MTH U201Midterm 2010: Answers

GRADING

Question 1: Best 5 of 6 answered: five points each, total: 25

Questions 2 and 4: best one, total 25

Question 3: total 25

Questions 5 and 6: best one, total 25.




  1. Chose any six of the following and write a brief (paragraph or so) identification with approximate dates (nearest century, say), and a statement of its significance in the history of mathematics. Be concise and specific and give examples if appropriate. (4 points each)MATH

    Eratosthenes (3rd century BCE): Was the head librarian in Alexandria; made one of the first maps of the world, using lines of latitude and longitude; developed the Sieve, an algorithm for finding all the primes in the first $N$ numbers; calculated the circumference of the earth very accurately.

    Pythagoreans (6th century BCE): Followers of Pythagoras --- many of their discoveries were credited to him; studied "figurate" numbers like the square and triangular numbers; believed in the transmigration of souls, hence were vegetarians; allowed women to learn the mysteries of their cult; believed that everything in the world is number, and that every object and concept is assigned a number; studied harmony using simple number ratios; discovered the incommensurability of the square and its diagonal; may have had a proof of the Pythagorean Theorem.

    Thales (7th-6th centuries BCE): Perhaps the first person to use deductive reasoning to prove mathematical theorems; studied geometrical figures such as circles and triangles; knew about similar triangles and used them to compute the height of the Great Pyramid and calculate distances.

    Alexandria (founded: 4th century BCE): Founded by Ptolemy I, one of Alexander the Great's generals; at the mouth of the Nile river in Egypt; major trading center but, more importantly, contained the Museum and Library, making it the greatest center of learning in the ancient world. Many famous mathematicians studied in Alexandria, including Eudoxos, Euclid, Eratosthenes and Archimedes.

    Ruler and Compass (5th century BCE?): The only "officially" allowable means of constructing geometric figures in ancient Greece. Although they were used with writing instruments, they were considered theoretically capable of drawing perfect circles and lines; ancient mathematicians sought to use them to Duplicate the Cube, Square the Circle, and Trisect the angle --- problems proved many centuries later to be impossible.

    The Parallel Postulate: (4th century BCE): Eulid's 5th and most controversial postulate: "If a transversal crosses two lines, and the consecutive interior angles on one side total less than two right angles, then the lines will meet on that side of the transversal." It made a prediction about how lines would behave when extended indefinitely; was considered at once too strong and at the same time deducible from the other axioms (it isn't). Attempts to prove it lead to the creation of non-Euclidean geometries centuries later. It must be used to prove that the sum of the angles of a triangle is two straight angles.

    Archimedes (3rd century BCE): The greatest mathematician of the ancient world. Was born and lived most of his life in Syracuse, but also studied in Alexandria and corresponded with Eratosthenes among others; discovered the principle of the lever and of specific gravity, and invented many machines based on mathematical and physical principles (e.g. pumps and pulleys and clever lever systems). Discovered and proved many area and volume formulas from geometry, including the areas of spirals and circles and the volume of the sphere. Estimated the number of grains of sand in the known universe and developed a notation system to express this huge number; made first careful estimate of $\pi $.

    Triangular numbers (6th century BCE?): numbers counting dots in triangular arrays, with each row having one more dot than its predecessor; examples: 1, MATH (the "tetractys"); studied by the Pythagoreans, who discovered facts such as: the sum of any two consecutive triangular numbers is a perfect square, and the sum $T_{n}+T_{n}=n(n+1)$, so that $T_{n}=n(n+1)/2.$

    Plato (4th century BCE): Major philosopher, student of Socrates and teacher of Aristotle. Founded his Academy in Athens in 388 BCE and put up a sign saying "Let no man enter who doesn't know Geometry." Believed logic and geometry the best training for the mind, and that numbers and geometric figures existed in a pure form outside our immediate experience; similarly, that only the compass and ruler constructions represented the ideal way of realizing geometrical forms.

    Euclid's Elements (4th century BCE): The thirteen books, edited by Euclid, that laid out the logical foundations and deduced most of the known theorems in Greek Geometry and Number Theory (see question 5 on its stucture). It is not clear if Euclid himself originated any of these theorems, but he masterfully combined all of the theory into a coherent whole. The Elements are one of the best-selling books of all time.

