Reading Questions and Preview Problems for Math 128

Section/Material Reading Questions
Berggren 5.5 0. Please bring a calcualtor to class on Monday.
1. What is the first/preliminary step in al-Biruni's method of calculating the radius of the earth?
2. What is the main theorem he uses in the determination of the radius of the earth?
3. Did he actually use his method? If so where and what value did he come up with?
4. What is the advantage of al-Biruni's method over more ancient ones?
5. Did you understand everything about this method?
Berggren 5.4 1. Write down the law of sine.
2. What application of the sine law from Nasir al-Din's work is given in this section?
3. Where is the "usual rule" for calculating the height of a triangle taken from?
4. Do you have any difficulty with the proof of the sine law and the applications that follow? Does anything need clarification?
Berggren 5.2, 5.3 1. What was the main contribution of Muslim mathematicians to trigonometry?
2. What is the main difference between original definitions of the 6 trig functions and the modern definitions of the same functions?
3. What mathematician may be regarded as the first to have calculated the modern trig functions?
4. Write down the addition formula (or theorem) for the sine function in modern notation.
5. Was there anything on this section that was not clear to you needing clarification?
Berggren 5.1 1. Where and when are the first traces of the subject of trigonometry found? What science is it most connected to?
2. What is the meaning and evolution of the term "Almagest"
3. Explain the meanings of the numbers on 14th row of the table on page 129.
4. What does it mean to solve a triangle?
5. How does Nasir al-Din al-Tusi use a chord table to solve a right triangle when all sides and one angle are known?
6. Was there anything on this section that was not clear to you needing clarification?
Berggren 4.7 1. How many types of cubic equations did Omar Khayyam classify? How many of those can be solved by Euclidean methods and how many by conic sections?
2. What are the possible number of roots for a cubic equation? What remarks did Khayyam make about possible number of roots of a cubic equation?
3. Did he explain his solutions using algebraic symbols?
4. Express the statement "cube and a number equal sides" as a cubic equation.
5. Was there anything on this section that was not clear to you needing clarification?
Berggren 4.6 1. Who is the first mathematician to develop algebra of expressions containing higher powers of an unknown?
2. According to the rules of exponents, what is x^m*x^n equal to (as a single power of x )
3. What is the simplified form of -a-(-b) ?
4. Write down the equivalents of the following terms as powers of x:   mm,   mc,   pm,   pmc 
5. Convert the expression "mal cube and 2 mals less 3 units" to a polynomial in x. 
6. Was there anything on this section that was not clear to you needing clarification? Did you watch the video, did it help?
Berggren 4.5 1. Who is the Muslim mathematician whose epithet is "the Egyptian calculator" 
2. In what way does Abu Kamil's  work on algebra go beyond  that of al-Khwarizmi?
3. Give an example to show the usefulness of the rule sqrt(a)/sqrt(b) = sqrt(a/b)
4. Was there anything on this section that was not clear to you needing clarification?
Berggren 4.4

Note: There are some typos in this section: i) In Figure 4.5,     ii) in the radical expression in the middle of page 107. The radical should not include the term -p/2

1. What is a difference between Thabit b. Qurra's work on algebra and that of al-Khwarizmi?
2. What are the modern algebraic equivalents of the following terms (in symbols)
      a) root(thing)  b) roots  c) number of roots   d) mal
3.In proving his solutions to quadratic equations, Thabit uses two theorems from Elements. Write the statement of one of the theorems.
4. Was there anything on this section that was not clear to you needing clarification?
Berggren 4.1-4.3 1. What is the reason, many scholars believe, for the Greek emphasis on geometry? 
2. What are the meanings of the Arabic words "al-jabr" and "al-muqabala"?
3. Translate the following sentence to a symbolic algebraic equation: "mal equals 5 roots and 3" (you can write x^2 to denote x² )
4. What distinguished al-Khwarizmi (and his followers) from earlier mathematicians who worked on algebraic problems?
5. Was there anything on this section that was not clear to you needing clarification?
Berggren 3.7

1. What aspect of Islamic civilization does section 3.7 talk about and how is it related to mathematics?
2. Where are the problems/constructions (1-5) at the end of the section taken from (the name of the book/treatise and the author)?
3. What sorts of other constructions does this treatise contain?
4. Do you understand the problems (constructions) 1-4 at the end of the section?
Berggren 3.5 and 3.6

1. Why does the ability to trisect an angle imply the construction of a regular nonagon?
2. Who discovered a method to trisect an angle without using "moving geometry" (vergings)?
3. Why was it useful for instrument makers to know how to construct hyperbolas?
4. Who invented a proof for the area of a segment of a parabola to rescue the family's scientific reputation?
5. Do you understand Ibrahim b. Sinan's constructions of a parabola and a hyperbola?
Berggren 3.4 1. What was the problem with Archimedes' construction of a regular heptagon that Abul-Jud complained about?
2. What is the method of analyis? What is the method of synthesis.
3. What are the major steps (not details) of Abu Sahl's analysis for the construction of a regular heptagon?
4. Was there anything on this section that was not clear to you needing clarification? Did you watch the video? Did it help?
Berggren 3.1-3.3 1. What are the properties of Euclid's straightedge and compass?
2. Why does Euclid bother showing that a collapsible compass can transfer lengths like a rigid compass does?
3. What are the main Greek sources of geometry for Islamic mathematicians?
4. What is a conic section? What is a sypmtom of a conic section?
5. Was there anything confusing or unclear to you in this section?
Berggren 2.7

1. Write at least 3 rules of Islamic inheritance law.
2. Why do you think the author makes the comment in parantheses in the penultimate paragraph on page 63?
3. What is zakat?
4. What is shahadah?
5. Did you understand all of the calculations in this section?
Berggren 2.5 1.  In the algorithm finding the square root of a number N, if r is the integer part of the answer and if u/v is the fractional part, what are u and v in terms of N and r?
2.  In finding/justifying the fractional part of the approximation, what did they effectively assume about the square root function? (without using the terminology)
3.  Al-Kashi's instruction that "double A, the previous digit of the root, (multiply by 10) then put the digit B next to it and multiply by B" is based on an algebraic identity. What is that identity?
Berggren 2.5 1. From where is the algorithm presented in 2.5 to find approximate square root of an integer taken?
2. Why does al-Kashi start by dividing the digits of the radicand into groups of two (that he called cycles)?
3. Suppose you want to find the square root of 82119 using this method. What would be the first (highest order) digit of the answer and why?
4. Was there anything on this section that was not clear to you needing clarification? Did you watch the video? Did it help?
Berggren 2.3 and 2.4 1. Who seems to have invented the decimal fractions? When does the decimal point first appear in history?
2. What observation or known rule did the invention come out of?
3. What are the contributions of Muslim mathematicians to the sexagesimal system?
4. What is the method of levelling used in sexagesimal multiplication?
5. Was there anything on this section that was not clear to you needing clarification? Did you watch the videos? Did they help?
Berggren 2.1 and 2.2

1. What exactly are the contributions of Hindus and Muslims to the modern number system (called the Hindu-Arabic system)? Be specific and explain each civilization's contributions separately.

2. How was the notion of zero in that system/time different from the modern notion of zero? Arabic word "sifr" is the source of two modern mathematical/technical terms. What are those?

3. What was the earliest Arabic work that explained the Hindu number system?

4. Did you have any trouble understanding any of the operations described in 2.2? Does any of them require clarification? Did you watch the videos? Did they help?
Berggren Preface and Chp 1

1. What are the sections called "Islamic Dimensions" at the end of chapters 2,3,4, and 6 about?

2. Berggren says a book titled "History of Mathematics in Medieval Islam" cannot be written yet. Why not?

3. Tell me something interesting or important about each one of the 4 Muslim scientists described in this chapter.

4. What are some obstacles to modern scholars who want to study scientific works of medieval Muslim scientists that exist in libraries in various parts of the Muslim world.

5. How is it that there are large collections of Arabic manuscripts in European libraries?

6.According to Berggren, what are some major contributions of the Islamic Civilization to mathematics between 750 and 1450?