Mars: Influential Mathematicians and Their Discoveries
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Questions and Answers

Who was the first to use the method of proof by mathematical induction?

  • Nasir Al-Din Al-Tusi (correct)
  • Thabit ibn Qurra
  • Fermat
  • Abul Wafa Buzjani

    • What major contribution did Nasir Al-Din Al-Tusi make to trigonometry?

  • He discovered the sine function.
  • He developed the concept of amicable numbers.
  • He formulated the law of sines for spherical triangles.
  • He treated trigonometry as a discipline distinct from astronomy. (correct)

    • Which mathematical formulation is attributed to Nasir Al-Din Al-Tusi?

  • The law of sines for plane triangles (correct)
  • The method of mathematical induction
  • The law of cosines
  • The Pythagorean theorem

    • Who discovered the sine law for spherical triangles before Al-Tusi?

    <p>Abul Wafa Buzjani and Abu Nasr Mansur</p> Signup and view all the answers

    What are amicable numbers?

    <p>Pairs of numbers for which the sum of the divisors of one equals the other</p> Signup and view all the answers

    What is a magic square?

    <p>A square with numbers that total the same in all rows, columns, and diagonals</p> Signup and view all the answers

    What total does each row, column, and diagonal of the Lo Shu Square sum to?

    <p>15</p> Signup and view all the answers

    Who is considered the greatest mathematician of ancient China?

    <p>Liu Hui</p> Signup and view all the answers

    When was the 'Nine Chapters on the Mathematical Art' written?

    <p>200 BCE onwards</p> Signup and view all the answers

    What mathematical method did the Nine Chapters influence that later appeared in the West?

    <p>Gaussian elimination</p> Signup and view all the answers

    What was one of Liu Hui's contributions to mathematics?

    <p>Leaving roots unevaluated for exact results</p> Signup and view all the answers

    What innovative representation did Yang Hui produce in the 13th century?

    <p>A triangular representation of binomial coefficients</p> Signup and view all the answers

    What weight do one plum and three peaches total according to the example given?

    <p>750g</p> Signup and view all the answers

    What was the significance of the Brahmasphutasiddhanta in the context of zero?

    <p>It was the first known text to treat zero as a number in its own right.</p> Signup and view all the answers

    Who is credited with establishing the mathematical rules for zero, including the operations 1 + 0 = 1 and 1 x 0 = 0?

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

    What was Brahmagupta's misconception regarding division by zero?

    <p>He believed that any number divided by zero equals zero.</p> Signup and view all the answers

    What innovation in the representation of numerals is attributed to medieval Arabic mathematicians?

    <p>The dissemination of the nine numerals we use today.</p> Signup and view all the answers

    How did Bhaskara II differ from Brahmagupta in his treatment of division by zero?

    <p>He argued that it should be considered undefined.</p> Signup and view all the answers

    What concept did Brahmagupta explore in relation to quadratic equations?

    <p>That they may have two possible solutions, including negative ones.</p> Signup and view all the answers

    What method did Brahmagupta use to represent unknowns in his equations?

    <p>Initials of the names of colors.</p> Signup and view all the answers

    Which of the following is a misconception about the use of zero in mathematics?

    <p>Zero is never treated as a number in any mathematical context.</p> Signup and view all the answers

    What is the weight of a single peach according to the calculation provided?

    <p>200g</p> Signup and view all the answers

    How much does a single plum weigh based on the calculations presented?

    <p>150g</p> Signup and view all the answers

    What mathematical concept is discussed regarding remainders and unknown numbers?

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

    Which mathematician is noted for exploring quadratic and cubic equations during the Golden Age of Chinese mathematics?

    <p>Qin Jiushao</p> Signup and view all the answers

    What practical application of the Chinese Remainder Theorem is mentioned?

    <p>Internet cryptography</p> Signup and view all the answers

    In the example of arranging balls, how many balls are left over when arranged in rows of 3?

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

    What number represents the smallest quantity of balls in the arranged example?

    <p>52</p> Signup and view all the answers

    Which mathematical era is defined as the Golden Age of Chinese mathematics?

    <p>13th Century</p> Signup and view all the answers

    Which mathematician developed an algorithm to calculate the value of phi as approximately 3.14159?

    <p>Liu Hui</p> Signup and view all the answers

    What was hieratic writing primarily used for?

    <p>Monumental writing</p> Signup and view all the answers

    Which group of people were known for using a positional number system?

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

    What aspect of knowledge did ancient Greeks focus on enhancing?

    <p>Philosophical methods</p> Signup and view all the answers

    What is the numeric equivalent of the Greek numeral ∅?

    <p>400</p> Signup and view all the answers

    What kind of writing did Egyptians use for monumental purposes?

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

    Which philosopher is NOT noted for contributions to mathematics?

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

    Which of the following best describes the nature of contributions from Mesopotamia to mathematics?

    <p>Foundational advancements</p> Signup and view all the answers

    Study Notes

    Nasir Al-Din Al-Tusi

    • 13th Century Persian astronomer, scientist and mathematician
    • First to treat trigonometry as a separate mathematical discipline, distinct from astronomy
    • Gave the first extensive exposition of spherical trigonometry, including the six distinct cases of a right triangle in spherical trigonometry
    • Formulated the famous law of sines for plane triangles: 𝑎/sin𝐴 = 𝑏/sin𝐵 = 𝑐/sin𝐶
    • The sine law for spherical triangles was discovered earlier by the 10th century Persians Abul Wafa Buzjani and Abu Nasr Mansur

    Thabit ibn Qurra

    • 9th Century Arab
    • Developed a general formula by which amicable numbers could be derived
    • Amicable numbers are pairs of numbers for which the sum of the divisors of one number equals the other number (e.g. 220 and 284)

    Brahmasphutasiddhanta

    • 7th Century Indian mathematician Brahmagupta
    • Earliest known text to treat zero as a number in its own right
    • Established the basic mathematical rules for dealing with zero: 1 + 0 = 1; 1 – 0 = 1; and 1 x 0 = 0
    • Brahmagupta’s understanding of division by zero was incomplete (he thought 1 ÷ 0 = 0)
    • Established rules for dealing with negative numbers
    • Pointed out that quadratic equations could in theory have two possible solutions

    Bhaskara II

    • 12th Century Indian mathematician
    • Showed that the answer to 1 ÷ 0 should be infinity, not zero
    • This was considered correct for centuries, but the modern view is that a number divided by zero is “undefined”
    • Bhaskara II would have been the one to solve the non-sensical operation 1 ÷ 0

    Magic Squares

    • Squares of numbers where each row, column and diagonal adds up to the same total were regarded as having great spiritual and religious significance
    • The Lo Shu Square is an order three square where each row, column and diagonal adds up to 15, dating back to around 650 BCE
    • Larger magic squares were being constructed in the 13th Century with even greater magical and mathematical powers

    Yang Hui

    • 13th Century Chinese mathematician
    • Produced a triangular representation of binomial coefficients identical to the later Pascals’ Triangle
    • Perhaps the first to use decimal fractions in the modern form

    Jiuzhang Suanshu” or “Nine Chapters on the Mathematical Art”

    • Written over a period of time from about 200 BCE onwards, by variety of authors
    • An important tool in the education of civil service, covering hundreds of problems in practical areas such as trade, taxation, engineering and payment of wages
    • It was particularly important as a guide to how to solve equations, using a sophisticated matrix-based method

    Liu Hui

    • Greatest mathematicians of ancient China.
    • Produced a detailed commentary on the “Nine Chapters” in 263 CE.
    • One of the first mathematicians known to leave roots unevaluated, giving more exact results instead of approximations.
    • Formulated an algorithm which calculated the value of π as 3.14159 (correct to five decimal places), as well as developing a very early form of both integral and differential calculus

    The Chinese Remainder Theorem

    • The Chinese went on to solve far more complex equations using far larger numbers than those outlined in the “Nine Chapters”
    • Used remainders after dividing an unknown number by a succession of smaller numbers, such as 3, 5 and 7, in order to calculate the smallest value of the unknown number.
    • A technique for solving such problems was being used to measure planetary movements by Chinese astronomers in the 6th Century AD, and even today it has practical uses, such as in Internet cryptography

    Qin Jiushao

    • 13th Century Chinese mathematician
    • Explored solutions to quadratic and even cubic equations using a method of repeated approximations very similar to that later devised in the West by Sir Isaac Newton in the 17th Century
    • Extended his technique to solve equations involving numbers up to the power of ten.

    The Chinese Remainder Theorem

    • Finds the smallest number of balls such that given the following information:
      • If a collection of balls are arranged in rows of 3, there is one ball left over
      • If arranged in rows of 5, there are two balls left over
      • If arranged in rows of 7, there are three balls left over
      • The smallest number of balls must be 52

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    Description

    Explore the contributions of remarkable mathematicians throughout history, including Nasir Al-Din Al-Tusi, Thabit ibn Qurra, and Brahmagupta. This quiz covers significant advancements in trigonometry, amicable numbers, and the concept of zero. Test your knowledge on how these scholars shaped modern mathematics.

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