the ancient and medieval: china and india · india the nine chapters rod numerals ˇ 13th century...
TRANSCRIPT
![Page 1: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/1.jpg)
ChinaIndia
The Ancient and Medieval: China and India
Douglas Pfeffer
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 2: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/2.jpg)
ChinaIndia
Table of contents
1 China
2 India
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 3: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/3.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Outline
1 China
2 India
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 4: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/4.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 5: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/5.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Civilizations along the Yangtze and Huanghe (or Yellow) riversare comparable in age to those of the Nile andTigris/Euphrates
Unfortunately, however, far fewer mathematical texts havesurvivedThe majority of what we have are not original texts, butreplicas
The oldest text we currently have is the Zhoubi Suanjing(Chou Pei Suan Ching)
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 6: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/6.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Civilizations along the Yangtze and Huanghe (or Yellow) riversare comparable in age to those of the Nile andTigris/Euphrates
Unfortunately, however, far fewer mathematical texts havesurvived
The majority of what we have are not original texts, butreplicas
The oldest text we currently have is the Zhoubi Suanjing(Chou Pei Suan Ching)
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 7: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/7.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Civilizations along the Yangtze and Huanghe (or Yellow) riversare comparable in age to those of the Nile andTigris/Euphrates
Unfortunately, however, far fewer mathematical texts havesurvivedThe majority of what we have are not original texts, butreplicas
The oldest text we currently have is the Zhoubi Suanjing(Chou Pei Suan Ching)
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 8: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/8.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Zhoubi Suanjing
The oldest of the mathematical classics, it is estimated to befrom 1200 BCE or 100 BCE
Modern scholars put it to be written a little after 300 BCEnear the Han dynasty (202 BCE)Concerns astronomical calculations, the Pythagorean theorem,and fractionsWritten in the form of a dialog between a prince and hisminister regarding the calendar
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 9: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/9.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Zhoubi Suanjing
The oldest of the mathematical classics, it is estimated to befrom 1200 BCE or 100 BCE
Modern scholars put it to be written a little after 300 BCEnear the Han dynasty (202 BCE)Concerns astronomical calculations, the Pythagorean theorem,and fractionsWritten in the form of a dialog between a prince and hisminister regarding the calendar
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 10: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/10.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Zhoubi Suanjing
The oldest of the mathematical classics, it is estimated to befrom 1200 BCE or 100 BCE
Modern scholars put it to be written a little after 300 BCEnear the Han dynasty (202 BCE)
Concerns astronomical calculations, the Pythagorean theorem,and fractionsWritten in the form of a dialog between a prince and hisminister regarding the calendar
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 11: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/11.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Zhoubi Suanjing
The oldest of the mathematical classics, it is estimated to befrom 1200 BCE or 100 BCE
Modern scholars put it to be written a little after 300 BCEnear the Han dynasty (202 BCE)Concerns astronomical calculations, the Pythagorean theorem,and fractions
Written in the form of a dialog between a prince and hisminister regarding the calendar
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 12: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/12.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Zhoubi Suanjing
The oldest of the mathematical classics, it is estimated to befrom 1200 BCE or 100 BCE
Modern scholars put it to be written a little after 300 BCEnear the Han dynasty (202 BCE)Concerns astronomical calculations, the Pythagorean theorem,and fractionsWritten in the form of a dialog between a prince and hisminister regarding the calendar
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 13: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/13.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Zhoubi Suanjing
Its demonstration of the Pythagorean theorem
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 14: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/14.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
The Nine Chapters
Jiuzhang Suanshu (or Nine Chapters on the Mathematical Arts )
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 15: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/15.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
The Nine Chapters
Jiuzhang Suanshu (or Nine Chapters on the Mathematical Arts )
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 16: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/16.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
The Nine Chapters
The most influential of all ancient Chinese texts – pennedaround the beginning of the common era
Contained 246 problems on surveying, agriculture, engineering,taxation, calculation, the solutions of equations, and propertiesof right triangles
Similar in style to the Babylonian ‘problem sets’, very unlikethe contemporary Greek systematically expository treatises
Chapter 8 tackled simultaneous linear equations consideringsolutions that were both positive and negative
The final problem was one considering 4 equations and 5unknowns
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 17: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/17.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
The Nine Chapters
The most influential of all ancient Chinese texts – pennedaround the beginning of the common era
Contained 246 problems on surveying, agriculture, engineering,taxation, calculation, the solutions of equations, and propertiesof right triangles
Similar in style to the Babylonian ‘problem sets’, very unlikethe contemporary Greek systematically expository treatises
Chapter 8 tackled simultaneous linear equations consideringsolutions that were both positive and negative
The final problem was one considering 4 equations and 5unknowns
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 18: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/18.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
The Nine Chapters
The most influential of all ancient Chinese texts – pennedaround the beginning of the common era
Contained 246 problems on surveying, agriculture, engineering,taxation, calculation, the solutions of equations, and propertiesof right triangles
Similar in style to the Babylonian ‘problem sets’, very unlikethe contemporary Greek systematically expository treatises
Chapter 8 tackled simultaneous linear equations consideringsolutions that were both positive and negative
The final problem was one considering 4 equations and 5unknowns
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 19: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/19.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
The Nine Chapters
The most influential of all ancient Chinese texts – pennedaround the beginning of the common era
Contained 246 problems on surveying, agriculture, engineering,taxation, calculation, the solutions of equations, and propertiesof right triangles
Similar in style to the Babylonian ‘problem sets’, very unlikethe contemporary Greek systematically expository treatises
Chapter 8 tackled simultaneous linear equations consideringsolutions that were both positive and negative
The final problem was one considering 4 equations and 5unknowns
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 20: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/20.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
The Nine Chapters
The most influential of all ancient Chinese texts – pennedaround the beginning of the common era
Contained 246 problems on surveying, agriculture, engineering,taxation, calculation, the solutions of equations, and propertiesof right triangles
Similar in style to the Babylonian ‘problem sets’, very unlikethe contemporary Greek systematically expository treatises
Chapter 8 tackled simultaneous linear equations consideringsolutions that were both positive and negative
The final problem was one considering 4 equations and 5unknowns
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 21: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/21.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
The Nine Chapters
The Chinese were very fond of patterns.
The very first recorded Magic Square is found in this treatise:
Reportedly, this square was brought to man by a turtle fromthe River Luo in the days of the legendary Emperor Yii
Fun fact: This is the smallest (and unique up to rotation andreflection) non-trivial case of a magic square, measuring 3× 3
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 22: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/22.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
The Nine Chapters
The Chinese were very fond of patterns.
The very first recorded Magic Square is found in this treatise:
Reportedly, this square was brought to man by a turtle fromthe River Luo in the days of the legendary Emperor Yii
Fun fact: This is the smallest (and unique up to rotation andreflection) non-trivial case of a magic square, measuring 3× 3
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 23: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/23.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
The Nine Chapters
The Chinese were very fond of patterns.
