Pythagorean Triples and Rational Points on the Unit Circle


 Jasmine Mitchell
 2 years ago
 Views:
Transcription
1 Pythagorean Triles and Rational Points on the Unit Circle Solutions Below are samle solutions to the roblems osed. You may find that your solutions are different in form and you may have found atterns not listed here. Exloring Triangles Algebraically Take a moment and list some Pythagorean triles that you know. Are there any atterns in your list? For instance, in any Pythagorean trile, how many even entries are ossible? Are 0, 1,, and 3 all ossible? Can the hyotenuse be the only even side length? Justify your answer. The key to this roblem is that the square of an even number is even and the square of an odd number is odd. So, we cannot have all of a, b, and c being odd since if the sum or difference of two odd numbers must be even. So, 0 even numbers is imossible. Clearly, having all of a, b, and c as even numbers is ossible as 6, 8, 10) is a Pythagorean trile. Also, a single even is ossible since 3, 4, 5) is a trile. Since the sum or difference of even numbers is also even, it is imossible to have exactly two numbers in the trile be even. Thus, there can only be triles with a single even number or three even numbers. In the case of a single even number in a Pythagorean trile, it is imossible for the hyotenuse to be even. If a = j + 1 and b = k + 1 are the odd triangle lengths and c = m is the even hyotenuse length, then we have j + 1) + k + 1) = m) = 4j + 4j k + 4k + 1 = 4m = 4j + j + k + k) + = 4m The lefthand side is two more than a multile of 4 while the righthand side is a multile of 4; this is imossible. 1
2 Notice that 3, 4, 5) and 6, 8, 10) are two triles that are, in some sense, very closely related. How? Can you find another trile that is related to 3, 4, 5) and to 6, 8, 10)? Can you use this to show that there are infinitely many right triangle triles? Exlain why your result is true. Multilying a given Pythagorean trile by a ositive integer, k yields another Pythagorean trile. That is, if a, b, c) is a Pythagorean trile, then so is ka, kb, kc) since ka) + kb) = k a + b ) = k c = kc). Since there are infinitely many ositive integers, each Pythagorean trile generates an infinite family of triles. Consider the following table of Pythagorean triles. a b c Send some time determining how these triles are related. Can you determine the next three triles in this attern? Be sure to check that your triles are Pythagorean triles!) The next three triles are 15, 11, 113), 17, 144, 145) and 19, 180, 181). It is obvious that c = b + 1 for each of these triles and that a is running through the odd numbers in order. Then, or b = 1 a 1). a + b = b + 1) = a = b + 1, Use the attern you found in the revious roblem to create a formula for these triles and will hence generate an infinite number of Pythagorean triles. Prove that your formula always gives a Pythagorean trile. From the revious roblem, we see that given an odd number a, we can create a Pythagorean trile: a, 1 a 1), 1 ) a 1) + 1 = a, 1 a 1), 1 ) a + 1).
3 We can even do a bit better and actually enumerate these triles by writing the odd number a as a = j + 1, with j = 1,, 3,... Then, we have j + 1, 1 j + 1) 1), 1 ) j + 1) + 1) = j + 1, j } {{ } + j, j } {{ } + j + 1 } {{ } a b c We check that this always is a Pythagorean trile: a + b = j + 1) + j + j) = 4j + 4j j 4 + 8j 3 + 4j = 4j 4 + 8j 3 + 8j + 4j + 1 = j + j + 1) = c A natural question at this oint is: Have now found ALL the Pythagorean triles? Unfortunately or fortunately, deending on your oint of view), the following table gives another attern of Pythagorean triles. a b c Send some time determining how these triles are related. Can you determine the next three triles in this attern? Be sure to check that your triles are Pythagorean triles!) The next three triles are 8, 195, 197), 3, 55, 57), and 36, 33, 35). The first thing we notice is that the a values are running through the ositive multiles of 4. We also notice that c = b + and that b and c are, resectively, 1 more than and 1 less than a erfect square. What erfect square? Well, c = b + imlies that a = b + ) b = 4b + 4 = b = 1 4 a 1. Then, for k = 1,, 3,..., we have a = 4k and so b = 1 4 4k) 1 = 4k 1 = k) 1 and c = k)
