The First Great Number Theorist Garner Ancient Fascination with Numbers Friendly Numbers Polygonal Numbers Diophantus and His Works Rhetorical Algebra Metrodorus Syncopated Algebra Diophantus Brahmagupta Pacioli Recorde Bombelli Viéte Homework The First Great Number Theorist The History of Mathematics, Part 11 Chuck Garner, Ph.D. Department of Mathematics Rockdale Magnet School for Science and Technology February 20, 2014
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The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
The First Great Number TheoristThe History of Mathematics, Part 11
Chuck Garner, Ph.D.
Department of Mathematics
Rockdale Magnet School for Science and Technology
February 20, 2014
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
OutlineAncient Fascination with Numbers
Friendly Numbers
Polygonal Numbers
Diophantus and His Works
Rhetorical AlgebraMetrodorus
Syncopated AlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
OutlineAncient Fascination with Numbers
Friendly Numbers
Polygonal Numbers
Diophantus and His Works
Rhetorical AlgebraMetrodorus
Syncopated AlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Two Aspects of Numbers
1. Relationship among numbers — “Arithmetic”
(Today we call this number theory)
2. Computing with numbers — “Logistic”
(Today we call this arithmetic)
Greek tendency to think of aspects in pairs:
I Even/Odd
I Measure/Number
I Magnitude/Multitude
I Arithmetic/Logistic
I Commensurable/Incommensurable
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Amicable Numbers
Pythagoreans defined amicable numbers: Two numbers
are amicable if each is the sum of the proper divisors of
fourth power ∆Υ∆ literally, square-squarefi�h power ∆KΥ square-cubesixth power KΥK cube-cube
minus Λ| leipis (ΛEIΨIΣ), lacking
the constant
◦M monades (MONA∆EΣ), units
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Diophantus
x3 + 13x2 + 8x would be found in Diophantus as
KΥα∆Υιγςη.
x3 − 8x2 + 2x − 3 would be found in Diophantus as
KΥαςβΛ|∆Υη◦
M γ.
I First instance of syncopated algebra
I Took a while to catch on
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Diophantus
x3 + 13x2 + 8x would be found in Diophantus as
KΥα∆Υιγςη.
x3 − 8x2 + 2x − 3 would be found in Diophantus as
KΥαςβΛ|∆Υη◦
M γ.
I First instance of syncopated algebra
I Took a while to catch on
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Brahmagupta
Brahmagupta
598-670
“As the sun eclipses the stars by its brilliancy, so the manof knowledge will eclipse the fame of others in assembliesof the people if he proposes algebraic problems, and still
more if he solves them.”
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Brahmagupta
I Addition indicated by juxtaposition
I Subtraction by a dot over the subtrahend
I Multiplication by “bha” wri�en a�er factors (from
bhavita, product)
I Division by writing divisor beneath dividend
I Square root by “ka” before number (from karana,
irrational)
I Unknown by “ya” (from yava�avat, so much as)
I Knowns prefixed by “ru” (from rupa, absolute
number)
I Additional unknowns by color; a second unknown
might be “ka” (from kalaka, black)
8xy +√
10− 7 may appear as ya ka 8 bha ka 10 ru 7.
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Brahmagupta
I Addition indicated by juxtaposition
I Subtraction by a dot over the subtrahend
I Multiplication by “bha” wri�en a�er factors (from
bhavita, product)
I Division by writing divisor beneath dividend
I Square root by “ka” before number (from karana,
irrational)
I Unknown by “ya” (from yava�avat, so much as)
I Knowns prefixed by “ru” (from rupa, absolute
number)
I Additional unknowns by color; a second unknown
might be “ka” (from kalaka, black)
8xy +√
10− 7 may appear as ya ka 8 bha ka 10 ru 7.
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Brahmagupta
I Addition indicated by juxtaposition
I Subtraction by a dot over the subtrahend
I Multiplication by “bha” wri�en a�er factors (from
bhavita, product)
I Division by writing divisor beneath dividend
I Square root by “ka” before number (from karana,
irrational)
I Unknown by “ya” (from yava�avat, so much as)
I Knowns prefixed by “ru” (from rupa, absolute
number)
I Additional unknowns by color; a second unknown
might be “ka” (from kalaka, black)
8xy +√
10− 7 may appear as ya ka 8 bha ka 10 ru 7.
