History of Mathematical Thought Exam 2 Flashcards
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Questions and Answers

Why is the book Nine Chapters on the Mathematical Art significant in the history of mathematics?

It is central to Chinese mathematical activity for hundreds of years.

The Nine Chapters on the Mathematical Art was central to the development of mathematics in which country?

China

The Chinese numeration system that required the use of a counting board and rods was not a positional system.

False

The method that Liu Hui used to approximate the value of pi was similar to the method used by which Greek mathematician?

<p>Archimedes</p> Signup and view all the answers

How was ancient Chinese mathematics different from ancient Greek mathematics?

<p>The Chinese mathematics did not have a proof-based axiomatic development.</p> Signup and view all the answers

Chinese Square Root Algorithm: If a, b, and c are the digits (respectively) of the square root of x, then which of the following is equal to sqrt(x)?

<p>100a+10b+c</p> Signup and view all the answers

Which set of numbers was used on the counting boards of the ancient Chinese?

<p>Integers -- both positive and negative</p> Signup and view all the answers

Why did Chinese mathematics develop independently of Western cultural influence for so long?

<p>The geography of China isolated the region from the rest of the world.</p> Signup and view all the answers

What letter of the alphabet is a gnomon shaped like?

<p>L</p> Signup and view all the answers

What strategy did the Chinese use on their counting boards to avoid confusion between consecutive places?

<p>Alternating the orientation of the rods</p> Signup and view all the answers

During the period of the Islamic Empire, negative numbers were acceptable solutions of quadratic equations.

<p>False</p> Signup and view all the answers

What was the significance of The Book on Addition and Subtraction?

<p>It explained the use of the Hindu decimal system of numerals.</p> Signup and view all the answers

Which mathematician/poet used geometric methods to solve cubic equations?

<p>Omar Khayyam</p> Signup and view all the answers

The mathematician, al-Khwarizmi, used a symbol for zero in his book about numeration.

<p>True</p> Signup and view all the answers

In what branch of mathematics were the most important contributions of the Islamic mathematicians?

<p>Algebra</p> Signup and view all the answers

What was the FIRST 'translation movement'?

<p>The movement to translate the world's knowledge into Arabic</p> Signup and view all the answers

To what did 'al-Jabr' and 'al-Muqabala' refer in the title of al-Khwarizmi's book?

<p>The operations used during the Islamic Empire to solve equations.</p> Signup and view all the answers

In what area did the mathematician, Al-Uqlidisi, make his contributions?

<p>Decimal fractions</p> Signup and view all the answers

The Islamic mathematician, al-Khwarizmi, made his most significant contributions to which topic in mathematics?

<p>Solving equations</p> Signup and view all the answers

Which city was the academic and intellectual center of the Islamic Empire and the home to the House of Wisdom?

<p>Baghdad</p> Signup and view all the answers

Who was the greatest European mathematician of the 11th and 12th centuries?

<p>Leonardo of Pisa</p> Signup and view all the answers

What was the source of most of the mathematics in Fibonacci's work?

<p>The procedures and concepts of tenth-century Islamic mathematics.</p> Signup and view all the answers

The Mathematical Classic of Master Sun is an important work from which country or empire?

<p>China</p> Signup and view all the answers

What is the 6th number in the Fibonacci sequence?

<p>8</p> Signup and view all the answers

What is the most famous problem in Liber Abaci?

<p>The rabbit problem</p> Signup and view all the answers

Who was the Indian mathematician who introduced negative numbers and the rules for computing with them?

<p>Brahmagupta</p> Signup and view all the answers

What role did the Islamic Empire play in the preservation of Greek knowledge?

<p>They preserved Greek knowledge by translating Greek works into Arabic.</p> Signup and view all the answers

How did the educational system in medieval India facilitate the transmission of mathematical knowledge from one generation to the next?

<p>The education system was family-based, and families would pass their libraries and traditions down to the next generation.</p> Signup and view all the answers

A Chinese text from the eleventh century provides evidence that they knew about what mathematical idea that is usually associated with a 17th century French mathematician?

<p>Pascal's triangle</p> Signup and view all the answers

What is zero divided by a negative number?

<p>Zero</p> Signup and view all the answers

Which book is considered the first book devoted entirely to algebra?

<p>Arithmetica</p> Signup and view all the answers

Bombelli wrote 'It was a wild thought in the judgment of many; and I too for a long time was of the same opinion.' Bombelli was referring to what set of numbers?

<p>Imaginary numbers</p> Signup and view all the answers

Who is sometimes called the 'father of algebra'?

<p>Diophantus</p> Signup and view all the answers

What was Viete's contribution to the development of the modern symbolism of algebra?

<p>He used letters to represent unknown quantities.</p> Signup and view all the answers

Who is the father of English cryptography?

<p>John Wallis</p> Signup and view all the answers

In which area did Girolamo Cardano make the most significant contributions?

<p>Algebra</p> Signup and view all the answers

What is 'syncopated' algebra?

<p>Shorthand way of writing algebraic expressions using abbreviations</p> Signup and view all the answers

In which stage of algebraic development did Diophantus make contributions?

<p>Syncopated</p> Signup and view all the answers

Which mathematician used the letters at the end of the alphabet to represent unknown quantities and the letters at the beginning of the alphabet to represent known quantities or constants?

<p>Rene Descartes</p> Signup and view all the answers

What were the primary contents of the book, Ars Magna?

<p>The solution of cubic and quartic equations.</p> Signup and view all the answers

According to Kepler's First Law, what is the shape of planetary orbits around the sun?

<p>Ellipse</p> Signup and view all the answers

For what operation did John Napier use Napier's bones (or Napier's rods)?

<p>Multiplication</p> Signup and view all the answers

Who was the first astronomer to put forward the theory that the sun is at the center of the universe?

<p>Aristarchus of Samos</p> Signup and view all the answers

Besides mathematics and astronomy, in which other branch of science did Ptolemy make significant contributions?

<p>Geography</p> Signup and view all the answers

Who wrote the mathematical text, Almagest?

<p>Ptolemy</p> Signup and view all the answers

What is a chord of a circle?

<p>A line segment that intersects a circle in two points</p> Signup and view all the answers

In the Ptolemaic model of the universe, the equant is the center of the epicycle that moves around the earth.

<p>False</p> Signup and view all the answers

What is the base in common logarithms?

<p>10</p> Signup and view all the answers

Why did John Napier invent logarithms?

<p>To make computation with large numbers less tedious.</p> Signup and view all the answers

The Ptolemaic model of the universe is a heliocentric model.

<p>False</p> Signup and view all the answers

How were Descartes' and Fermat's coordinate systems different from ours today?

<p>They never referred to a vertical axis.</p> Signup and view all the answers

Who was the first mathematician to solve a maximum/minimum problem?

<p>Pierre de Fermat</p> Signup and view all the answers

Who was the Queen of Sweden when Descartes died?

<p>Christina</p> Signup and view all the answers

What does 'Cogito ergo sum' mean?

<p>I think therefore I am.</p> Signup and view all the answers

Which two mathematicians co-developed analytic geometry?

<p>Rene Descartes and Pierre de Fermat</p> Signup and view all the answers

Which mathematician is known as the father of modern philosophy?

<p>Rene Descartes</p> Signup and view all the answers

In which country were Descartes and Fermat born?

<p>France</p> Signup and view all the answers

What method did Fermat use to solve maximum/minimum problems?

<p>Pseudo-equality method</p> Signup and view all the answers

What was the major contribution of Father Marin Mersenne to the development of mathematics?

<p>He organized regular meetings of scholars that allowed them to share their ideas.</p> Signup and view all the answers

What branch of mathematics was introduced in the book, La Geometrie?

<p>Analytic Geometry</p> Signup and view all the answers

Which one of the ancient civilizations knew about the triangle that we call 'Pascal's Triangle?'

<p>Chinese</p> Signup and view all the answers

What is true about the sum of the numbers on each row of Pascal's triangle?

<p>The sums are all powers of two.</p> Signup and view all the answers

Which mathematician was the 'greatest might have been' in mathematics?

<p>Blaise Pascal</p> Signup and view all the answers

Who founded the theory of games?

<p>John von Neumann</p> Signup and view all the answers

According to Pascal, what does Pascal's Wager prove?

<p>Pascal's Wager proves that belief in God is rational.</p> Signup and view all the answers

Who wrote the first book on games of chance titled The Book on Games of Chance?

<p>Girolamo Cardano</p> Signup and view all the answers

What was the problem of the points?

<p>How should the stakes be distributed fairly when a game is stopped early?</p> Signup and view all the answers

Who were the co-developers of probability?

<p>Blaise Pascal and Pierre de Fermat</p> Signup and view all the answers

If a fair coin is tossed twice, what is the probability of getting two heads?

<p>1/4</p> Signup and view all the answers

The two men who co-developed probability used different strategies to solve the problem of the points. Which one of the two made a list of all the possible ways that the game could be completed to solve the problem?

<p>Pierre de Fermat</p> Signup and view all the answers

Who proved Fermat's Last Theorem?

<p>Andrew Wiles</p> Signup and view all the answers

What was Marin Mersenne's contribution to the development of mathematics?

<p>He hosted regular gatherings of mathematicians who exchanged ideas.</p> Signup and view all the answers

Which of the following is an application of number theory?

<p>Cryptography</p> Signup and view all the answers

Why is 6 a perfect number?

<p>Because 6=1+2+3</p> Signup and view all the answers

What are amicable numbers?

<p>Pairs of numbers that are equal to the sum of the proper factors of the other number.</p> Signup and view all the answers

Study Notes

Significance of Mathematical Works

  • The "Nine Chapters on the Mathematical Art" was crucial to Chinese mathematics for centuries.
  • "The Book on Addition and Subtraction" outlined the Hindu decimal system, impacting numeral use in mathematics.

Mathematics in Different Cultures

  • Chinese mathematics differed from Greek mathematics in that it lacked a proof-based axiomatic approach.
  • Ancient Chinese used counting boards to perform calculations with integers, positive and negative.

Developments in Chinese Mathematics

  • Liu Hui approximated pi similarly to Archimedes, showcasing early methods of approximation.
  • Chinese mathematicians demonstrated knowledge of concepts like Pascal's Triangle as early as the 11th century.

Islamic Contributions to Mathematics

  • The Islamic Empire preserved Greek mathematical knowledge by translating texts into Arabic.
  • Al-Khwarizmi's book introduced algebraic terminology, with "al-Jabr" referring to operations for solving equations.

Key Islamic Mathematicians

  • Omar Khayyam was noted for using geometric methods to solve cubic equations.
  • Brahmagupta introduced negative numbers and rules for computing with them.

Mathematics in Europe

  • Leonardo of Pisa, also known as Fibonacci, was a key figure whose work was influenced by Islamic mathematics.
  • The "rabbit problem" from Fibonacci's "Liber Abaci" is his most famous problem.

Algebraic Advances

  • Diophantus is recognized as the "father of algebra" for early algebraic contributions.
  • Symbolism in algebra was advanced by Viete, who utilized letters for unknowns, influencing later algebraic notation.

Probability and Games

  • Blaise Pascal and Pierre de Fermat co-developed the field of probability, particularly through their correspondence on games of chance.
  • Pascal's Wager presents a rational argument for belief in God, connecting philosophy with probability theory.

Key Theorems and Numbers

  • Andrew Wiles is credited with proving Fermat's Last Theorem, a significant breakthrough in number theory.
  • Perfect numbers, such as 6, are defined as those equal to the sum of their proper factors.

Methodologies and Concepts

  • Syncopated algebra involves shorthand notation for algebraic expressions, an evolution in the representation of mathematical ideas.
  • Pascal's Triangle demonstrates that the sum of each row equals powers of two, showcasing relationships in combinatorial mathematics.

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Description

This quiz covers key concepts from Exam 2 of the History of Mathematical Thought course. You will explore significant texts and their impact on mathematical development, focusing particularly on the Nine Chapters on the Mathematical Art and its importance in Chinese mathematics.

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