4 Use the attern you found in the revious roblem to create a formula for these triles and will hence generate another infinite list of Pythagorean triles. Prove that your formula always gives a Pythagorean trile. As we saw in the revious roblem, these triles are given by We check: }{{} 4k, k) 1, k) + 1). } {{ } } {{ } a b c a + b = 4k) + k) 1) = 16k + 16k 4 8k + 1 = 16k 4 + 8k + 1 = k) + 1) = c A General Formula for Pythagorean Triles We will see how the following formula is derived in the Rational Points on the Unit Circle section, but for now we will work with this. q, q, + q ) is a Pythagorean trile whenever and q are ositive integers with q >. Furthermore, every Pythagorean trile is similar to a trile of this form. Use this formula to find a few Pythagorean triles that you have not yet seen in this investigation. Many ossible answers, such as Prove that, if, q and k are ositive integers with q >, then kq ), kq, k + q )) is a Pythagorean trile. kq )) + kq) = k q ) + 4k q = k q 4 q q ) = k q 4 + q + 4 ) = k q + ) = k + q )) 4
5 Rational Points on the Unit Circle In this section, we investigate rational oints on the unit circle and make the link to Pythagorean triles. A oint x, y), on the unit circle is rational if x and y are rational numbers satisfying x + y = 1. Similarly, x, y) is an integer oint on the unit circle if x and y are integers with x + y = 1. Find all integer oints on the unit circle. How do you know that these are all of them? The only integer oints on the unit circle are 1, 0), 0, 1), 1, 0) and 0, 1). These are the only integer oints there are only four integer oints within distance 1 of the origin. Find some rational oints on the unit circle. For instance, can you have x = 1/ for a rational oint on the unit circle? Exlain why or why not. If x = 1/, then y = ± 1 1/) = ± 3/ is not rational. Some rational oints include 3/5, 4/5), 5/13, 1/13), and 8/17, 15/17). Do you notice anything familiar about these oints? Now, consider the integer oint 1, 0) on the unit circle. Draw a line through 1, 0) and any other oint x, y) on the unit circle. Go to htt://faculty.ithaca.edu/dabrown/docs/ythagorean/rational/ and download the file shown. Doubleclick to decomress and then load the file in a browser window. Drag the sloe slider to change the sloe of the this line. If the sloe of this line is m, then write down the equation of this line. The line between 1, 0) and x, y), with sloe m has equation y = mx + 1). Show that if x, y) is a rational oint on the unit circle, the sloe of the line above must also be rational if x 1. The sloe, m, of this line is m = y 0 x 1) = y x + 1. So, as long as x 1, this is the quotient of two rational numbers with nonzero denominator) and hence is rational. Set the sloe equal to 0.5 and observe the x and y coordinates of the intersection oint. If you lug these coordinates into the equation of the unit circle and clear the denominators, what do you get? Click the the box show triangle for a clue. Try this with a few other oints. 5
6 When m =.5, the oint of intersection with the unit circle is 3/5, 4/5). Plugging into the equation of the circle and clearing the denominator yields ) which is a Pythagorean trile! ) 4 = 1 = = 5 5 Show that if a, b, c) is a Pythagorean trile, then the oint a/c, b/c) is a rational oint on the unit circle. a + b = c = a c + b a ) ) b c = 1 = + = 1. c c Since a, b, and c are integers, a/c and b/c are rational, and a/c, b/c) lies on the unit circle. Now, show that if x, y) is the oint of intersection of the above line with the unit circle and the sloe m is rational, then x and y are also rational. You can do this concretely by showing that x = 1 m 1 + m and y = m 1 + m. We intersect y = mx + 1) with x + y = 1 in the first quadrant. x + y = 1 = x + mx + 1)) = 1 = x + m x + m x + m 1 = 0 = 1 + m )x + m )x + m 1) = 0 = x = m ± 4m m )m 1) 1 + m ) = x = m ± 4m 4 4m 4 1) 1 + m ) = x = m ± m ) = x = m ± 1 + m ) = x = m m, 1 m 1 + m = x = 1, 1 m 1 + m 6
7 Working in the first quadrant, x = 1 m 1 + m and ) 1 m y = mx + 1) = m 1 + m + 1 = m 1 m 1 + m ) m = m 1 + m 1 + m. Recall that a rational number m can be exressed as /q for integers and q 0. Use this fact and the revious roblem to exress rational oints x, y) on the unit circle in terms of and q. Reduce the fractions as much as ossible and recover the Pythagorean Triles Formula from earlier. Setting m = q, x = 1 m 1 + m = ) q q ) = q + q and Thus, ) y = m 1 + m = q 1 + q ) = q + q. ) q q + ) = 1 = q ) + q) = + q ). + q + q So, q, q, + q ) gives the general formula for a Pythagorean trile. And, the geometric construction shows that there is a onetoone corresondence between the Pythagorean triles and the rational oints on the unit circle in the first quadrant). 7
PYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More information1 Gambler s Ruin Problem
Coyright c 2009 by Karl Sigman 1 Gambler s Ruin Problem Let N 2 be an integer and let 1 i N 1. Consider a gambler who starts with an initial fortune of $i and then on each successive gamble either wins
More informationPythagorean Triples. Chapter 2. a 2 + b 2 = c 2
Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the
More informationWe now explore a third method of proof: proof by contradiction.
CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement
More informationPythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers
Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers Amnon Yekutieli Department of Mathematics Ben Gurion University email: amyekut@math.bgu.ac.il Notes available at http://www.math.bgu.ac.il/~amyekut/lectures
More informationSample Problems. 2. Rationalize the denominator in each of the following expressions Simplify each of the following expressions.
Lecture Notes Radical Exressions age Samle Problems. Simlify each of the following exressions. Assume that a reresents a ositive number. a) b) g) 7 + 7 c) h) 7 x y d) 0 + i) j) x x + e) k) ( x) ( + x)
More informationNumber Theory Homework.
Number Theory Homework. 1. Pythagorean triples and rational points on quadratics and cubics. 1.1. Pythagorean triples. Recall the Pythagorean theorem which is that in a right triangle with legs of length
More information8.7 Mathematical Induction
8.7. MATHEMATICAL INDUCTION 8135 8.7 Mathematical Induction Objective Prove a statement by mathematical induction Many mathematical facts are established by first observing a pattern, then making a conjecture
More informationThis is Radical Expressions and Equations, chapter 8 from the book Beginning Algebra (index.html) (v. 1.0).
This is Radical Expressions and Equations, chapter 8 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More informationCoordinate Transformation
Coordinate Transformation Coordinate Transformations In this chater, we exlore maings where a maing is a function that "mas" one set to another, usually in a way that reserves at least some of the underlyign
More informationPartial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:
Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than
More information1 Review of complex numbers
1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely
More informationContinued fractions and good approximations.
Continued fractions and good approximations We will study how to find good approximations for important real life constants A good approximation must be both accurate and easy to use For instance, our
More informationFactoring Patterns in the Gaussian Plane
Factoring Patterns in the Gaussian Plane Steve Phelps Introduction This paper describes discoveries made at the Park City Mathematics Institute, 00, as well as some proofs. Before the summer I understood
More informationMath 115 Spring 2011 Written Homework 5 Solutions
. Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence
More informationApplications of Fermat s Little Theorem and Congruences
Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More informationSample Problems. 6. The hypotenuse of a right triangle is 68 cm. The di erence between the other two sides is 28 cm. Find the sides of the triangle.
Lecture Notes The Pythagorean Theorem age 1 Samle Problems 1. Could the three line segments given below be the three sides of a right triangle? Exlain your answer. a) 6 cm; 10 cm; and 8 cm b) 7 ft, 15
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationLogic, Sets, and Proofs
Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Statements. A logical statement is a mathematical statement that is either true or false. Here we denote logical
More informationComplex Numbers and the Complex Exponential
Complex Numbers and the Complex Exponential Frank R. Kschischang The Edward S. Rogers Sr. Department of Electrical and Computer Engineering University of Toronto September 5, 2005 Numbers and Equations
More information5 =5. Since 5 > 0 Since 4 7 < 0 Since 0 0
a p p e n d i x e ABSOLUTE VALUE ABSOLUTE VALUE E.1 definition. The absolute value or magnitude of a real number a is denoted by a and is defined by { a if a 0 a = a if a
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationPRINCIPLES OF PROBLEM SOLVING
PRINCIPLES OF PROBLEM SOLVING There are no hard and fast rules that will ensure success in solving problems. However, it is possible to outline some general steps in the problemsolving process and to
More informationFourier Synthesis Tutorial v.2.1
Fourier Synthesis Tutorial v.2.1 This tutorial is designed for students from many different academic levels and backgrounds. It occasionally uses terms from disciplinespecific fields that you may not
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationSOME PROPERTIES OF EXTENSIONS OF SMALL DEGREE OVER Q. 1. Quadratic Extensions
SOME PROPERTIES OF EXTENSIONS OF SMALL DEGREE OVER Q TREVOR ARNOLD Abstract This aer demonstrates a few characteristics of finite extensions of small degree over the rational numbers Q It comrises attemts
More informationIntroduction Proof by unique factorization in Z Proof with Gaussian integers Proof by geometry Applications. Pythagorean Triples
