Math 460 Final Exam
Discuss the two historical and mathematical significances of the discovery that the diagonal and the side of a square are incommensurable.
The Greeks believed that any two given segments were always commensurable. Many geometric theorems (e.g., the theory of proportion and similarity of triangles) were based on this incorrect assumption. The discovery of incommensurable segments caused Greeks to rebuild theories based on such assumption. The belief that any two segments were commensurable implied that all numbers were rational, that is, any number could be represented as the quotient of two natural numbers. This supported their philosophy that all human affairs can be explained in terms of natural numbers. Their whole philosophy collapsed with the discovery of incommensurable segments (that is, irrational numbers).
State the two fundamental problems of analytic geometry.
(1) Find the equation of the set of points that satisfy a certain condition. (2) Describe the set of points that satisfy a given equation.
Describe briefly but thoroughly the four phases in the quest for numerical approximations to π
(a) Empirical phase involved using experiential measurements. (b) Archimedean phase involved finding a lower and an upper bound to the numerical value of p by inscribing and circumscribing regular polygons with a large number of sides. (c) Series phase involved using series that converges to a constant involving π. (d) Computer phase involves using the power of the computer.
State the two main theorems proved by Archimedes in which he used double reduction to absurdity.
(a) The area of any circle is equal to the area of a right triangle in which one of the legs is equal to the radius, and the other to the circumference of the circle. (b) The surface area of any sphere is equal to four times the area of its greatest circle.
Write five unsolved mathematical problems or questions related to prime numbers that we discussed in class. State your problems in self-contained form.
1) Is the set of Mersenne primes infinite? 2) Is the set of perfect numbers infinite? 3) Is the set of twin primes infinite? 4) Are there odd perfect numbers? 5) Is every even number greater than 2 equal to the sum of two prime numbers?
State two related significant historical mathematical problems that cannot be solved related to the material covered in this test that we discussed in class. State your problems in self-contained form. (e.g., Construct with straightedge and compass the angle bisector for an arbitrary angle).
1) Prove that the diagonal and the side of a square are commensurable. 2) Prove that √2 is rational.
Briefly describe the four main mathematical features of the Babylonian numeration system.
1) The system lacked the notion of complete cipherization. The numerical notation used two-wedge characters and to represent the digits 1-59. 2) It is a partial positional numeration system with base 60 (Sexagesimal). The value of each digit depends on its position on the number. 4 3) It lacked the concept of zero as a number. Therefore, it did not have a symbol for zero. 4) The system lacks a sexagesimal point and, therefore, it is difficult to distinguish the fractional part of a number from its whole number
Formulate two open questions involving π
1. Does each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 appear infinitely often in π? 2. Is there a place in the decimal expansion of π where a thousand consecutive digits are all zero? (Brouwer's question) 3. Is π normal? That is, does every block of digits of a given length appear equally often in the expansion in every base in an asymptotic sense? 4. Is π normal to base 10? That is, does every block of digits of a given length appear equally often in its decimal expansion in an asymptotic sense?
Discuss two problems associated with Euclid's fifth postulate.
1. We cannot verify this postulate experimentally. 2. It looks more like a theorem because it is not brief, simple, and self-evident.
Formulate an open question or problem related to prime numbers that was mentioned in Chapter 4
A Fermat prime is a prime that can be represented as 2^(2^k) + 1, where k is a nonnegative integer. 1. Are F_0(3), F_1(5), F_2(17), F_3(257), F_4 and F_5 the only Fermat primes? 2. Is the set of Fermat primes infinite?
Provide the Newtonian definitions of fluent and fluxion.
A fluent is a changing quantity. A fluxion of the fluent is its ratio of change.
Define what a normal number is.
A number in which every block of digits of a given length appear equally often in the expansion in every base in an asymptotic sense.
Define perfect number. Use the definition of perfect number to justify whether a given number is perfect.
A perfect number is a number that is equal to the sum of its proper divisors. For example, 28 is a perfect number because it can be represented as the sum of its proper divisors: 28 = 1 + 2 + 4 + 7 + 14.
Define Mersenne prime. Provide the first four Mersenne primes.
A prime of the form 1 + 2 +2^2 + ... + 2^n for some natural number n is called a Mersenne prime. The first four Mersenne primes are 3 (1+ 2), 7 (1 + 2 + 4), 31 (1 + 2 + 4 + 8 + 16), and 127 (1 + 2 + 4 + 8 + 16 + 32 + 64).