The very first recorded Magic Square is found in this treatise:
Reportedly, this square was brought to man by a turtle fromthe River Luo in the days of the legendary Emperor Yii
Fun fact: This is the smallest (and unique up to rotation andreflection) non-trivial case of a magic square, measuring 3× 3
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 24: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/24.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
The Nine Chapters
The Chinese were very fond of patterns.
The very first recorded Magic Square is found in this treatise:
Reportedly, this square was brought to man by a turtle fromthe River Luo in the days of the legendary Emperor Yii
Fun fact: This is the smallest (and unique up to rotation andreflection) non-trivial case of a magic square, measuring 3× 3
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 25: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/25.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
The Nine Chapters
Additionally, it seemed to suggest early matrix computations.
In solving the system:3x + 2y + z = 39
2x + 3y + z = 34
x + 2y + 3z = 26
It started by drafting up the following grid
1 2 32 3 23 1 1
26 34 39
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 26: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/26.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
The Nine Chapters
Additionally, it seemed to suggest early matrix computations.In solving the system:
3x + 2y + z = 39
2x + 3y + z = 34
x + 2y + 3z = 26
It started by drafting up the following grid
1 2 32 3 23 1 1
26 34 39
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 27: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/27.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
The Nine Chapters
Additionally, it seemed to suggest early matrix computations.In solving the system:
3x + 2y + z = 39
2x + 3y + z = 34
x + 2y + 3z = 26
It started by drafting up the following grid
1 2 32 3 23 1 1
26 34 39
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 28: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/28.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
The Nine Chapters
Through tedious descriptions:
it reduced the grid to the following:1 2 32 3 23 1 1
26 34 39
→
0 0 30 5 2
36 1 199 24 39
They finished the problem by back solving the equations36z = 99, 5y + z = 24, and 3x + 2y + z = 39.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 29: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/29.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
The Nine Chapters
Through tedious descriptions:
it reduced the grid to the following:1 2 32 3 23 1 1
26 34 39
→
0 0 30 5 2
36 1 199 24 39
They finished the problem by back solving the equations36z = 99, 5y + z = 24, and 3x + 2y + z = 39.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 30: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/30.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
The Nine Chapters
Through tedious descriptions:
it reduced the grid to the following:1 2 32 3 23 1 1
26 34 39
→
0 0 30 5 2
36 1 199 24 39
They finished the problem by back solving the equations36z = 99, 5y + z = 24, and 3x + 2y + z = 39.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 31: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/31.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Rod Numerals
Chinese numeration essentially consisted of two systems:The first (and less popular) was ‘multiplicative’: It had ciphersfor 1− 10 and then for powers of 10.
678 = 61007108
The second (and more popular) was the so-called RodNumerals
This system was positional and had ciphers for 1− 9 andmultiples of 10 up to 90Numbers, as in the first system, were read in pairs.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 32: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/32.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Rod Numerals
Chinese numeration essentially consisted of two systems:
The first (and less popular) was ‘multiplicative’: It had ciphersfor 1− 10 and then for powers of 10.
678 = 61007108
The second (and more popular) was the so-called RodNumerals
This system was positional and had ciphers for 1− 9 andmultiples of 10 up to 90Numbers, as in the first system, were read in pairs.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 33: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/33.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Rod Numerals
Chinese numeration essentially consisted of two systems:The first (and less popular) was ‘multiplicative’: It had ciphersfor 1− 10 and then for powers of 10.
678 = 61007108
The second (and more popular) was the so-called RodNumerals
This system was positional and had ciphers for 1− 9 andmultiples of 10 up to 90Numbers, as in the first system, were read in pairs.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 34: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/34.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Rod Numerals
Chinese numeration essentially consisted of two systems:The first (and less popular) was ‘multiplicative’: It had ciphersfor 1− 10 and then for powers of 10.
678 = 61007108
The second (and more popular) was the so-called RodNumerals
This system was positional and had ciphers for 1− 9 andmultiples of 10 up to 90Numbers, as in the first system, were read in pairs.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 35: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/35.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Rod Numerals
Ex: 56, 789
As in Babylonia, the ‘empty position’ came later with a roundO symbol
In a 1247 CE text the value 1, 405, 536 is given:
This was the most common system and, since it wascentesimal, was useful for computation.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 36: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/36.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Rod Numerals
Ex: 56, 789
As in Babylonia, the ‘empty position’ came later with a roundO symbol
In a 1247 CE text the value 1, 405, 536 is given:
This was the most common system and, since it wascentesimal, was useful for computation.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 37: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/37.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Rod Numerals
Ex: 56, 789
As in Babylonia, the ‘empty position’ came later with a roundO symbol
In a 1247 CE text the value 1, 405, 536 is given:
This was the most common system and, since it wascentesimal, was useful for computation.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 38: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/38.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
The Abacus
Rod numerals weren’t just a notation for computation: Actualbamboo rods were carried about in a bag by administratorsand used as calculation devices on ‘counting boards’
So dexterous were these counters that an 11th century writerdescribed them as “flying so quickly that the eye could notfollow their movement.”
These counting boards anticipated the abacus
The modern abacus are relatively new (c. 1500s), but theconcept dates back to the 500s
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 39: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/39.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
The Abacus
Rod numerals weren’t just a notation for computation: Actualbamboo rods were carried about in a bag by administratorsand used as calculation devices on ‘counting boards’
So dexterous were these counters that an 11th century writerdescribed them as “flying so quickly that the eye could notfollow their movement.”
These counting boards anticipated the abacus
The modern abacus are relatively new (c. 1500s), but theconcept dates back to the 500s
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 40: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/40.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
The Abacus
Rod numerals weren’t just a notation for computation: Actualbamboo rods were carried about in a bag by administratorsand used as calculation devices on ‘counting boards’
So dexterous were these counters that an 11th century writerdescribed them as “flying so quickly that the eye could notfollow their movement.”
These counting boards anticipated the abacus
The modern abacus are relatively new (c. 1500s), but theconcept dates back to the 500s
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 41: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/41.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Chinese Remainder Theorem
The earliest known statement of the theorem appears in the3rd-century book Sunzi Suanjing by Sunzi:
“There are certain things whose number is unknown. If wecount them by threes, we have two left over; by fives, we havethree left over; and by sevens, two are left over. How manythings are there?”
Sunzi’s work contains neither a proof nor a full algorithm.
Much later, algorithms would be developed by Indianmathematicians Aryabhata (6th century) and Brahmagupta(7th century), and in Fibonacci’s Liber Abaci (1202)
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 42: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/42.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Chinese Remainder Theorem
The earliest known statement of the theorem appears in the3rd-century book Sunzi Suanjing by Sunzi:
“There are certain things whose number is unknown. If wecount them by threes, we have two left over; by fives, we havethree left over; and by sevens, two are left over. How manythings are there?”
Sunzi’s work contains neither a proof nor a full algorithm.
Much later, algorithms would be developed by Indianmathematicians Aryabhata (6th century) and Brahmagupta(7th century), and in Fibonacci’s Liber Abaci (1202)
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 43: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/43.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Chinese Remainder Theorem
The earliest known statement of the theorem appears in the3rd-century book Sunzi Suanjing by Sunzi:
“There are certain things whose number is unknown. If wecount them by threes, we have two left over; by fives, we havethree left over; and by sevens, two are left over. How manythings are there?”
Sunzi’s work contains neither a proof nor a full algorithm.
Much later, algorithms would be developed by Indianmathematicians Aryabhata (6th century) and Brahmagupta(7th century), and in Fibonacci’s Liber Abaci (1202)
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 44: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/44.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Chinese Remainder Theorem
The earliest known statement of the theorem appears in the3rd-century book Sunzi Suanjing by Sunzi:
“There are certain things whose number is unknown. If wecount them by threes, we have two left over; by fives, we havethree left over; and by sevens, two are left over. How manythings are there?”
Sunzi’s work contains neither a proof nor a full algorithm.
Much later, algorithms would be developed by Indianmathematicians Aryabhata (6th century) and Brahmagupta(7th century), and in Fibonacci’s Liber Abaci (1202)
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 45: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/45.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
π
Early Chinese works used various approximations for pi :3, 3.1547,
√10, 92
29 ,14245
In 200 CE, Liu Hui reworked the Nine Chapters and, using a96−gon, achieved π ≈ 3.14. Then used a 3072−gon to getπ ≈ 3.14159
Of interest is that Liu Hui also, much as the Babylonians hadmuch earlier, correctly calculated the volume of the frustrumof a right-pyramid
Oddly enough, when he tackled the frustrum of a cone, he justused π = 3.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 46: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/46.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
π
Early Chinese works used various approximations for pi :3, 3.1547,
√10, 92
29 ,14245
In 200 CE, Liu Hui reworked the Nine Chapters and, using a96−gon, achieved π ≈ 3.14. Then used a 3072−gon to getπ ≈ 3.14159
Of interest is that Liu Hui also, much as the Babylonians hadmuch earlier, correctly calculated the volume of the frustrumof a right-pyramid
Oddly enough, when he tackled the frustrum of a cone, he justused π = 3.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 47: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/47.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
π
Early Chinese works used various approximations for pi :3, 3.1547,
√10, 92
29 ,14245
In 200 CE, Liu Hui reworked the Nine Chapters and, using a96−gon, achieved π ≈ 3.14. Then used a 3072−gon to getπ ≈ 3.14159
Of interest is that Liu Hui also, much as the Babylonians hadmuch earlier, correctly calculated the volume of the frustrumof a right-pyramid
Oddly enough, when he tackled the frustrum of a cone, he justused π = 3.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 48: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/48.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
π
Early Chinese works used various approximations for pi :3, 3.1547,
√10, 92
29 ,14245
In 200 CE, Liu Hui reworked the Nine Chapters and, using a96−gon, achieved π ≈ 3.14. Then used a 3072−gon to getπ ≈ 3.14159
Of interest is that Liu Hui also, much as the Babylonians hadmuch earlier, correctly calculated the volume of the frustrumof a right-pyramid
Oddly enough, when he tackled the frustrum of a cone, he justused π = 3.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 49: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/49.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
π
Early Chinese works used various approximations for pi :3, 3.1547,
√10, 92
29 ,14245
In 200 CE, Liu Hui reworked the Nine Chapters and, using a96−gon, achieved π ≈ 3.14. Then used a 3072−gon to getπ ≈ 3.14159
Of interest is that Liu Hui also, much as the Babylonians hadmuch earlier, correctly calculated the volume of the frustrumof a right-pyramid
Oddly enough, when he tackled the frustrum of a cone, he justused π = 3.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 50: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/50.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
π
The 400s saw the rise of mathematician Zu Chongzhi
He achieved: π ≈ 355113 ≈ 3.14159292
This approximation was not matched until the 1400sHow Changzhi achieved this approximation is not known
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 51: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/51.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
π
The 400s saw the rise of mathematician Zu Chongzhi
He achieved: π ≈ 355113 ≈ 3.14159292
This approximation was not matched until the 1400sHow Changzhi achieved this approximation is not known
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 52: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/52.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
From the 6th to the 10th century, a group of a dozen‘classics’ served as the foundation for mathematics taught inthe “School for the Sons of the State”
Primarily arithmetic and number theoryIncludes the Zhoughbi, Nine Chapters, and Liu Hui’s works
From the 10th to the 13th centuries, no new mathematicalbreakthroughs seem to have occurred
This is interesting since these centuries saw the invention ofpaper, the compass, and gunpowder
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 53: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/53.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
From the 6th to the 10th century, a group of a dozen‘classics’ served as the foundation for mathematics taught inthe “School for the Sons of the State”
Primarily arithmetic and number theoryIncludes the Zhoughbi, Nine Chapters, and Liu Hui’s works
From the 10th to the 13th centuries, no new mathematicalbreakthroughs seem to have occurred
This is interesting since these centuries saw the invention ofpaper, the compass, and gunpowder
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 54: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/54.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
13th Century
As the Sung dynasty ended, China saw the Mongol expansionand increased contact with Islam
Li Zhi of Peking (Beijing) was a hermit, scholar, andacamedician
He wrote Ceyuan Haijing (Sea-Mirror of the CircleMeasurements)Contained 170 problems about circles inscribed within (orcircumscribed without) a right triangle and determining therelationships between the sides and the radii
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 55: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/55.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
13th Century
As the Sung dynasty ended, China saw the Mongol expansionand increased contact with Islam
Li Zhi of Peking (Beijing) was a hermit, scholar, andacamedician
He wrote Ceyuan Haijing (Sea-Mirror of the CircleMeasurements)Contained 170 problems about circles inscribed within (orcircumscribed without) a right triangle and determining therelationships between the sides and the radii
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 56: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/56.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
13th Century
As the Sung dynasty ended, China saw the Mongol expansionand increased contact with Islam
Li Zhi of Peking (Beijing) was a hermit, scholar, andacamedician
He wrote Ceyuan Haijing (Sea-Mirror of the CircleMeasurements)Contained 170 problems about circles inscribed within (orcircumscribed without) a right triangle and determining therelationships between the sides and the radii
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 57: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/57.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
13th Century
As the Sung dynasty ended, China saw the Mongol expansionand increased contact with Islam
Li Zhi of Peking (Beijing) was a hermit, scholar, andacamedician
He wrote Ceyuan Haijing (Sea-Mirror of the CircleMeasurements)
Contained 170 problems about circles inscribed within (orcircumscribed without) a right triangle and determining therelationships between the sides and the radii
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 58: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/58.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
13th Century
As the Sung dynasty ended, China saw the Mongol expansionand increased contact with Islam
Li Zhi of Peking (Beijing) was a hermit, scholar, andacamedician
He wrote Ceyuan Haijing (Sea-Mirror of the CircleMeasurements)Contained 170 problems about circles inscribed within (orcircumscribed without) a right triangle and determining therelationships between the sides and the radii
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 59: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/59.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
13th Century
Yang Hui was a prolific arithmetician. Among hisachievements are advancements on:
Magic Squares of order > 3Summation of SeriesBinomial coefficients (Pascals Triangle)
Of interest, however, is that these last two contributions areseldom attributed to Hui. Similar results were published in themore popular text Precious Mirror by Zhu Shiji
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 60: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/60.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
13th Century
Yang Hui was a prolific arithmetician. Among hisachievements are advancements on:
Magic Squares of order > 3
Summation of SeriesBinomial coefficients (Pascals Triangle)
Of interest, however, is that these last two contributions areseldom attributed to Hui. Similar results were published in themore popular text Precious Mirror by Zhu Shiji
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 61: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/61.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
13th Century
Yang Hui was a prolific arithmetician. Among hisachievements are advancements on:
Magic Squares of order > 3Summation of Series
Binomial coefficients (Pascals Triangle)
Of interest, however, is that these last two contributions areseldom attributed to Hui. Similar results were published in themore popular text Precious Mirror by Zhu Shiji
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 62: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/62.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
13th Century
Yang Hui was a prolific arithmetician. Among hisachievements are advancements on:
Magic Squares of order > 3Summation of SeriesBinomial coefficients (Pascals Triangle)
Of interest, however, is that these last two contributions areseldom attributed to Hui. Similar results were published in themore popular text Precious Mirror by Zhu Shiji
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 63: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/63.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
13th Century
Yang Hui was a prolific arithmetician. Among hisachievements are advancements on:
Magic Squares of order > 3Summation of SeriesBinomial coefficients (Pascals Triangle)
Of interest, however, is that these last two contributions areseldom attributed to Hui. Similar results were published in themore popular text Precious Mirror by Zhu Shiji
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 64: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/64.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Zhu Shiji
Zhu Shiji was the last and greatest of the Sungmathematicians
Lived near Peking and eventually wandered China for 20 yearsteaching mathIn 1299, he wrote Suanxue qimeng (Introduction toMathematical Studies)
An elementary work that heavily influenced Korea and JapanWas lost until a replica was found in the 1800s
In 1303, he wrote Siyuan yujian (Jade Mirror of the FourOrigins)
The four origins were heaven, earth, man, and matter. Theyrepresented the four unknown quantities in a given equation.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 65: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/65.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Zhu Shiji
Zhu Shiji was the last and greatest of the Sungmathematicians
Lived near Peking and eventually wandered China for 20 yearsteaching math
In 1299, he wrote Suanxue qimeng (Introduction toMathematical Studies)
An elementary work that heavily influenced Korea and JapanWas lost until a replica was found in the 1800s
In 1303, he wrote Siyuan yujian (Jade Mirror of the FourOrigins)
The four origins were heaven, earth, man, and matter. Theyrepresented the four unknown quantities in a given equation.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 66: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/66.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Zhu Shiji
Zhu Shiji was the last and greatest of the Sungmathematicians
Lived near Peking and eventually wandered China for 20 yearsteaching mathIn 1299, he wrote Suanxue qimeng (Introduction toMathematical Studies)
An elementary work that heavily influenced Korea and JapanWas lost until a replica was found in the 1800s
In 1303, he wrote Siyuan yujian (Jade Mirror of the FourOrigins)
The four origins were heaven, earth, man, and matter. Theyrepresented the four unknown quantities in a given equation.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 67: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/67.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Zhu Shiji
Zhu Shiji was the last and greatest of the Sungmathematicians
Lived near Peking and eventually wandered China for 20 yearsteaching mathIn 1299, he wrote Suanxue qimeng (Introduction toMathematical Studies)
An elementary work that heavily influenced Korea and JapanWas lost until a replica was found in the 1800s
In 1303, he wrote Siyuan yujian (Jade Mirror of the FourOrigins)
The four origins were heaven, earth, man, and matter. Theyrepresented the four unknown quantities in a given equation.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 68: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/68.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Jade Mirror of the Four Origins
In this work we find the pinnacle of Chinese algebra –considering equations with degrees as high as 14
Also included a study of series:
12 + 22 + . . .+ n2 = n(n+1)(2n+1)3!
1 + 8 + 30 + 80 + . . .+ n2(n+1)(n+2)3! = n(n+1)(n+2)(n+3)(4n+1)
5!
No proofs are given for formulas like these and, interestingly,the subject died until that 1800s in China
Also included a systematic study of binomial coefficients andexhibited what is now mistakenly attributed to Blaise Pascal(1600s)
Featured Rod Numerals with the empty position OZhu himself claims the triangle is ‘known’ and ‘old’ and,notably, not his discovery
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 69: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/69.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Jade Mirror of the Four Origins
In this work we find the pinnacle of Chinese algebra –considering equations with degrees as high as 14
Also included a study of series:
12 + 22 + . . .+ n2 = n(n+1)(2n+1)3!
1 + 8 + 30 + 80 + . . .+ n2(n+1)(n+2)3! = n(n+1)(n+2)(n+3)(4n+1)
5!
No proofs are given for formulas like these and, interestingly,the subject died until that 1800s in China
Also included a systematic study of binomial coefficients andexhibited what is now mistakenly attributed to Blaise Pascal(1600s)
Featured Rod Numerals with the empty position OZhu himself claims the triangle is ‘known’ and ‘old’ and,notably, not his discovery
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 70: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/70.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Jade Mirror of the Four Origins
In this work we find the pinnacle of Chinese algebra –considering equations with degrees as high as 14
Also included a study of series:
12 + 22 + . . .+ n2 = n(n+1)(2n+1)3!
1 + 8 + 30 + 80 + . . .+ n2(n+1)(n+2)3! = n(n+1)(n+2)(n+3)(4n+1)
5!
No proofs are given for formulas like these and, interestingly,the subject died until that 1800s in China
Also included a systematic study of binomial coefficients andexhibited what is now mistakenly attributed to Blaise Pascal(1600s)
Featured Rod Numerals with the empty position OZhu himself claims the triangle is ‘known’ and ‘old’ and,notably, not his discovery
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 71: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/71.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Jade Mirror of the Four Origins
In this work we find the pinnacle of Chinese algebra –considering equations with degrees as high as 14
Also included a study of series:
12 + 22 + . . .+ n2 = n(n+1)(2n+1)3!
1 + 8 + 30 + 80 + . . .+ n2(n+1)(n+2)3! = n(n+1)(n+2)(n+3)(4n+1)
5!
No proofs are given for formulas like these and, interestingly,the subject died until that 1800s in China
Also included a systematic study of binomial coefficients andexhibited what is now mistakenly attributed to Blaise Pascal(1600s)
Featured Rod Numerals with the empty position OZhu himself claims the triangle is ‘known’ and ‘old’ and,notably, not his discovery
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 72: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/72.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Jade Mirror of the Four Origins
In this work we find the pinnacle of Chinese algebra –considering equations with degrees as high as 14
Also included a study of series:
12 + 22 + . . .+ n2 = n(n+1)(2n+1)3!
1 + 8 + 30 + 80 + . . .+ n2(n+1)(n+2)3! = n(n+1)(n+2)(n+3)(4n+1)
5!
No proofs are given for formulas like these and, interestingly,the subject died until that 1800s in China
Also included a systematic study of binomial coefficients andexhibited what is now mistakenly attributed to Blaise Pascal(1600s)
Featured Rod Numerals with the empty position OZhu himself claims the triangle is ‘known’ and ‘old’ and,notably, not his discovery
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 73: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/73.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Chinese ‘Pascals Triangle’
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 74: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/74.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Closing on the Chinese scene
After the 13th century, mathematics in China declined backinto commercial arithmetic and routine study of the NineChapters
We now turn our eyes westward toward the Indiansubcontinent...
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 75: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/75.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Closing on the Chinese scene
After the 13th century, mathematics in China declined backinto commercial arithmetic and routine study of the NineChapters
We now turn our eyes westward toward the Indiansubcontinent...
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 76: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/76.jpg)
ChinaIndia
The Nine ChaptersRod Numeralsπ13th Century
Closing on the Chinese scene
After the 13th century, mathematics in China declined backinto commercial arithmetic and routine study of the NineChapters
We now turn our eyes westward toward the Indiansubcontinent...
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 77: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/77.jpg)
ChinaIndia
Outline
1 China
2 India
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 78: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/78.jpg)
ChinaIndia
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 79: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/79.jpg)
ChinaIndia
Archaeological excavations at Mohenjo Daro and Harappagive evidence to an old and highly cultured civilization in theIndus Valley around 2650 BCE
Unfortunately, no mathematical documents
The area, however, was volatile through movement andconquest. Even Indian languages were not entirely uniform.
The Vedas, a group of ancient religious texts, do givendetailed building prescriptions for altars and the like
These prescriptions came in the form of the Sulbasutras or“rules of the chord”Eerily similar to Egyptian geometry (though few scholarssupport a strong connection to Egypt due to a lack ofcontinuity in Indian mathematics)
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 80: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/80.jpg)
ChinaIndia
Archaeological excavations at Mohenjo Daro and Harappagive evidence to an old and highly cultured civilization in theIndus Valley around 2650 BCE
Unfortunately, no mathematical documents
The area, however, was volatile through movement andconquest. Even Indian languages were not entirely uniform.
The Vedas, a group of ancient religious texts, do givendetailed building prescriptions for altars and the like
These prescriptions came in the form of the Sulbasutras or“rules of the chord”Eerily similar to Egyptian geometry (though few scholarssupport a strong connection to Egypt due to a lack ofcontinuity in Indian mathematics)
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 81: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/81.jpg)
ChinaIndia
Archaeological excavations at Mohenjo Daro and Harappagive evidence to an old and highly cultured civilization in theIndus Valley around 2650 BCE
Unfortunately, no mathematical documents
The area, however, was volatile through movement andconquest. Even Indian languages were not entirely uniform.
The Vedas, a group of ancient religious texts, do givendetailed building prescriptions for altars and the like
These prescriptions came in the form of the Sulbasutras or“rules of the chord”Eerily similar to Egyptian geometry (though few scholarssupport a strong connection to Egypt due to a lack ofcontinuity in Indian mathematics)
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 82: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/82.jpg)
ChinaIndia
Sulbasutras
Written by many authors (all in verse) as early as 1000 BCE
Contains, in part, of Pythagorean triples (although there islittle concrete evidence of Mesopotamian influence)
One such author, Apastamba, gave a number of geometricarguments
For example, a construction on how to, given a rectangle, draftup a square with the same area
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 83: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/83.jpg)
ChinaIndia
Sulbasutras
Written by many authors (all in verse) as early as 1000 BCE
Contains, in part, of Pythagorean triples (although there islittle concrete evidence of Mesopotamian influence)
One such author, Apastamba, gave a number of geometricarguments
For example, a construction on how to, given a rectangle, draftup a square with the same area
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 84: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/84.jpg)
ChinaIndia
Aryabhata
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 85: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/85.jpg)
ChinaIndia
Aryabhata
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 86: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/86.jpg)
ChinaIndia
Aryabhata
In 499 CE, Aryabhata wrote Aryabhatiya, a text on astronomyand mathematics
Written entirely in verseAlthough the names of Indian mathematicians exist beforehim, none of their work survived
Aryabhatiya, much like Euclid’s Elemets, was a summary ofearlier developments compiled by a single authorUnlike Elements, however, Aryabhatiya exhibited no deductivemethodology
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 87: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/87.jpg)
ChinaIndia
Aryabhata
In 499 CE, Aryabhata wrote Aryabhatiya, a text on astronomyand mathematics
Written entirely in verseAlthough the names of Indian mathematicians exist beforehim, none of their work survived
Aryabhatiya, much like Euclid’s Elemets, was a summary ofearlier developments compiled by a single authorUnlike Elements, however, Aryabhatiya exhibited no deductivemethodology
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 88: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/88.jpg)
ChinaIndia
Aryabhata
In 499 CE, Aryabhata wrote Aryabhatiya, a text on astronomyand mathematics
Written entirely in verseAlthough the names of Indian mathematicians exist beforehim, none of their work survived
Aryabhatiya, much like Euclid’s Elemets, was a summary ofearlier developments compiled by a single authorUnlike Elements, however, Aryabhatiya exhibited no deductivemethodology
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 89: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/89.jpg)
ChinaIndia
Aryabhatiya
The mathematics portion opened with the powers of 10 andrules on obtaining the square and cube roots of integers
It contained a mixed bag of correct and incorrect results:
Area of a triangle = half the product of the base and thealtitude (correct)Volume of a pyramid = half of the product of the bnase andthe altitude (incorrect)
Other examples include:
area of a circle vs. the volume of a spherearea of a trapezoid vs. area of an arbitrary plane figure
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 90: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/90.jpg)
ChinaIndia
Aryabhatiya
The mathematics portion opened with the powers of 10 andrules on obtaining the square and cube roots of integers
It contained a mixed bag of correct and incorrect results:
Area of a triangle = half the product of the base and thealtitude (correct)
Volume of a pyramid = half of the product of the bnase andthe altitude (incorrect)
Other examples include:
area of a circle vs. the volume of a spherearea of a trapezoid vs. area of an arbitrary plane figure
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 91: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/91.jpg)
ChinaIndia
Aryabhatiya
The mathematics portion opened with the powers of 10 andrules on obtaining the square and cube roots of integers
It contained a mixed bag of correct and incorrect results:
Area of a triangle = half the product of the base and thealtitude (correct)Volume of a pyramid = half of the product of the bnase andthe altitude (incorrect)
Other examples include:
area of a circle vs. the volume of a spherearea of a trapezoid vs. area of an arbitrary plane figure
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 92: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/92.jpg)
ChinaIndia
Aryabhatiya
The mathematics portion opened with the powers of 10 andrules on obtaining the square and cube roots of integers
It contained a mixed bag of correct and incorrect results:
Area of a triangle = half the product of the base and thealtitude (correct)Volume of a pyramid = half of the product of the bnase andthe altitude (incorrect)
Other examples include:
area of a circle vs. the volume of a sphere
area of a trapezoid vs. area of an arbitrary plane figure
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 93: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/93.jpg)
ChinaIndia
Aryabhatiya
The mathematics portion opened with the powers of 10 andrules on obtaining the square and cube roots of integers
It contained a mixed bag of correct and incorrect results:
Area of a triangle = half the product of the base and thealtitude (correct)Volume of a pyramid = half of the product of the bnase andthe altitude (incorrect)
Other examples include:
area of a circle vs. the volume of a spherearea of a trapezoid vs. area of an arbitrary plane figure
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 94: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/94.jpg)
ChinaIndia
Aryabhatiya
At one point, Aryabhata writes:
“Add 4 to 100, multiply by 8, and add 62,000. The result isapproximately the circumference of a circle of which thediameter is 20,000”
By this estimate, π ≈ 3.1416
Some scholars use this to argue the successes found within theAryabhatiyaHowever, it should be noted that Ptolemy had a similarapproximation 400 years prior
The fact that Aryabhata may have been influenced by theGreeks is further supported by his adoption of the myriad as aunit length in geometry
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 95: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/95.jpg)
ChinaIndia
Aryabhatiya
At one point, Aryabhata writes:
“Add 4 to 100, multiply by 8, and add 62,000. The result isapproximately the circumference of a circle of which thediameter is 20,000”
By this estimate, π ≈ 3.1416
Some scholars use this to argue the successes found within theAryabhatiya
However, it should be noted that Ptolemy had a similarapproximation 400 years prior
The fact that Aryabhata may have been influenced by theGreeks is further supported by his adoption of the myriad as aunit length in geometry
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 96: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/96.jpg)
ChinaIndia
Aryabhatiya
At one point, Aryabhata writes:
“Add 4 to 100, multiply by 8, and add 62,000. The result isapproximately the circumference of a circle of which thediameter is 20,000”
By this estimate, π ≈ 3.1416
Some scholars use this to argue the successes found within theAryabhatiyaHowever, it should be noted that Ptolemy had a similarapproximation 400 years prior
The fact that Aryabhata may have been influenced by theGreeks is further supported by his adoption of the myriad as aunit length in geometry
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 97: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/97.jpg)
ChinaIndia
Aryabhatiya
At one point, Aryabhata writes:
“Add 4 to 100, multiply by 8, and add 62,000. The result isapproximately the circumference of a circle of which thediameter is 20,000”
By this estimate, π ≈ 3.1416
Some scholars use this to argue the successes found within theAryabhatiyaHowever, it should be noted that Ptolemy had a similarapproximation 400 years prior
The fact that Aryabhata may have been influenced by theGreeks is further supported by his adoption of the myriad as aunit length in geometry
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 98: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/98.jpg)
ChinaIndia
Aryabhatiya
Oftentimes Aryabhata wrote in colorful prose.
For example, when solving the equation ab = c
x for x(supposing a, b, and c are known), he writes:
“In the rule of three multiply the fruit by the desire and divideby the measure. The result will be the fruit of the desire.”
Here,
a = ‘measure’,b = ‘fruit’,c = ‘desire′, andx = ‘fruit of the desire’
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 99: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/99.jpg)
ChinaIndia
Aryabhatiya
Oftentimes Aryabhata wrote in colorful prose.
For example, when solving the equation ab = c
x for x(supposing a, b, and c are known), he writes:
“In the rule of three multiply the fruit by the desire and divideby the measure. The result will be the fruit of the desire.”
Here,
a = ‘measure’,b = ‘fruit’,c = ‘desire′, andx = ‘fruit of the desire’
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 100: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/100.jpg)
ChinaIndia
Aryabhatiya
Oftentimes Aryabhata wrote in colorful prose.
For example, when solving the equation ab = c
x for x(supposing a, b, and c are known), he writes:
“In the rule of three multiply the fruit by the desire and divideby the measure. The result will be the fruit of the desire.”
Here,
a = ‘measure’,b = ‘fruit’,c = ‘desire′, andx = ‘fruit of the desire’
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 101: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/101.jpg)
ChinaIndia
Multiplication
Indian mathematicians seemed to be influenced little by Greekgeometry – they focused mainly on numbers
Addition and multiplication were carried out similar to modernday, but written right-to-left
They did invent lattice multiplication:
Example: 456× 34
This method was hypothesized to be formulated c. 12thcentury in India and then disseminated to China and Arabiaand from Arabia to Italy in the 14th and 15th centuries
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 102: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/102.jpg)
ChinaIndia
Multiplication
Indian mathematicians seemed to be influenced little by Greekgeometry – they focused mainly on numbers
Addition and multiplication were carried out similar to modernday, but written right-to-left
They did invent lattice multiplication:
Example: 456× 34
This method was hypothesized to be formulated c. 12thcentury in India and then disseminated to China and Arabiaand from Arabia to Italy in the 14th and 15th centuries
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 103: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/103.jpg)
ChinaIndia
Multiplication
Indian mathematicians seemed to be influenced little by Greekgeometry – they focused mainly on numbers
Addition and multiplication were carried out similar to modernday, but written right-to-left
They did invent lattice multiplication:
Example: 456× 34
This method was hypothesized to be formulated c. 12thcentury in India and then disseminated to China and Arabiaand from Arabia to Italy in the 14th and 15th centuries
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 104: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/104.jpg)
ChinaIndia
Multiplication
Indian mathematicians seemed to be influenced little by Greekgeometry – they focused mainly on numbers
Addition and multiplication were carried out similar to modernday, but written right-to-left
They did invent lattice multiplication:
Example: 456× 34
This method was hypothesized to be formulated c. 12thcentury in India and then disseminated to China and Arabiaand from Arabia to Italy in the 14th and 15th centuries
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 105: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/105.jpg)
ChinaIndia
Brahmagupta
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 106: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/106.jpg)
ChinaIndia
Brahmagupta
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 107: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/107.jpg)
ChinaIndia
Brahmagupta
Lived in 7th century in Central India (about a century afterAryabhata)
In his works, he mentions two values for π
The ‘practical value’ π ≈ 3The ‘neat value’ π ≈
√10
Notably, no mention to Aryabhata’s estimate – suggestinglittle inheritance
Best known work was Brahmasphuta Siddhanta
Much like Aryabhatiya, it contained a mixture of correct andincorrect results
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 108: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/108.jpg)
