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Mathematics in the Modern World > Group 3 > Flashcards

Group 3 Flashcards

1
Q
  • used small bamboo rods arranged to represent numbers 1 to 9 which were placed in columns representing units, tens, hundreds, thousands, etc.
  • decimal place value system
A

Ancient Chinese Number System

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2
Q

the Ancient Chinese Number System dates back to at least the

A

2nd millennium BCE

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3
Q
  • similar to the counting board
  • used beads sliding on wire
A

Abacus

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4
Q

first Chinese abacus

A

suanpan

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5
Q

first Chinese abacus dates back to about

A

2nd century BCE

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6
Q

one of the first known examples of a magic square

A

Lo Shu Square

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7
Q

Same sum in magic squares is called

A

magic constant

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8
Q

magic constant of the Lo Shu Square

A

15

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9
Q

Lo Shu Square dates back to

A

650 BCE

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10
Q

legend of Emperor Yu’s discovery of the square on the back of a turtle is set as taking place in about

A

2800 BCE

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11
Q

bigger magic squares culminated in the elaborate magic squares, circles, and triangles of who

A

Yang Hui (13th century)

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12
Q

Yang Hui’s triangle is identical to the later

A

Pascal’s triangle

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13
Q
  • written over a period of time probably by a variety of authors
  • became an important tool in the education of civil service
  • covered hundreds of problems in practical areas such as trade, taxation, engineering, and payment of wages
A

Jiuzhang Suanshu or Nine Chapters on the Mathematical Art

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14
Q

example of equation on the Nine Chapters on the mathematical art

A

deduction of an unknown number from other known information

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15
Q

Who re-discover the sophisticated matrix-based method found in the Nine Chapters on the Mathematical Art

A

Carl Friedrich Gauss

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16
Q
  • First mathematician known to leave roots unevaluated giving more exact results
  • Formulated an algorithm using a regular polygon with 192 sides which calculated the value of π as 3.14159
  • Developed early forms of integral and differential calculus
  • Among the greatest mathematicians in Ancient China
  • produced a detailed commentary on the “Nine Chapters”
A

Liu Hui

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17
Q

What did Liu Hui formulate

A

algorithm using a regular polygon with 192 sides which calculated the value of π as 3.14159

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18
Q

What did Liu Hui develop

A

integral and differential calculus’ early forms

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19
Q

To calculate the smallest value of the unknown number, it uses the remainders after dividing the unknown number by a succession of smaller numbers, such as 3, 5 and 7

A

Chinese Remainder Theorem

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20
Q

who initially posed the Chinese Remainder Theorem

A

Sun Tzu (3rd century CE)

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21
Q

Technique for solving problems in the Chinese Remainder Theorem is used in

A
  • measure planetary movements (Chinese astronomers in 6th century CE)
  • internet cryptography (today)
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22
Q

When is the golden age of chinese mathematics

A

13th century CE

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23
Q
  • explored solutions to quadratic and even cubic equations using method of repeated approximations
  • extended his technique to solve equations involving numbers up to the power of ten
  • wrote “Shushu Jiuzhang” or “Mathematical Treatise in Nine Sections”
A

Qin Jiushao

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24
Q

Qin Jiushao’s method of repeated approximations is very similar to the later devised by

A

Sir Isaac Newton (17th century)

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25
Q

Important study that Qin Jiushao wrote

A

“Shushu Jiuzhang” or “Mathematical Treatise in Nine Sections”

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26
Q
  • invented by mathematics in India
  • created between the 1st century to 4th centuries and later adopted in Arabic mathematics by the 9th century
A

Hindu-Arabic numeral system

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27
Q

before 1000 BCE in India

A

Vedic period

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28
Q

what was created in Vedic Period

A

arithmetic operations

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29
Q

text that lists several simple Pythagorean triples, as well as a statement of the simplified Pythagorean theorem for the sides of a square and for a rectangle

A

Sulba Sutras (or “Sulva Sutras”)