  2. Define: $a$ divides $b$, and prime number. Prove that "there are an infinite number of primes," i.e., for any collection of primes, there is always another one. You may assume without proof that (1) $p$ never divides $K\cdot p+1$ and (2) the smallest divisor of any number $N>1$ is always prime. (Remember that, for the Greeks, number meant what we now call a whole positive number.)

    $a$ divides $b$ means that there is a third number $k$ such that $b=a\cdot k$. A number is prime if its only divisors are itself and $1$. For a proof of the infinitude of primes, see our website.

  3. Use the Sieve of Eratosthenes to find the primes between $2$ and $100$. Explain exactly what you are doing.MATH

    REPEAT THIS STEP: Circle the first number $P$ not crossed out --- it will be prime --- and cross out every $P$th number after it; ie.cross out all of its multiples. (For example, circle $2$ and cross out every $2$nd number: $4,6,8,$etc.)

    DO THIS UNTIL $P^{2}$ is bigger than $100$. (In this case, stop after crossing out all multiples of $7$, since the next $P$ will be $11$, whose square is bigger than $100$.)The numbers not crossed out will be all the primes between $2$ and $100$.

  4. Prove that $\sqrt{2}$ is not rational. How is this related to commensurability?

    See our website for the proof of this famous theorem. Proving that $\sqrt{2}$ is not rational is equivalent to showing that there is no unit $U$ which goes evenly into the side of a square and its diagonal: this follows from the Pythagorean Theorem; indeed, if $S=BU$ and $D=AU$, then $S^{2}+S^{2}=D^{2}$, soMATHthus, the rational number $A/B$ would be the squareroot of $2$.

  5. Discuss the structure of the axiomatic system, as used by Euclid. Give examples.

    The basis for an axiomatic system consists of the following:

    1. Undefined terms and defined terms; any new term may be defined in terms of these. Euclid, erroneously, did not use any undefined terms.

    2. Axioms or Postulates: statements assumed without proof. Euclid had 10 of these; the last five he called "common notions". First 5 axioms are: A line may be drawn between two points; A line may be extended indefinitely; A circle may be drawn with any center and radius; All right angles are equal; The Parallel Postulate. Some common notions are: Equals added to equals are equal and Things equal to the same thing are equal to each other.

    3. Theorems, which are statements proved using the above, together with logic. Examples are: "Base angles of an isosceles triangle are equal", "An exterior angle of a triange is greater than either remote interior angle" (it's equal to their sum if the Parallel Postulate is used).

  6. Greek mathematics had its strengths and weaknesses. Discuss them; give comparisons with Egyptian and Babylonian mathematics. Be specific, and discuss number systems, arithmetic and geometry.

    A major strength of Greek mathematics was its careful logical structure, which made it possible to prove correct statements. The Greeks were very strong in geometry and number theory, where these methods were very effective. They believed, with Plato, that mathematics was correct because is reflected the perfect world that we couldn't see but existed in ideal form. Many of the most famous Greek mathematicians such as Plato and Archimedes strongly favored theory over practicality; this was both an advantage and disadvantage.

    One great weakness of Greek mathematics was its very primitive number system. The Greeks only recognized whole positive numbers, and used a non-positional base 10 system --- similar to the unwieldy Egyptian system --- based on their alphabet. They never considered fractions as numbers, though they did use ratios and proportions. Since their system was non-positional, they never developed decimal fractions either. Finally, they made no important contributions to algebra; they never even got to the point of denoting numbers by letters or using "$=$" signs.

    The Egyptians and Babylonians never developed ideas of proving theorems; consequently, they had many wrong formulas (expressed in words of course) --- mostly in geometry. It is unclear that the early Egyptians (2000 BCE) even knew the Pythagorean Theorem, or that they were ever able to prove it. The Babylonians had a superior number system and knew many facts about numbers, such as how to generate many Pythagorean triples; they could also solve quadratic equations, which the Greeks couldn't do until nearly a thousand years later. However, they also never developed the formal ideas of abstract systems like Euclid's Elements, and never seem to have given general proofs. Both the Egyptians and Babylonian seemed to be more interested in practical applications than in abstract theory.