ChinaIndia
Brahmagupta
Lived in 7th century in Central India (about a century afterAryabhata)
In his works, he mentions two values for π
The ‘practical value’ π ≈ 3The ‘neat value’ π ≈
√10
Notably, no mention to Aryabhata’s estimate – suggestinglittle inheritance
Best known work was Brahmasphuta Siddhanta
Much like Aryabhatiya, it contained a mixture of correct andincorrect results
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 109: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/109.jpg)
ChinaIndia
Brahmagupta
Lived in 7th century in Central India (about a century afterAryabhata)
In his works, he mentions two values for π
The ‘practical value’ π ≈ 3The ‘neat value’ π ≈
√10
Notably, no mention to Aryabhata’s estimate – suggestinglittle inheritance
Best known work was Brahmasphuta Siddhanta
Much like Aryabhatiya, it contained a mixture of correct andincorrect results
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 110: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/110.jpg)
ChinaIndia
Brahmagupta
Lived in 7th century in Central India (about a century afterAryabhata)
In his works, he mentions two values for π
The ‘practical value’ π ≈ 3The ‘neat value’ π ≈
√10
Notably, no mention to Aryabhata’s estimate – suggestinglittle inheritance
Best known work was Brahmasphuta Siddhanta
Much like Aryabhatiya, it contained a mixture of correct andincorrect results
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 111: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/111.jpg)
ChinaIndia
Brahmagupta
Lived in 7th century in Central India (about a century afterAryabhata)
In his works, he mentions two values for π
The ‘practical value’ π ≈ 3The ‘neat value’ π ≈
√10
Notably, no mention to Aryabhata’s estimate – suggestinglittle inheritance
Best known work was Brahmasphuta Siddhanta
Much like Aryabhatiya, it contained a mixture of correct andincorrect results
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 112: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/112.jpg)
ChinaIndia
Brahmasphuta Siddhanta
Brahmagupta systematized the arithmetic of negativenumbers and the value zero
Greeks knew how to subtract via geometric magnitudes, butBrahmagupta formalized these into numerical rules for positiveand negative numbersGreeks, likewise, had a concept of ‘nothingness’, but theynever made it into a genuine number
On the matter of zero, Brahmagupta seemed to have made afalse start:
He argued that 00 = 0, but on a
0 for a 6= 0, he avoided it...“Positive divided by positive, or negative by negative, isaffirmative. Cipher divided by cipher is naught. Positivedivided by negative is negative. Negative divided by affirmativeis negative. Positive or negative divided by cipher is a fractionwith that for denominator”
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 113: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/113.jpg)
ChinaIndia
Brahmasphuta Siddhanta
Brahmagupta systematized the arithmetic of negativenumbers and the value zero
Greeks knew how to subtract via geometric magnitudes, butBrahmagupta formalized these into numerical rules for positiveand negative numbersGreeks, likewise, had a concept of ‘nothingness’, but theynever made it into a genuine number
On the matter of zero, Brahmagupta seemed to have made afalse start:
He argued that 00 = 0, but on a
0 for a 6= 0, he avoided it...“Positive divided by positive, or negative by negative, isaffirmative. Cipher divided by cipher is naught. Positivedivided by negative is negative. Negative divided by affirmativeis negative. Positive or negative divided by cipher is a fractionwith that for denominator”
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 114: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/114.jpg)
ChinaIndia
Brahmasphuta Siddhanta
Brahmagupta systematized the arithmetic of negativenumbers and the value zero
Greeks knew how to subtract via geometric magnitudes, butBrahmagupta formalized these into numerical rules for positiveand negative numbersGreeks, likewise, had a concept of ‘nothingness’, but theynever made it into a genuine number
On the matter of zero, Brahmagupta seemed to have made afalse start:
He argued that 00 = 0, but on a
0 for a 6= 0, he avoided it...“Positive divided by positive, or negative by negative, isaffirmative. Cipher divided by cipher is naught. Positivedivided by negative is negative. Negative divided by affirmativeis negative. Positive or negative divided by cipher is a fractionwith that for denominator”
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 115: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/115.jpg)
ChinaIndia
Indeterminate Equations
Brahmagupta was apparently the first to provide a generalsolution to the linear Diophantine equation ax + by = c wherea, b, c ∈ ZHe also suggested the quadratic Diophantine equationx2 = 1 + py2
This equation is often (mistakenly) attributed to the 1600smathematician John Pell – the so-called Pells equationSpecial cases of Pells equation was solved by the nextprominent Indian mathematician Bhaskara in c. 1100
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 116: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/116.jpg)
ChinaIndia
Indeterminate Equations
Brahmagupta was apparently the first to provide a generalsolution to the linear Diophantine equation ax + by = c wherea, b, c ∈ Z
He also suggested the quadratic Diophantine equationx2 = 1 + py2
This equation is often (mistakenly) attributed to the 1600smathematician John Pell – the so-called Pells equationSpecial cases of Pells equation was solved by the nextprominent Indian mathematician Bhaskara in c. 1100
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 117: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/117.jpg)
ChinaIndia
Indeterminate Equations
Brahmagupta was apparently the first to provide a generalsolution to the linear Diophantine equation ax + by = c wherea, b, c ∈ ZHe also suggested the quadratic Diophantine equationx2 = 1 + py2
This equation is often (mistakenly) attributed to the 1600smathematician John Pell – the so-called Pells equationSpecial cases of Pells equation was solved by the nextprominent Indian mathematician Bhaskara in c. 1100
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 118: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/118.jpg)
ChinaIndia
Indeterminate Equations
Brahmagupta was apparently the first to provide a generalsolution to the linear Diophantine equation ax + by = c wherea, b, c ∈ ZHe also suggested the quadratic Diophantine equationx2 = 1 + py2
This equation is often (mistakenly) attributed to the 1600smathematician John Pell – the so-called Pells equation
Special cases of Pells equation was solved by the nextprominent Indian mathematician Bhaskara in c. 1100
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 119: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/119.jpg)
ChinaIndia
Indeterminate Equations
Brahmagupta was apparently the first to provide a generalsolution to the linear Diophantine equation ax + by = c wherea, b, c ∈ ZHe also suggested the quadratic Diophantine equationx2 = 1 + py2
This equation is often (mistakenly) attributed to the 1600smathematician John Pell – the so-called Pells equationSpecial cases of Pells equation was solved by the nextprominent Indian mathematician Bhaskara in c. 1100
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 120: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/120.jpg)
ChinaIndia
Bhaskara
Bhaskara improved on other aspects of Brahmagupta’s workas well.
In addressing the a0 for a 6= 0 debacle, he writes:
“Statement: Dividend 3. Divisor 0. Quotient the fraction 3/0.This fraction of which the denominator is cipher, is termed aninfinite quantity. In this quantity consisting of that which hascipher for a divisor, there is no alteration, though many beinserted or extracted; as no change takes place in the infiniteand immutable God.”
He goes on, however, to note that a0 · 0 = a, so a full
understanding had yet to be achieved.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 121: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/121.jpg)
ChinaIndia
Bhaskara
Bhaskara improved on other aspects of Brahmagupta’s workas well.
In addressing the a0 for a 6= 0 debacle, he writes:
“Statement: Dividend 3. Divisor 0. Quotient the fraction 3/0.This fraction of which the denominator is cipher, is termed aninfinite quantity. In this quantity consisting of that which hascipher for a divisor, there is no alteration, though many beinserted or extracted; as no change takes place in the infiniteand immutable God.”
