Group 3 Flashcards
- 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
Ancient Chinese Number System
the Ancient Chinese Number System dates back to at least the
2nd millennium BCE
- similar to the counting board
- used beads sliding on wire
Abacus
first Chinese abacus
suanpan
first Chinese abacus dates back to about
2nd century BCE
one of the first known examples of a magic square
Lo Shu Square
Same sum in magic squares is called
magic constant
magic constant of the Lo Shu Square
15
Lo Shu Square dates back to
650 BCE
legend of Emperor Yu’s discovery of the square on the back of a turtle is set as taking place in about
2800 BCE
bigger magic squares culminated in the elaborate magic squares, circles, and triangles of who
Yang Hui (13th century)
Yang Hui’s triangle is identical to the later
Pascal’s triangle
- 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
Jiuzhang Suanshu or Nine Chapters on the Mathematical Art
example of equation on the Nine Chapters on the mathematical art
deduction of an unknown number from other known information
Who re-discover the sophisticated matrix-based method found in the Nine Chapters on the Mathematical Art
Carl Friedrich Gauss
- 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”
Liu Hui
What did Liu Hui formulate
algorithm using a regular polygon with 192 sides which calculated the value of π as 3.14159
What did Liu Hui develop
integral and differential calculus’ early forms
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
Chinese Remainder Theorem
who initially posed the Chinese Remainder Theorem
Sun Tzu (3rd century CE)
Technique for solving problems in the Chinese Remainder Theorem is used in
- measure planetary movements (Chinese astronomers in 6th century CE)
- internet cryptography (today)
When is the golden age of chinese mathematics
13th century CE
- 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”
Qin Jiushao
Qin Jiushao’s method of repeated approximations is very similar to the later devised by
Sir Isaac Newton (17th century)
Important study that Qin Jiushao wrote
“Shushu Jiuzhang” or “Mathematical Treatise in Nine Sections”
- invented by mathematics in India
- created between the 1st century to 4th centuries and later adopted in Arabic mathematics by the 9th century
Hindu-Arabic numeral system
before 1000 BCE in India
Vedic period
what was created in Vedic Period
arithmetic operations
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
Sulba Sutras (or “Sulva Sutras”)
Sulba Sutras (or “Sulva Sutras”) when
8th century BCE
- 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
Jain Mathematics
- 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
Decimal Place Value Number System
What was the first character used for the number zero
circle
Include zero as a number was credited to who
- Brahmagupta or possibly
- Bhaskara I
Golden Age of Indian Mathematics
- 5th - 12th century
- created fundamental advances in the theory of trigonometry
- 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
Aryabhata
summarizes Hindu mathematics up to the 6th century
Aryabhatiya
- also called Bhaskaracharya
- Indian mathematician and astronomer who extended Brahmagupta’s work on number systems
- made important contributions to many different areas of mathematics
Bhaskara II
Bhaskara II is credited with
explaining previously misunderstood operation of division by zero
- great 7th Century Indian mathematician and astronomer
- wrote some important works on both mathematics and astronomy
Brahmagupta
Brahmagupta is a great __ century Indian mathematician and astronomer
7th
Brahmagupta wrote some important works on both what
mathematics and astronomy
Most of Brahmagupta’s works are composed in __ __, a common practice in Indian mathematics at the time
elliptic verse
What is Brahmagupta’s most famous text
Brahmasphutasiddhanta
Brahmagupta explained how to find what
cube and cube root of integer
Brahmagupta gave rules facilitating the computation of what
squares and square roots
Brahmagupta gave rules for dealing with what
five types of combinations of fractions
Brahmagupta gave the sum of what
squares and cubes of the first natural numbers
Brahmagupta established the basic mathematical rules for __ _ __
dealing with zero
Brahmagupta’s different view of numbers allowed him to realize what
there could be such a thing as a negative number
Brahmagupta established __ as a good practical approximation of π (3.141593)
√10 (3.162277)
Brahmagupta area of a cyclic quadrilateral
Brahmagupta’s formula
Brahmagupta’s celebrated theorem on the diagonals of a cyclic quadrilateral
Brahmagupta’s Theorem
- 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
Madhava
Madhava is sometimes called as
greatest mathematician astronomer of medieval India
Madhava is a source for several
infinite series expansions
What did Madhava by successively adding and substracting different odd number fractions to infinity
could home in on an exact formula for π
- 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
Islamic Mathematics
Islamic mathematics made significant contributions toward what century
8th century onwards
Islamic mathematics were able to draw on and fuse together the development of both what
Greek and Indian mathematics
Islamic mathematics made use of __ __ __ to decorate their buildings, raising mathematics to the form of an art
complex geometric patterns
What was set up in Baghdad around 810
House of Wisdom
Where was the House of Wisdom set up
Baghdad
When was the House of Wisdom in Baghdad set up
around 810
Why did Islamic mathematics stagnate
stifling influence of Turkish Ottoman Empire from 14th - 15th century onwards
- 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
Muhammad al-Khwarizmi
Where is the word algorithm derived from
Latinization of Muhammad al-Khwarizmi
Where is the word algebra derived from
Latinization of “al-jabr”
What is Muhammad al-Khwarizmi most important contribution to mathematics
strong advocacy of the Hindu numerical system
- considered the foundational text of modern algebra
- Muhammad al-Khwarizmi’s book
The Compendious Book on Calculation by Completion and Balancing
- alternative to long multiplication for numbers
- later introduced to Europe by Fibonacci
lattice method
Muhammad al-Khwarizmi developed the first what
first quadrant
- 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
Muhammad Al-Karaji
- 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
Omar Khayyam
Omar Khayyam generalized Indian methods for extracting what
square and cube roots to include 4th, 5th and higher roots
Omar Khayyam carried out a systematic analysis of what
cubic problems
Omar Khayyam is usually credited with identifying what
foundations of algebraic geometry
- 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
Nasir Al- Din Al-Tusi
Nasir Al- Din Al-Tusi gave the first extensive exposition of what
spherical trigonometry
What is Nasir Al- Din Al-Tusi one major contribution
formulation of famous law of sines for plane triangles
pioneer in the field of trigonometry
Nasir Al- Din Al-Tusi
developed a general formula by which amicable numbers could be derived
9th Century
Arab Thabit ibn Qurra
wrote the earliest surviving texts showing the positional use of Arabic numerals, and particularly the use of decimals instead of fractions
10th Century Arab
Abul Hasan al-Uqlidisi
continued Archimedes’ investigations of areas and volumes, as well as on tangents of a circle
10th Century Arab geometer
Ibrahim ibn Sinan
established the beginnings of the link between algebra and geometry, and devised what is now known as “Alhazen’s problem”
11th Century Persian
Ibn al-Haytham
Ibn al-Haytham devised what is now known as
Alhazen’s problem
- applied the theory of conic sections to solve optical problems
- pursued work in number theory such as on amicable numbers, factorization and combinatorial methods
13th Century Persian
Kamal al-Din al-Farisi
- 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
13th Century Moroccan
Ibn al-Banna al-Marrakushi
