1. Negative Numbers

2. Chinese Mathematics 200 BCE: Chinese Rod System
Commercial calculations
Red rods cancelled black rods
Amount Sold: Positive
Amount Spent: Negative

3. Negative Numbers Brahmagupta – 7th Century Mathematician
1st wrote of negative numbers
Zero already had a value
Developed rules for negative numbers
Developed the Integers we know

4. Arithmetic rules with Integers
Brahmagupta’s work
Translation to modern day
A debt minus zero is a debt
A fortune minus zero is a fortune
Zero minus zero is zero
A debt subtracted from zero is a fortune
A fortune subtracted from zero is a debt
Negative – 0 = negative
Positive – 0 = positive
0 – 0 = 0
0 – negative = positive
0 – positive = negative

5. Arithmetic rules with Integers – cont’d
Brahmagupta’s work
A product of zero multiplied by a debt or fortune is zero
The product of zero multiplied by zero is zero
The product or quotient of two fortunes is a fortune
The product or quotient of two debts is a fortune
The product or quotient of a debt and a fortune is a debt
The product or quotient of a fortune and a debt is a debt
YOUR TURN –
Write an example that illustrates each rule on your white board

6. Negative numbers in greece
Ignored and Neglected by Greeks
On your white board:
Mathematics in Greece established through Geometry
300 CE: Diophantus wrote Arithmetica
4 = 4x + 20
“Absurd result”
Why would problems arising from Geometry cause Greeks to ignore negative numbers?

7. Arabian mathematics Also ignored negatives
Al-Khwarizami’s Algebra book –
780 CE
6 forms of linears and quadratics
Acknowledged Brahmagupta
Heaviily influenced by the Greeks
Called Negative Results “meaningless”

8. Arabian mathematics – cont’d
His contribution to math
Al-Samawal (1130 – 1180 CE)
Produced statements regarding algebra
“If we subtract a positive number from an empty power, the same negative number remains”
Also had statements about products and quotients
al-Samawal is said to have been developing algebra of polynomials Which one of the statements to the left suggests this? What do you think he was trying to say?

9. European mathematics 15th century Arabs brought negatives to Europe
Translated ancient Islamic and Byzantine texts
Spurred solutions to quadratics and cubics

10. European mathematics Luca Pacioli (1445 – 1517) Italian Summa
Double Entry Book-Keeping
John Wallis ( )
English
Invented Number Line

11. European mathematics 1758: Francis Maseres British
“ (negative numbers) darken the very whole doctrines of the equations and made dark the things which are in their nature excessively obvious and simple”

12. European mathematics 1770: Euler Swiss
“Since negative numbers may be considered as debts ... We say that negative numbers are less that nothing. Thus, when a man has nothing of his own, and owes 50 crowns, it is certain that he has 50 crowns less than nothing; though if any were to make a present of 50 crowns to pay his debt, he would still have nothing, though really richer than before.”

13. SOURCES History of Negative Numbers: http://nrich.maths.org/5961
Brahmagupta:
The History of Mathematics:
Negative Numbers:
MacTutor History of Mathematics: