1. Irrational Numbers

2. When it came to measuring quantities in dissimilar vessels, such a proportion could only be found by finding a unit of measure by which both vessels could be measured as a whole number  Anthyphairesis

3. Anthyphairesis
GO TO MATH HISTORY LESSON TO SEE PROCESS!!!!

4. InComMensurability Egyptiona and Babylonians calculated square roots
These were approximated
Not appreciated
Hippasus of Metapontum
Credited for discovering Irrationals
Died for revealing the discovery

5. InComMesurability First recorded proof that is irrational
Euclid’s Elements
Here is the most popular proof

6. The History of pi Approximation of Pi
1650 BC: Rhind Papyrus x =
950 BC Temple of Solomon: π = 3

7. The History of pi Approximation of Pi 250 BC: Archimedes 3.1418
150 CE: Ptolemy used a 360 – gon
263 CE: Liu Hiu used a 192 regular inscribed polygon
480 CE: Zu Chongzhi used a gon

8. The History of pi
Definition of Pi
Ratio of

9. The history of Sometimes known as Euler’s constant.
The first references to “e” were in the appendix of a work by John Napier
The discovery of the constant itself is credited to Jacob Bernoulli
This is what Bernoulli was trying to solve when he discovered e

10. Negative Numbers

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

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

13. 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

14. 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

15. Negative numbers in greece
Ignored and Neglected by Greeks
300 CE: Diophantus wrote Arithmetica
4 = 4x + 20
“Absurd result”
Why would problems arising from Geometry cause Greeks to ignore negative numbers?

16. Arabian mathematics Also ignored negatives
Al-Khwarizami’s Algebra book –
780 CE
Acknowledged Brahmagupta
Heaviily influenced by the Greeks
Called Negative Results “meaningless”

17. Arabian mathematics – cont’d
His contribution to math
Al-Samaw’al (1130 – 1180 CE)
Shining Book of Calculations
Produced statements regarding algebra
Had no difficulty handling negative expressions
al-Samawal is said to have been developing algebra of polynomials He introduced decimals, well before its appearance in Europe

18. Al-Samawal’s Algebra
If we subtract a positive number from an ‘empty power’, the same negative number remains.
If we subtract the negative number from an ‘empty power’, the same positive number remains.
The product of a negative number by a positive number is negative, and be a negative number is positive.

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

20. European mathematics Luca Pacioli (1445 – 1517)
Summa de arithmetica, geometria
Double Entry Book-Keeping
He kept the use of negatives alive
John Wallis ( )
English
Invented Number Line

21. 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”

22. European mathematics 1770: Euler Swiss
“Since negative numbers may be considered as debts ... We say that negative numbers are less than 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.”

23. Potential Infinity vs Actual Infinity

24. SOURCES History of Negative Numbers: http://nrich.maths.org/5961
MacTutor History of Mathematics: