Description
Please complete question belowGiven that we have no documents from the time of Greek mathematics—the earliest manuscripts we have are medieval—how can we be sure that the texts we have are actually what the authors wrote?Were the copyists who wrote the early medieval manuscripts simply concerned with reproducing the text faithfully, or is it possible that they tried to improve it by revisions they thought of themselves?How could we know if they did?
1. Given that we have no documents from the time of Greek mathematics—the earliest manuscripts we have are medieval—how can we be sure that the texts we have are actually what the authors wrote? Were the copyists who wrote the early medieval manuscripts simply concerned with reproducing the text faithfully, or is it possible that they tried to improve it by revisions they thought of themselves? How could we know if they did?
2. What are the main topics investigated in ancient Greek number theory? How much of number theory has a practical application nowadays? (If you don’t know about RSA codes, for example, look them up on-line.)
3. What mathematical ideas were ascribed to the Pythagoreans by ancient commentators?
4. For what reason would the ancient Greeks have been investigating figurate numbers, perfect numbers, and the like? Did they have a practical application for these ideas?
5. Assuming that there are two square integers whose ratio is 5, derive a contradiction using the principle that underlies Knorr’s conjecture. (If the integers are relatively prime, then both must be odd. Use that fact and the fact that the square of any odd number is one unit larger than a multiple of 8 to derive a contradiction.)
6. How do you resolve the paradoxes of Zeno?
7. Summarize the progress made on each of the three classical problems during the fourth century BCE.
8. Why was it important to Menaechmus’ solution of the problem of two mean proportionals that the plane cutting the cone be at right angles to one of its generators?
9. How would you establish that two triangles with equal altitudes and equal bases are equal (“in area,” as we would say, although Euclid would not)?
10. Why is the problem of squaring the circle much more difficult than the problem of doubling the cube or trisecting the angle?
11. It appears that the Greeks overlooked a simple point that might have led them to break out of the confining circle of Euclidean methods. If only they had realized that composite ratios represent multiplication, they would have been freed from the need for dimensional consistency, since their ratios were dimensionless. They could, for example, multiply and number of ratios, whereas interpreting the product of two lines as a rectangle precluded the possibility of any geometric interpretation of product containing more than three factors. Could they have developed analytic geometry if they had made this realization? What else would they have needed?
12. Granting that if two lines perpendicular to the same transversal line meet on one side of that line, reflection about the midpoint of the interval between the two points where the lines meet the transversal shows that they must also meet on the other side. How do you know that these two points of intersection are not the same point? What other assumption must you introduce in order to establish that they are different?
13. Was the Elements an exposition of the most advanced mathematics of its time?
14.What advances in geometry, beyond the basic results found in the Elements, are due to Archimedes?
15. Show from Apollonius’ definition of the foci that the product of the distances from each focus to the ends of the major axis of an ellipse equals the square on half of the minor axis.
Explanation & Answer
View attached explanation and answer. Let me know if you have any questions.
Running head: APPLIED MATHEMATICS ESSAY QUESTIONS
Applied Mathematics Essay Questions
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APPLIED MATHEMATICS ESSAY QUESTIONS
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Applied Mathematics Essay Questions
Question 1
While it is correct to note that documents directly related to ancient Greek mathematics
are lacking, it is still possible to deduce whether what was written in copied manuscripts was
actually what the authors wrote. This is because the copyists were from different areas such as
Syria, Byzantine, and Arabs (Ornes, 2017). If these individuals were indeed copying the same
thing, then they would write the same thing or very similar things regardless of when they wrote
them. The fact that they wrote things that complemented one another and were in synchrony, is
an indication that they indeed were reading from similar scripts and that they represented them
properly. It is likely that the ones who reproduced the manuscripts were focused on doing it
accurately so as to represent the true mathematical concepts. If they did not do this, there would
be inconsistencies in different translations of the original manuscripts that would indicate bias.
Question 2
The ancient Greek number theory focused on a number of topics, the most common ones
were the divisibility properties of integers, figurative numbers, and the substitution of digits by
letters through algebra. Nowadays in the era of information technology, the number theory is still
applicable in several areas (Agarwal, 2021). For example, when companies create encryption
systems and keys, they often use the concept of prime numbers in doing so. An example of such
systems that use prime numbers are the RSA codes that use prime numbers in their algorithms
(Zaid & Hassan, 2018). In the same vein, number theories are very much applicable in the
classroom environment where teachers have to teach students the basic tenets of mathematics.
Question 3
APPLIED MATHEMATICS ESSAY QUESTIONS
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There are multiple ideas that were ascribed to the Pythagoreans by commentators. One
such idea was the Pythagorean triple. In this representation of numbers, a2 + b2 is equal to c2,
where the letters in the question represent positive integers in the numerical system (Ochieng et
al., 2019). Another idea that was attributed to the Pythagoreans is that of representing numbers as
an arrangement of dots. In this configuration, numbers can be classified as square numbers, and
even triangular numbers depending on whether their arrangement yields a square or a triangle.
Finally, musical harmony was also attributed to Pythagoreans since they were deemed to have
been the ones who understood that musical notes were heavily connected to whole numbers or
integer ratios (Anderson, 1983).
Question 4
One of the primary things that made the ancient Greeks study numbers was their desire
for knowledge and absolute truth and not necessarily for the practical application of those
numbers or formulas. The second reason that they studied numbers was because of their mystical
nature (Agarwal, 2021). Individuals such as Pythagoras studied perfect numbers because he
believed they had mystical properties and represented things such as ba...