Briefly describe, in general terms, a proof by double reduction to absurdity
A proof by double reduction to absurdity involves eliminating two out of three possibilities by reduction to absurdity. In other words, given three possibilities, if two of these are impossible, then the third possibility must be true.
Provide the definition of a birectangle ABCD. Indicate its properties included on the definition on the following figure with appropriate marks.
A quadrilateral ABCD in which AD = BC and angles A and B are right.
State Gauss-Wantzel constructibility theorem.
A regular polygon of n sides is constructible if, and only if, n is of one of the following forms 1. n = (2^r)(4), r = 0, 1, 2, ... 2. n = (2^r)(p), where p is a fermat prime and r = 0, 1, 2, ...; 3. n = (2^r)(p_1) (p_2)...(p_m)), where p_1, p_2, ... and p_m) are distinct Fermat primes and r = 0, 1, 2, ....
Mention the main contribution of Euclid to mathematics
Euclid did not only construct proofs, he organized the proofs within an axiomatic system.
Discuss the strengths and weakness of Euclid's axiomatic treatment of geometry
Euclid organized the mathematical theorems into an axiomatic structure. Euclid began his Elements with 23 definitions, 5 postulates, and 5 common notions. With those tools he proved his first proposition. Then, with his first proposition and the definitions, postulates and common notions he proved his second proposition and son on. So, he did not only construct proofs, he provided the proofs within an axiomatic organization. His definitions of points and lines are not mathematical definitions His list of axioms is incomplete Euclid's axiomatic geometry has many holes as he sometimes uses visual clues to support his arguments. Some of those unstated or assumed statements are axioms or theorems in modern Euclidean geometry.
Describe Euclid and Euler's contributions related to the field of perfect numbers.
Euclid's theorem about perfect numbers: If 1 + 2 +2^2 + ... + 2^n is prime for some whole number n, then N = 2^n(1 + 2 +2^2 + ... + 2^n) is a perfect number. Euler's theorem about perfect numbers: If N is an even perfect number then there is a natural number n such that N = 2 (1 + 2 +2^2 + ... + 2^n) where 1 + 2 +2^2 + ... + 2^n is a (Mersenne) prime
State and illustrate the fundamental theorem of arithmetic with an example.
Every positive integer greater than 1 is either prime or has a unique representation, apart from order, as the product of primes). As an example, consider 2474346. Since 2474346 is divisible by 2, it is a composite number that has a unique prime factorization, namely, 2474346 = 2×3×7×58913, where 2, 3, 7, and 58913 are primes.
State Fermat's theorem. Mention Fermat, Euler, and Wiles's mathematical contributions to Fermat's last theorem
Fermat theorem: The equation x^n + y^n = z^n has no non-trivial integer solutions for n > 2. Euler proved the theorem for n = 3, and Wiles proved it for any natural number.
State the method of exhaustion. Explain it using the given quantity 0.0003
For inscribed regular polygons: Given an area of 0.0003 cm^2 , there is a regular polygon Q such that Area (circle) - Area (Q) < .0003cm^2 For circumscribed regular polygons given an area of 0.0003 cm^2 , there is a regular polygon Q such that Area (Q) - Area (circle) < 0.0003 cm^2
State a significant historical mathematical problem that cannot be solved related to graph theory that we discussed in class. State your problems in self-contained form. (e.g., Construct with straightedge and compass the angle bisector for an arbitrary angle).
For the following network, is there a path that traces each edge once and only once?)
State and discuss the significance of the four-color problem.
Formulation of the problem: How many colors are needed to color any map on a plane so that neighboring regions sharing a boundary have different colors if neighboring regions share a boundary? Significance of the problem: The four-color problem is the first example of a mathematical problem that may be impossible to solve without computers. This problem may be the first example that shows the limitations of the human mind. The solution of the problem raises questions about the nature of mathematical proof.
State and illustrate with a diagram the following equivalent propositions of Euclid's fifth postulate
Proclus's axiom If a line intersects one of two parallel lines, then intersects the other also. Playfair's axiom Through a point not on a given line, there is only one parallel to the given line Equidistance axiom Parallel lines are everywhere equidistant.
Use Euler's theorem to justify whether a given number is perfect.