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30
Q

Sulba Sutras (or “Sulva Sutras”) when

A

8th century BCE

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31
Q
  • early as the 3rd or 2nd century BCE
  • recognized five different types of infinities: infinite in one direction, in two directions, in area, infinite everywhere, and perpetually infinite
A

Jain Mathematics

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32
Q
  • 3rd century CE
  • Indians refined and perfected the system, particularly the written representation of the numerals that is used across the world today
  • sometimes considered one of the greatest intellectual innovations of all time
A

Decimal Place Value Number System

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33
Q

What was the first character used for the number zero

A

circle

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34
Q

Include zero as a number was credited to who

A
  • Brahmagupta or possibly
  • Bhaskara I
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35
Q

Golden Age of Indian Mathematics

A
  • 5th - 12th century
  • created fundamental advances in the theory of trigonometry
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36
Q
  • produced categorical definitions of sine, cosine, versine, and inverse sine, and specified complete sine and versine tables
  • wrote the Aryabhatiya which summarizes Hindu mathematics up to the 6th Century
A

Aryabhata

37
Q

summarizes Hindu mathematics up to the 6th century

A

Aryabhatiya

38
Q
  • also called Bhaskaracharya
  • Indian mathematician and astronomer who extended Brahmagupta’s work on number systems
  • made important contributions to many different areas of mathematics
A

Bhaskara II

39
Q

Bhaskara II is credited with

A

explaining previously misunderstood operation of division by zero

40
Q
  • great 7th Century Indian mathematician and astronomer
  • wrote some important works on both mathematics and astronomy
A

Brahmagupta

41
Q

Brahmagupta is a great __ century Indian mathematician and astronomer

A

7th

42
Q

Brahmagupta wrote some important works on both what

A

mathematics and astronomy

43
Q

Most of Brahmagupta’s works are composed in __ __, a common practice in Indian mathematics at the time

A

elliptic verse

44
Q

What is Brahmagupta’s most famous text

A

Brahmasphutasiddhanta

45
Q

Brahmagupta explained how to find what

A

cube and cube root of integer

46
Q

Brahmagupta gave rules facilitating the computation of what

A

squares and square roots

47
Q

Brahmagupta gave rules for dealing with what

A

five types of combinations of fractions

48
Q

Brahmagupta gave the sum of what

A

squares and cubes of the first natural numbers

49
Q

Brahmagupta established the basic mathematical rules for __ _ __

A

dealing with zero

50
Q

Brahmagupta’s different view of numbers allowed him to realize what

A

there could be such a thing as a negative number

51
Q

Brahmagupta established __ as a good practical approximation of π (3.141593)

A

√10 (3.162277)

52
Q

Brahmagupta area of a cyclic quadrilateral

A

Brahmagupta’s formula

53
Q

Brahmagupta’s celebrated theorem on the diagonals of a cyclic quadrilateral

A

Brahmagupta’s Theorem

54
Q
  • sometimes called the greatest mathematician astronomer of medieval India - source for several infinite series expansions
  • happy to play around with infinity, particularly the infinite series when most previous cultures are rather nervous about the concept of infinity
A

Madhava

55
Q

Madhava is sometimes called as

A

greatest mathematician astronomer of medieval India

56
Q

Madhava is a source for several

A

infinite series expansions

57
Q

What did Madhava by successively adding and substracting different odd number fractions to infinity

A

could home in on an exact formula for π

58
Q
  • Made significant contributions towards mathematics during the 8th Century onwards
  • Were able to draw on and fuse together the mathematical Development of both Greek and Indian
A

Islamic Mathematics

59
Q

Islamic mathematics made significant contributions toward what century

A

8th century onwards

60
Q

Islamic mathematics were able to draw on and fuse together the development of both what

A

Greek and Indian mathematics

61
Q

Islamic mathematics made use of __ __ __ to decorate their buildings, raising mathematics to the form of an art

A

complex geometric patterns

62
Q

What was set up in Baghdad around 810

A

House of Wisdom

63
Q

Where was the House of Wisdom set up

A

Baghdad

64
Q

When was the House of Wisdom in Baghdad set up

A

around 810

65
Q

Why did Islamic mathematics stagnate

A

stifling influence of Turkish Ottoman Empire from 14th - 15th century onwards

66
Q
  • One of the directors of the house of wisdom in the early 9th century
  • Oversaw the translation of the major Greek and Indian mathematical and astronomy works into Arabic
  • word “algorithm” is derived from the Latinization of his name, and the word “algebra” is derived from the Latinization of “al-jabr”
  • Most important contribution to mathematics was his strong advocacy of the Hindu numerical system
A

Muhammad al-Khwarizmi

67
Q

Where is the word algorithm derived from

A

Latinization of Muhammad al-Khwarizmi

68
Q

Where is the word algebra derived from

A

Latinization of “al-jabr”

69
Q

What is Muhammad al-Khwarizmi most important contribution to mathematics

A

strong advocacy of the Hindu numerical system

70
Q
  • considered the foundational text of modern algebra
  • Muhammad al-Khwarizmi’s book
A

The Compendious Book on Calculation by Completion and Balancing

71
Q
  • alternative to long multiplication for numbers
  • later introduced to Europe by Fibonacci
A

lattice method

72
Q

Muhammad al-Khwarizmi developed the first what

A

first quadrant

73
Q
  • introduced the theory of algebraic calculus
  • used mathematical induction to prove the binomial theorem
  • a 10th Century Persian mathematician
  • worked to extend algebra still further
A

Muhammad Al-Karaji

74
Q
  • Generalized Indian methods for extracting square and cube roots to include fourth, fifth and higher roots in the early 12th Century
  • carried out a systematic analysis of cubic problems
  • Usually credited with identifying the foundations of algebraic geometry
A

Omar Khayyam

75
Q

Omar Khayyam generalized Indian methods for extracting what

A

square and cube roots to include 4th, 5th and higher roots

76
Q

Omar Khayyam carried out a systematic analysis of what

A

cubic problems

77
Q

Omar Khayyam is usually credited with identifying what

A

foundations of algebraic geometry

78
Q
  • first to treat trigonometry as a separate mathematical discipline, distinct from astronomy
  • gave the first extensive exposition of spherical trigonometry
  • one of major contributions were the formulation of the famous law of sines for plane triangles
A

Nasir Al- Din Al-Tusi

79
Q

Nasir Al- Din Al-Tusi gave the first extensive exposition of what

A

spherical trigonometry

80
Q

What is Nasir Al- Din Al-Tusi one major contribution

A

formulation of famous law of sines for plane triangles

81
Q

pioneer in the field of trigonometry

A

Nasir Al- Din Al-Tusi

82
Q

developed a general formula by which amicable numbers could be derived

A

9th Century
Arab Thabit ibn Qurra

83
Q

wrote the earliest surviving texts showing the positional use of Arabic numerals, and particularly the use of decimals instead of fractions

A

10th Century Arab
Abul Hasan al-Uqlidisi

84
Q

continued Archimedes’ investigations of areas and volumes, as well as on tangents of a circle

A

10th Century Arab geometer
Ibrahim ibn Sinan

85
Q

established the beginnings of the link between algebra and geometry, and devised what is now known as “Alhazen’s problem”

A

11th Century Persian
Ibn al-Haytham

86
Q

Ibn al-Haytham devised what is now known as

A

Alhazen’s problem

87
Q
  • applied the theory of conic sections to solve optical problems
  • pursued work in number theory such as on amicable numbers, factorization and combinatorial methods
A

13th Century Persian
Kamal al-Din al-Farisi

88
Q
  • works included topics such as computing square roots and the theory of continued fractions
  • discovery of the first new pair of amicable numbers since ancient times
  • first use of algebraic notation since Brahmagupta
A

13th Century Moroccan
Ibn al-Banna al-Marrakushi

Mathematics in the Modern World (13 decks)

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