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Brahmagupta
I Addition indicated by juxtaposition
I Subtraction by a dot over the subtrahend
I Multiplication by “bha” wri�en a�er factors (from
bhavita, product)
I Division by writing divisor beneath dividend
I Square root by “ka” before number (from karana,
irrational)
I Unknown by “ya” (from yava�avat, so much as)
I Knowns prefixed by “ru” (from rupa, absolute
number)
I Additional unknowns by color; a second unknown
might be “ka” (from kalaka, black)
8xy +√
10− 7 may appear as ya ka 8 bha ka 10 ru 7.
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Brahmagupta
I Addition indicated by juxtaposition
I Subtraction by a dot over the subtrahend
I Multiplication by “bha” wri�en a�er factors (from
bhavita, product)
I Division by writing divisor beneath dividend
I Square root by “ka” before number (from karana,
irrational)
I Unknown by “ya” (from yava�avat, so much as)
I Knowns prefixed by “ru” (from rupa, absolute
number)
I Additional unknowns by color; a second unknown
might be “ka” (from kalaka, black)
8xy +√
10− 7 may appear as ya ka 8 bha ka 10 ru 7.
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Brahmagupta
I Addition indicated by juxtaposition
I Subtraction by a dot over the subtrahend
I Multiplication by “bha” wri�en a�er factors (from
bhavita, product)
I Division by writing divisor beneath dividend
I Square root by “ka” before number (from karana,
irrational)
I Unknown by “ya” (from yava�avat, so much as)
I Knowns prefixed by “ru” (from rupa, absolute
number)
I Additional unknowns by color; a second unknown
might be “ka” (from kalaka, black)
8xy +√
10− 7 may appear as ya ka 8 bha ka 10 ru 7.
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Brahmagupta
I Addition indicated by juxtaposition
I Subtraction by a dot over the subtrahend
I Multiplication by “bha” wri�en a�er factors (from
bhavita, product)
I Division by writing divisor beneath dividend
I Square root by “ka” before number (from karana,
irrational)
I Unknown by “ya” (from yava�avat, so much as)
I Knowns prefixed by “ru” (from rupa, absolute
number)
I Additional unknowns by color; a second unknown
might be “ka” (from kalaka, black)
8xy +√
10− 7 may appear as ya ka 8 bha ka 10 ru 7.
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Brahmagupta
I Addition indicated by juxtaposition
I Subtraction by a dot over the subtrahend
I Multiplication by “bha” wri�en a�er factors (from
bhavita, product)
I Division by writing divisor beneath dividend
I Square root by “ka” before number (from karana,
irrational)
I Unknown by “ya” (from yava�avat, so much as)
I Knowns prefixed by “ru” (from rupa, absolute
number)
I Additional unknowns by color; a second unknown
might be “ka” (from kalaka, black)
8xy +√
10− 7 may appear as ya ka 8 bha ka 10 ru 7.
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Brahmagupta
I Addition indicated by juxtaposition
I Subtraction by a dot over the subtrahend
I Multiplication by “bha” wri�en a�er factors (from
bhavita, product)
I Division by writing divisor beneath dividend
I Square root by “ka” before number (from karana,
irrational)
I Unknown by “ya” (from yava�avat, so much as)
I Knowns prefixed by “ru” (from rupa, absolute
number)
I Additional unknowns by color; a second unknown
might be “ka” (from kalaka, black)
8xy +√
10− 7 may appear as ya ka 8 bha ka 10 ru 7.
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Brahmagupta
I Addition indicated by juxtaposition
I Subtraction by a dot over the subtrahend
I Multiplication by “bha” wri�en a�er factors (from
bhavita, product)
I Division by writing divisor beneath dividend
I Square root by “ka” before number (from karana,
irrational)
I Unknown by “ya” (from yava�avat, so much as)
I Knowns prefixed by “ru” (from rupa, absolute
number)
I Additional unknowns by color; a second unknown
might be “ka” (from kalaka, black)
8xy +√
10− 7 may appear as ya ka 8 bha ka 10 ru 7.
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Pacioli
Luca Pacioli
1445-1517
“The second thing necessary in business is to be a goodbookkeeper and ready mathematician.”
