbers started to become Negative numbers started to become accepted. accepted.

Along with the acceptance, came the Along with the acceptance, came the realization of problems with negative realization of problems with negative numbers. numbers. I.e. square roots of negativesI.e. square roots of negatives

Rene Descartes partially accepted Rene Descartes partially accepted these numbers. these numbers.

Question:Question:

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

IMAGINARY!!IMAGINARY!!

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

17th century continued…17th century continued…

Many mathematicians who started Many mathematicians who started accepting negatives didn’t know where accepting negatives didn’t know where they belonged in relation to positives.they belonged in relation to positives. One math guy, John Wallis, thought that One math guy, John Wallis, thought that

negatives were larger than infinity. negatives were larger than infinity. Isaac Newton wrote a book in 1707 Isaac Newton wrote a book in 1707

called called Universal ArithmetickUniversal Arithmetick. In this . In this book he states, “Quantities are either book he states, “Quantities are either Affirmative or greater than nothing, or Affirmative or greater than nothing, or Negative, or less than nothing.” Negative, or less than nothing.”

Questions for thought…Questions for thought…

How can a quantity of something be How can a quantity of something be negative and less than nothing?negative and less than nothing?

Can you have a negative quantity of Can you have a negative quantity of books, food, clothing, or money?books, food, clothing, or money?

It was hard for people to grasp the It was hard for people to grasp the concept of negative numbers being concept of negative numbers being debt.debt.

Europe Middle 18th centuryEurope Middle 18th century

Negatives are officially accepted as real Negatives are officially accepted as real numbers!! numbers!!

Euler was fine with negatives during the Euler was fine with negatives during the writing of his book writing of his book Elements of AlgebraElements of Algebra. .

Even though negative numbers were known Even though negative numbers were known and used, it was common for people to still and used, it was common for people to still ignore them as results to equation systems. ignore them as results to equation systems.

It was still common practice to ignore a It was still common practice to ignore a negative results in any system of equations. negative results in any system of equations.

Europe 19th centuryEurope 19th century

Negatives finally become important Negatives finally become important enough to not be ignored. enough to not be ignored.

The works of Gauss, Galois, and The works of Gauss, Galois, and Abel really had a big impact on Abel really had a big impact on equation systems with negative equation systems with negative results. results.

Doubts of negative numbers finally Doubts of negative numbers finally disappear. disappear.

Summary Summary

Although negative numbers were Although negative numbers were “discovered” in BCE, negative numbers “discovered” in BCE, negative numbers were not completely accepted until the were not completely accepted until the 1800’s.1800’s.

Still, generally, mathematicians used Still, generally, mathematicians used negative numbers in computations, but did negative numbers in computations, but did not understand the concept of them.not understand the concept of them.

TimelineTimeline 44 thth century BCE- Aristotle made a distinction century BCE- Aristotle made a distinction

between numbers and magnitude.between numbers and magnitude. 100 BCE- In the 100 BCE- In the Nine Chapters of Mathematical Nine Chapters of Mathematical

ArtArt, the Chinese used negative numbers in , the Chinese used negative numbers in solving systems of equations.solving systems of equations.

33rdrd century CE- Diophantus solved equations with century CE- Diophantus solved equations with negative numbers in negative numbers in ArithmeticaArithmetica, but then , but then rejected the equation itself.rejected the equation itself.

77 thth century CE- Indians used negative numbers to century CE- Indians used negative numbers to represent debt.represent debt.

99 thth century CE – Arabs were familiar with negative century CE – Arabs were familiar with negative numbers, but rejected them.numbers, but rejected them.

1212 thth century CE- Bhaskara (Indian) gives negative century CE- Bhaskara (Indian) gives negative roots for quadratic equations, but rejects the roots for quadratic equations, but rejects the roots because people do not approve of negative roots because people do not approve of negative roots.roots.

Timeline continued…Timeline continued… 1616thth Century CE- European Mathematicians Century CE- European Mathematicians

thought of negative numbers as “fictitious” or thought of negative numbers as “fictitious” or “absurd.”“absurd.”

1717thth Century CE- Rene Descartes claims the Century CE- Rene Descartes claims the result of negative square roots as result of negative square roots as “imaginary.”“imaginary.”

1818thth Century CE- Negatives start to become Century CE- Negatives start to become accepted in Europe even though they are still accepted in Europe even though they are still commonly ignored.commonly ignored.

1919thth Century CE- Doubts of negative numbers Century CE- Doubts of negative numbers finally disappear and negatives are known finally disappear and negatives are known now as real numbers.now as real numbers.

ReferencesReferences Berlinghoff, William P. , and Fernando Q. Gouvea. Berlinghoff, William P. , and Fernando Q. Gouvea.

Math through the Ages A Gentle History for Teachers Math through the Ages A Gentle History for Teachers and Othersand Others. 1st ed. Farmington, Maine: Oxton House . 1st ed. Farmington, Maine: Oxton House Publishers, 2002.Publishers, 2002.

Katz, Victor J.. Katz, Victor J.. A History of MathematicsA History of Mathematics. New York: . New York: Pearson/Addison Wesley, 2004.Pearson/Addison Wesley, 2004.

Negative and non-negative numbers." Negative and non-negative numbers." WikipediaWikipedia. . 2006. 7 Sep 2006 2006. 7 Sep 2006 <http://en.wikipedia.org/wiki/Negative_numbers>.<http://en.wikipedia.org/wiki/Negative_numbers>.

"Number." "Number." WikipediaWikipedia. 2006. 7 Sep 2006 . 2006. 7 Sep 2006 <http://en.wikipedia.org/wiki/Number>.<http://en.wikipedia.org/wiki/Number>.

Smith, Martha K.. "History of Negative Numbers." Smith, Martha K.. "History of Negative Numbers." University of Texas at AustinUniversity of Texas at Austin. 19 Feb 2001. 9 Sep . 19 Feb 2001. 9 Sep 2006 2006 <http://www.ma.utexas.edu/users/mks/326K/Negnos.<http://www.ma.utexas.edu/users/mks/326K/Negnos.html>.html>.