He goes on, however, to note that a0 · 0 = a, so a full
understanding had yet to be achieved.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 122: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/122.jpg)
ChinaIndia
Bhaskara
Bhaskara improved on other aspects of Brahmagupta’s workas well.
In addressing the a0 for a 6= 0 debacle, he writes:
“Statement: Dividend 3. Divisor 0. Quotient the fraction 3/0.This fraction of which the denominator is cipher, is termed aninfinite quantity. In this quantity consisting of that which hascipher for a divisor, there is no alteration, though many beinserted or extracted; as no change takes place in the infiniteand immutable God.”
He goes on, however, to note that a0 · 0 = a, so a full
understanding had yet to be achieved.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 123: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/123.jpg)
ChinaIndia
Bhaskara
Bhaskara improved on other aspects of Brahmagupta’s workas well.
In addressing the a0 for a 6= 0 debacle, he writes:
“Statement: Dividend 3. Divisor 0. Quotient the fraction 3/0.This fraction of which the denominator is cipher, is termed aninfinite quantity. In this quantity consisting of that which hascipher for a divisor, there is no alteration, though many beinserted or extracted; as no change takes place in the infiniteand immutable God.”
He goes on, however, to note that a0 · 0 = a, so a full
understanding had yet to be achieved.
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 124: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/124.jpg)
ChinaIndia
Bhaskara
His major work was the Lilavata
This was one of the last significant texts in medieval Indianmathematics
In this text, he finds a solution for the Pells equationx2 = 1 + 61y2
His solution:
x = 1, 776, 319, 049y = 22, 615, 390
How he achieved this feat, no one knows...
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 125: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/125.jpg)
ChinaIndia
Bhaskara
His major work was the Lilavata
This was one of the last significant texts in medieval Indianmathematics
In this text, he finds a solution for the Pells equationx2 = 1 + 61y2
His solution:
x = 1, 776, 319, 049y = 22, 615, 390
How he achieved this feat, no one knows...
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 126: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/126.jpg)
ChinaIndia
Bhaskara
His major work was the Lilavata
This was one of the last significant texts in medieval Indianmathematics
In this text, he finds a solution for the Pells equationx2 = 1 + 61y2
His solution:
x = 1, 776, 319, 049y = 22, 615, 390
How he achieved this feat, no one knows...
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 127: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/127.jpg)
ChinaIndia
Bhaskara
His major work was the Lilavata
This was one of the last significant texts in medieval Indianmathematics
In this text, he finds a solution for the Pells equationx2 = 1 + 61y2
His solution:
x = 1, 776, 319, 049y = 22, 615, 390
How he achieved this feat, no one knows...
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 128: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/128.jpg)
ChinaIndia
Bhaskara
His major work was the Lilavata
This was one of the last significant texts in medieval Indianmathematics
In this text, he finds a solution for the Pells equationx2 = 1 + 61y2
His solution:
x = 1, 776, 319, 049y = 22, 615, 390
How he achieved this feat, no one knows...
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 129: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/129.jpg)
ChinaIndia
Madhava and the Keralese School
In the late 14th century, in southwest India, the KeraleseSchool began under the direction of Madhava
This school is most well-known for its early (and most likelyoriginal) discovery of what we would now call:
Power series expansions for sin and cos
Often attributed to Newton
The series for π4
Often attributed to Liebniz
Power series for arctangent
Often attributed to James Gregory
With the dawn on the Renaissance, however, the center formathematical discoveries again began to find its waywestward...
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 130: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/130.jpg)
ChinaIndia
Madhava and the Keralese School
In the late 14th century, in southwest India, the KeraleseSchool began under the direction of Madhava
This school is most well-known for its early (and most likelyoriginal) discovery of what we would now call:
Power series expansions for sin and cos
Often attributed to Newton
The series for π4
Often attributed to Liebniz
Power series for arctangent
Often attributed to James Gregory
With the dawn on the Renaissance, however, the center formathematical discoveries again began to find its waywestward...
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 131: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/131.jpg)
ChinaIndia
Madhava and the Keralese School
In the late 14th century, in southwest India, the KeraleseSchool began under the direction of Madhava
This school is most well-known for its early (and most likelyoriginal) discovery of what we would now call:
Power series expansions for sin and cos
Often attributed to Newton
The series for π4
Often attributed to Liebniz
Power series for arctangent
Often attributed to James Gregory
With the dawn on the Renaissance, however, the center formathematical discoveries again began to find its waywestward...
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 132: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/132.jpg)
ChinaIndia
Madhava and the Keralese School
In the late 14th century, in southwest India, the KeraleseSchool began under the direction of Madhava
This school is most well-known for its early (and most likelyoriginal) discovery of what we would now call:
Power series expansions for sin and cos
Often attributed to Newton
The series for π4
Often attributed to Liebniz
Power series for arctangent
Often attributed to James Gregory
With the dawn on the Renaissance, however, the center formathematical discoveries again began to find its waywestward...
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 133: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/133.jpg)
ChinaIndia
Madhava and the Keralese School
In the late 14th century, in southwest India, the KeraleseSchool began under the direction of Madhava
This school is most well-known for its early (and most likelyoriginal) discovery of what we would now call:
Power series expansions for sin and cos
Often attributed to Newton
The series for π4
Often attributed to Liebniz
Power series for arctangent
Often attributed to James Gregory
With the dawn on the Renaissance, however, the center formathematical discoveries again began to find its waywestward...
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 134: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/134.jpg)
ChinaIndia
Madhava and the Keralese School
In the late 14th century, in southwest India, the KeraleseSchool began under the direction of Madhava
This school is most well-known for its early (and most likelyoriginal) discovery of what we would now call:
Power series expansions for sin and cos
Often attributed to Newton
The series for π4
Often attributed to Liebniz
Power series for arctangent
Often attributed to James Gregory
With the dawn on the Renaissance, however, the center formathematical discoveries again began to find its waywestward...
Douglas Pfeffer The Ancient and Medieval: China and India
![Page 135: The Ancient and Medieval: China and India · India The Nine Chapters Rod Numerals ˇ 13th Century The Nine Chapters The Chinese were very fond of patterns. The very rst recorded Magic](https://reader034.vdocuments.net/reader034/viewer/2022043023/5f3e905d453aac2ab802387a/html5/thumbnails/135.jpg)
ChinaIndia
Madhava and the Keralese School
In the late 14th century, in southwest India, the KeraleseSchool began under the direction of Madhava
This school is most well-known for its early (and most likelyoriginal) discovery of what we would now call:
Power series expansions for sin and cos
Often attributed to Newton
The series for π4
Often attributed to Liebniz
Power series for arctangent
Often attributed to James Gregory
With the dawn on the Renaissance, however, the center formathematical discoveries again began to find its waywestward...
Douglas Pfeffer The Ancient and Medieval: China and India