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 ID: 614756
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Slide1
Irrational NumbersSlide2
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
AnthyphairesisSlide3
Anthyphairesis
GO TO MATH HISTORY LESSON TO SEE PROCESS!!!! Slide4
InComMensurability
Egyptiona
and Babylonians calculated square roots
These were approximated
Not appreciated
Hippasus
of Metapontum Credited for discovering IrrationalsDied for revealing the discoverySlide5
InComMesurability
First recorded proof that is irrational
Euclid’s Elements
Here is the most popular proofSlide6
The History of pi
Approximation of Pi
1650 BC:
Rhind
Papyrus x = 3.16045
950 BC Temple of Solomon:
π = 3Slide7
The History of pi
Approximation of Pi
2
50 BC: Archimedes 3.1418
150 CE: Ptolemy used a 360 – gon
3.14166
263 CE: Liu Hiu
used a 192 regular inscribed polygon
3.14159
480 CE:
Zu
Chongzhi used a 24576-gon 3.141929265Slide8
The History of pi
Definition of Pi
Ratio of Slide9
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 Slide10
Negative NumbersSlide11
Chinese Mathematics
200 BCE: Chinese Rod System
Commercial calculations
Red rods cancelled black rods
Amount Sold: Positive
Amount Spent: NegativeSlide12
Negative Numbers in India
Brahmagupta
– 7
th
Century Mathematician
1
st
wrote of negative numbers
Zero already had a value
Developed rules for negative numbers
Developed the Integers we know Slide13
Arithmetic rules with Integers
Brahmagupta’s
work
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 fortuneA fortune subtracted from zero is a debt
Translation to modern day
Negative – 0 = negative
Positive – 0 = positive
0 – 0 = 0
0 – negative = positive
0 – positive = negativeSlide14
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 debtSlide15
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?Slide16
Arabian mathematics
Also ignored negatives
Al-
Khwarizami’s
Algebra book –
780 CE
Acknowledged BrahmaguptaHeaviily influenced by the Greeks
Called Negative Results “meaningless”Slide17
Arabian mathematics – cont’d
Al-
Samaw’al
(1130 – 1180 CE)
Shining Book of Calculations
Produced statements regarding algebra
Had no difficulty handling negative expressions
His contribution to math
al-
Samawal
is said to have been developing algebra of polynomials
He introduced decimals, well before its appearance in EuropeSlide18
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.Slide19
European mathematics
15
th
century
Arabs brought negatives to Europe
Translated ancient Islamic and Byzantine texts
Spurred solutions to quadratics and cubicsSlide20
European mathematics
Luca
Pacioli
(1445 – 1517)
Summa de
arithmetica
, geometriaDouble Entry Book-KeepingHe kept the use of negatives alive
John Wallis ( 1616-1703)
English
Invented Number LineSlide21
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”Slide22
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.”Slide23
Potential Infinity vs
Actual InfinitySlide24
History of Negative Numbers:
http
://nrich.maths.org/5961
https://brilliant.org/discussions/thread/discovery-of-irrational-numbers/
https
://www.math.rutgers.edu/~cherlin/History/Papers2000/wilson.html
MacTutor
History of Mathematics:
http
://
www-history.mcs.st-and.ac.uk
SOURCES