Consider 496. Notice that 496 = 16(1 + 2 + 4 + 8 + 16) and (1 + 2 + 4 + 8 + 16) = 31 is prime so 496 is a perfect number
State Euler's Traversability Theorem
a) A connected network is traversable if, and only if, it has either no odd vertices or two odd vertices. b) If a connected network has no odd vertices, then any Euler path is a closed curve whose beginning and ending is the same vertex. c) If a connected network has two odd vertices, then those vertices are the endpoints of any Euler Path
State both versions of Cavalieri's principle.
a) If two planar pieces are included between a pair of parallel lines, and if the lengths of the two segments cut by them on any line parallel to the including lines are always in a given ratio, then the areas of the two planar pieces are also in this ratio. b) If two solids are included between a pair of parallel planes, and if the areas of the two sections cut by them on any plane parallel to the including planes are always in a given ratio, then the volumes of the two solids are also in this ratio.
Discuss whether the following propositions hold. Justify your responses a) Postulate 1: Given two points on the plane, there is a line that goes through them. . b) Euclid's fifth postulate (Playfair's version) c) The sum of the interior angles of any triangle is 180°. d) The length of a line is infinite.
a) Yes b) No, lines always intersect, so given a line l and a point P not on the line there is not a line that contains point P and is parallel to l. c) No. The sum of the measurements of the interior angles of a triangle is greater than 180°. d) No. See the first set of class notes for further diagrams.
Discuss whether the following propositions hold. Justify your responses a) Postulate 1: Given two points on the plane, there is a line that goes through them. In the test, make sure the endpoints of the chord are indicated with open circles. b) Euclid's fifth postulate (Playfair's version) c) The sum of the interior angles of any triangle is 180° d) A line segment can be extended beyond its endpoints.
a) Yes. b) No. Below is an example. Lines m, n and p are parallel to line l and go through the same point P c) No. No diagram is required for this. d) Yes, since the endpoints of the chord do not belong to the line, we can always extend a line segment no matter how close its endpoints are to the circle.
Let a, b, c, and d positive integers. If d = gcd(a, b) and if c | a and c | b, a) What can you say about a, b and d? b) What can you say about c and d?
a) d | a and d |b [and d = gcd(a, b)] b) c ≤ d
Let N = 2 x 5 x 7 x 11 x 13 + 1. a) Use the division algorithm theorem to explain why N is not divisible by any of the numbers 2, 5, 7, 11, and 13 b) Use the theorem that any composite number is divisible by some prime number to find another prime number different from 2, 5, 7, and 13. (In other words, if N = 2 x 5 x 7 x 11 x 13 + 1 is composite, find the prime that divides N. If N is prime, then N is the prime that we are looking for).
a). N = 2 (5 x 7 x 11 x 13) + 1, so N ÷ 2 = 5 x 7 x 11 x 13 + 1. In other words, 1 is the remainder of N divided by 2. N = 5 (2 x 7 x 11 x 13) + 1, so N ÷ 5 = 2 x 7 x 11 x 13 + 1. In other words, 1 is the remainder of N divided by 5. Similarly for 7, 11, and 13: 1 is the remainder of N divided by each of these numbers. b). N = 10011. If N is composite, then there is a prime number p such that p | N. In this case p = 3, 47, or 71.
b) State three historical theorems involving π.
π is irrational π is transcendental A segment of length π cannot be constructed using only straightedge and compass
a) Provide the (geometric) definition of π.
π is the ratio of the circumference to the diameter of a circle
Provide the first three numbers that satisfy the following theorem: If 1 + 2 +2^2 + ... + 2^n is a prime number, then n + 1 must be a prime number.
3 = 1 + 2 and 2 (1 + 1) is prime 7 = 1 + 2 + 2^2 and 3(2 + 1) is prime 31 = 1 + 2 + 2^2 + 2^3 + 2^4 and 5 (4 + 1) is prime
State and verify Goldbach's conjecture for the first appropriate 5 even numbers.
Goldbach's conjectures affirms that every even number greater than 2 equals to the sum of two prime numbers 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 5 + 5, 12 = 5 + 7
State Euclid's fifth postulate
If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles The fifth postulate (Playfair's version). Given a line l and a point P not on l, there is only, and only one, line parallel to l that contains P.
Provide the modern definition of limit.
L is the limit of the function f(x) as x approaches a [if] for every ε > 0, there exists a δ > 0 so that, if 0 < | x - a | < δ, then |f(x) - L| < ε.
Define the greatest common divisor of two integers
Let a and b two integers, with either a or b different from zero. The greatest common divisor of a and b, denoted by gcd(a, b), is the positive integer d satisfying (1) d | a and d | b (2) if c | a and c | b, then c ≤ d.
State and illustrate the division algorithm theorem with an example.
Let a, and b be two integers. There exist unique integers q and r such that a = bq + r, 0 ≤ r < b. For example, if n is a positive integer, then n = 6q + r, 0 ≤ r < 6.
State the theorem in which Euclid used his fifth postulate for the first time.
Proposition I.29. If two parallel lines are intersected by a transversal, then the alternate interior angles are congruent.
State the first two main propositions that depend on Euclid's fifth postulate.
Proposition I.29. If two parallel lines are intersected by a transversal, then the alternate interior angles are congruent. Proposition I.32(b). The sum of the measurements of the interior angles of any triangle is 180˚.
State the most significant historical impossible mathematical problems related to the material covered in this test that we discussed in class. State your problems in self-contained form. (e.g., Construct with straightedge and compass the angle bisector for an arbitrary angle).
Prove that there is an algebraic formula that represents the solution to the equation: a_nx^n+a_n-1x^n-1+....+a_2x^2+a_1x+a_0=0, n>_5, a_n doesn't equal 0
Discuss the relationship between the existence and uniqueness of parallel lines and Euclid's fifth postulate.
The existence of parallel lines depends mainly on proposition 1.27 (If a straight line falling on two straight lines makes the alternate [interior angles] equal to one another, the two straight lines will be parallel) while Euclid's fifth postulate guarantees its uniqueness.
Discuss the difference between an existence proof and a constructive proof using the fundamental theorem of algebra to illustrate the difference.
The fundamental theorem of algebra states that any polynomial equation with complex coefficients in one variable of degree n has n or less complex roots. An existence proof of the FTA involves proving that any polynomial equation with complex coefficients in one variable of degree n has n or less complex roots without providing an algorithm to find the roots. In contrast, a constructive proof of the FTA provides an algorithm to find the roots
State Abel-Ruffini's theorem
The general polynomial equation in one variable (a_nx^n+a_n-1x^n-1+....+a_2x^2+a_1x+a_0=0, n>_5, a_n doesn't equal 0) has not solution by radicals.
Define twin primes. Provide the first 5 pairs of twin primes.
Twin primes are two prime numbers whose difference is either 1 or 2. The first pair of twin numbers are (2, 3), (3, 5), (5, 7), (11, 13), and (17, 19).
The number of twin primes is infinite.
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Given segments of length 1, a and b, what segments can be constructed (with straightedge and compass)?
We can construct segments of length a + b, a - b (provided a > b), ab, a/b, √a, √a, and √ab, among others.
Define what it means for an equation to be solved by radicals. Illustrate your response with the quadratic equation.
We say that an equation can be solved by radicals if its solution can be represented with a formula involving only the coefficients of the equation and algebraic operations (addition, subtraction, multiplication, division, powers, and extraction of roots) used a finite number of times. For example, the solution of the quadratic equation in one variable, ax^2+bx+c=0 is given by the formula x= (-b +-√(b^2-4ac))/2a Notice that this formula (1) Involves only the coefficients of the equation (a, b, and c). (2) Involves only algebraic operations (addition, subtraction, multiplication, division, powers, and extraction of roots). (3) The operations are used only a finite number of times.
Consider Klein's model of Hyperbolic Geometry i) Define plane: ii) Define line.
i) A plane is the set of interior points of a circle. ii) Draw an example of a line. Line is a chord of a circle without the endpoints In the test, make sure the endpoints of the chord are indicated with open circles.
State two of four important scientific problems of the 17th century that gave origin to Calculus.
i) Given the formula for the distance of traveling object as a function of time, find the instantaneous velocity and acceleration. Given the acceleration of a traveling object as a function of time, find the velocity and distance. ii) Finding the tangent to a curve (pure geometry, optics [design of lenses], motion, and the concept of tangent itself) iii) Finding the maxima and minima values of a function (cannonball, heights reached by projectiles, planetary motion) iv) Finding the lengths of curves (e.g., distance traveled by a planet in a determined time period), areas bounded by curves, volumes bounded by surfaces, centers of gravity of objects, gravitational forces between bodies.
For Spherical Geometry i) Define plane & ii) Define line
i) The set of all points on the surface of a sphere ii) A circle of the sphere that contains the endpoints of a diameter of the sphere.