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Pacioli
I Revolutionized bookkeeping
I Wrote Summa de arithmetica in 1494
I Used abbreviations:
I p for plus (from piu, more)
I m for minus (from meno, less)
I co for the unknown (from cosa, thing)
I ce for square (from censo)
I cu for cube (from cuba)
I cece for fourth power
I ae for equals (from aequalis)
x3 − 8x2 + 2x − 3 = 9 would be cu m 8 ce p 2 co m 3 ae 9.
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Pacioli
I Revolutionized bookkeeping
I Wrote Summa de arithmetica in 1494
I Used abbreviations:
I p for plus (from piu, more)
I m for minus (from meno, less)
I co for the unknown (from cosa, thing)
I ce for square (from censo)
I cu for cube (from cuba)
I cece for fourth power
I ae for equals (from aequalis)
x3 − 8x2 + 2x − 3 = 9 would be cu m 8 ce p 2 co m 3 ae 9.
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Pacioli
I Used R for square root (from radix)
I RV as a grouping symbol (from radix universalis)
√7 +√
15 would be R V 7 p R 15.
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Pacioli
I Used R for square root (from radix)
I RV as a grouping symbol (from radix universalis)√7 +√
15 would be R V 7 p R 15.
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Recorde
Robert Recorde
1510-1558
“Besides the mathematical arts there is no infallibleknowledge, except that it be borrowed from them.”
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Recorde
I Wrote clear, readable math texts in English
I Worked as surveyor of the mines and the mint in
Ireland
I Wrote Whetstone of Wi�e in 1557
I Gave us our modern equals sign (=); took over a
century to be established
I Also used + and −
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Recorde’s The Whetstone of Wi�e
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Recorde’s The Whetstone of Wi�e
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Bombelli
Rafael Bombelli
1526-1572
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Bombelli
I Improved algebraic notation
I Gave rules for arithmetic with negatives
I Used grouping symbols
I Rq for square root and Rc for cube root
√7 +√
15 would be Rq b7 p Rq 15c.
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Bombelli
I Improved algebraic notation
I Gave rules for arithmetic with negatives
I Used grouping symbols
I Rq for square root and Rc for cube root√7 +√
15 would be Rq b7 p Rq 15c.
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Viéte
François Viéte
1540-1603
“There is no problem that cannot be solved.”
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Viéte
I Before Viéte, di�erent le�ers used for di�erent
powers
I Viéte used the same le�er properly qualified
I Our x is AI Our x2
is A quadratum, also A qI Our x3
is A cubum, also A c
I Viéte used vowels for unknowns and consonants
for knowns
x3− 8x2 + 2x− 3 = 9 would be A c− 8 A q + 2 A− 3 ae 9.
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Viéte
I Before Viéte, di�erent le�ers used for di�erent
powers
I Viéte used the same le�er properly qualified
I Our x is AI Our x2
is A quadratum, also A qI Our x3
is A cubum, also A c
I Viéte used vowels for unknowns and consonants
for knowns
x3− 8x2 + 2x− 3 = 9 would be A c− 8 A q + 2 A− 3 ae 9.
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
Viéte
I Before Viéte, di�erent le�ers used for di�erent
powers
I Viéte used the same le�er properly qualified
I Our x is AI Our x2
is A quadratum, also A qI Our x3
is A cubum, also A c
I Viéte used vowels for unknowns and consonants
for knowns
x3− 8x2 + 2x− 3 = 9 would be A c− 8 A q + 2 A− 3 ae 9.
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
OutlineAncient Fascination with Numbers
Friendly Numbers
Polygonal Numbers
Diophantus and His Works
Rhetorical AlgebraMetrodorus
Syncopated AlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
The First GreatNumber Theorist
Garner
AncientFascination withNumbersFriendly Numbers
Polygonal Numbers
Diophantus andHis Works
RhetoricalAlgebraMetrodorus
SyncopatedAlgebraDiophantus
Brahmagupta
Pacioli
Recorde
Bombelli
Viéte
Homework
HomeworkI Reminder: Weekender #4 due February 24
I Weekender #5 due February 28
I The same problem, 3000 years apart;
Reader, 5.C5
I A selection of 10 problems from the Arithmetica;
Reader, 5.D
I Old English mathematics from the Whetstone ofWi�e;