Pythagorean Triples Keith Conrad University of Connecticut August 4, 008 Introduction We seek positive integers a, b, and c such that a + b = c. Plimpton 3 Babylonian table of Pythagorean triples (1800
More informationAlgebra Practice Problems for Precalculus and Calculus
Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials
More informationCHAPTER 3 Numbers and Numeral Systems
CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,
More informationSolving Systems of Two Equations Algebraically
8 MODULE 3. EQUATIONS 3b Solving Systems of Two Equations Algebraically Solving Systems by Substitution In this section we introduce an algebraic technique for solving systems of two equations in two unknowns
More informationMAT2400 Analysis I. A brief introduction to proofs, sets, and functions
MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take
More informationUnited Arab Emirates University College of Sciences Department of Mathematical Sciences HOMEWORK 1 SOLUTION. Section 10.1 Vectors in the Plane
United Arab Emirates University College of Sciences Deartment of Mathematical Sciences HOMEWORK 1 SOLUTION Section 10.1 Vectors in the Plane Calculus II for Engineering MATH 110 SECTION 0 CRN 510 :00 :00
More informationIntroduction to Diophantine Equations
Introduction to Diophantine Equations Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles September, 2006 Abstract In this article we will only touch on a few tiny parts of the field
More informationStanford Math Circle: Sunday, May 9, 2010 SquareTriangular Numbers, Pell s Equation, and Continued Fractions
Stanford Math Circle: Sunday, May 9, 00 SquareTriangular Numbers, Pell s Equation, and Continued Fractions Recall that triangular numbers are numbers of the form T m = numbers that can be arranged in
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More informationAlgebra I Notes Review Real Numbers and Closure Unit 00a
Big Idea(s): Operations on sets of numbers are performed according to properties or rules. An operation works to change numbers. There are six operations in arithmetic that "work on" numbers: addition,
More informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
More informationMath Review Large Print (18 point) Edition Chapter 2: Algebra
GRADUATE RECORD EXAMINATIONS Math Review Large Print (18 point) Edition Chapter : Algebra Copyright 010 by Educational Testing Service. All rights reserved. ETS, the ETS logo, GRADUATE RECORD EXAMINATIONS,
More informationCOWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level 2
COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level This study guide is for students trying to test into College Algebra. There are three levels of math study guides. 1. If x and y 1, what
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More informationMath 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) =
Math 50 Eam # Practice Problem Solutions. Find the vertical asymptotes (if any) of the functions g() = + 4, h() = 4. Answer: The only number not in the domain of g is = 0, so the only place where g could
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION So far, we have been concerned with some particular aspects of curve sketching: Domain, range, and symmetry (Chapter 1) Limits, continuity,
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent
More informationMATH Fundamental Mathematics II.
MATH 10032 Fundamental Mathematics II http://www.math.kent.edu/ebooks/10032/funmath2.pdf Department of Mathematical Sciences Kent State University December 29, 2008 2 Contents 1 Fundamental Mathematics
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationThe Product Property of Square Roots states: For any real numbers a and b, where a 0 and b 0, ab = a b.
Chapter 9. Simplify Radical Expressions Any term under a radical sign is called a radical or a square root expression. The number or expression under the the radical sign is called the radicand. The radicand
More informationThe Deadly Sins of Algebra
The Deadly Sins of Algebra There are some algebraic misconceptions that are so damaging to your quantitative and formal reasoning ability, you might as well be said not to have any such reasoning ability.
More informationMATH 65 NOTEBOOK CERTIFICATIONS
MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1
More informationTRIANGLES ON THE LATTICE OF INTEGERS. Department of Mathematics Rowan University Glassboro, NJ Andrew Roibal and Abdulkadir Hassen
TRIANGLES ON THE LATTICE OF INTEGERS Andrew Roibal and Abdulkadir Hassen Department of Mathematics Rowan University Glassboro, NJ 08028 I. Introduction In this article we will be studying triangles whose
More informationRoots of Real Numbers
Roots of Real Numbers Math 97 Supplement LEARNING OBJECTIVES. Calculate the exact and approximate value of the square root of a real number.. Calculate the exact and approximate value of the cube root
More informationMATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.
MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P
More informationMORE TRIGONOMETRIC FUNCTIONS
CHAPTER MORE TRIGONOMETRIC FUNCTIONS The relationshis among the lengths of the sides of an isosceles right triangle or of the right triangles formed by the altitude to a side of an equilateral triangle
More informationM3 PRECALCULUS PACKET 1 FOR UNIT 5 SECTIONS 5.1 TO = to see another form of this identity.
M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO 5.3 5. USING FUNDAMENTAL IDENTITIES 5. Part : Pythagorean Identities. Recall the Pythagorean Identity sin θ cos θ + =. a. Subtract cos θ from both sides
More information1. Introduction identity algbriac factoring identities
1. Introduction An identity is an equality relationship between two mathematical expressions. For example, in basic algebra students are expected to master various algbriac factoring identities such as
More information7.7 Solving Rational Equations
Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate
More informationSolution to Exercise 2.2. Both m and n are divisible by d, som = dk and n = dk. Thus m ± n = dk ± dk = d(k ± k ),som + n and m n are divisible by d.
[Chap. ] Pythagorean Triples 6 (b) The table suggests that in every primitive Pythagorean triple, exactly one of a, b,orc is a multiple of 5. To verify this, we use the Pythagorean Triples Theorem to write
More information(This result should be familiar, since if the probability to remain in a state is 1 p, then the average number of steps to leave the state is
How many coin flis on average does it take to get n consecutive heads? 1 The rocess of fliing n consecutive heads can be described by a Markov chain in which the states corresond to the number of consecutive
More informationDiscrete Mathematics: Homework 7 solution. Due: 2011.6.03
EE 2060 Discrete Mathematics spring 2011 Discrete Mathematics: Homework 7 solution Due: 2011.6.03 1. Let a n = 2 n + 5 3 n for n = 0, 1, 2,... (a) (2%) Find a 0, a 1, a 2, a 3 and a 4. (b) (2%) Show that
More informationEquations and Inequalities
Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.
More informationWentzville School District Curriculum Development Template Stage 1 Desired Results
Wentzville School District Curriculum Development Template Stage 1 Desired Results Unit Title: Radicals and Radical Expressions Course: Middle School Algebra 1 Unit 10 Radicals and Radical Expressions
More informationMath 319 Problem Set #3 Solution 21 February 2002
Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod
More informationThis is Solving Quadratic Equations and Graphing Parabolas, chapter 9 from the book Beginning Algebra (index.html) (v. 1.0).
This is Solving Quadratic Equations and Graphing Parabolas, chapter 9 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More informationTrigonometric Identities and Conditional Equations C
Trigonometric Identities and Conditional Equations C TRIGONOMETRIC functions are widely used in solving realworld problems and in the development of mathematics. Whatever their use, it is often of value
More informationTHE PROOF IS IN THE PICTURE
THE PROOF IS IN THE PICTURE (an introduction to proofs without words) LAMC INTERMEDIATE GROUP  11/24/13 WARM UP (PAPER FOLDING) Theorem: The square root of 2 is irrational. We prove this via the mathematical
More informationEXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS
To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires
More informationMath 018 Review Sheet v.3
Math 018 Review Sheet v.3 Tyrone Crisp Spring 007 1.1  Slopes and Equations of Lines Slopes: Find slopes of lines using the slope formula m y y 1 x x 1. Positive slope the line slopes up to the right.
More informationMore Properties of Limits: Order of Operations
math 30 day 5: calculating its 6 More Proerties of Limits: Order of Oerations THEOREM 45 (Order of Oerations, Continued) Assume that!a f () L and that m and n are ositive integers Then 5 (Power)!a [ f
More informationAnalytic Geometry Section 26: Circles
Analytic Geometry Section 26: Circles Objective: To find equations of circles and to find the coordinates of any points where circles and lines meet. Page 81 Definition of a Circle A circle is the set
More informationSEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION
CHAPTER 5 SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION Alessandro Artale UniBZ  http://www.inf.unibz.it/ artale/ SECTION 5.2 Mathematical Induction I Copyright Cengage Learning. All rights reserved.
More informationObjectives. By the time the student is finished with this section of the workbook, he/she should be able
QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a
More informationFractions and Decimals
Fractions and Decimals Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles December 1, 2005 1 Introduction If you divide 1 by 81, you will find that 1/81 =.012345679012345679... The first
More informationTrigonometry. Week 1 Right Triangle Trigonometry
Trigonometry Introduction Trigonometry is the study of triangle measurement, but it has expanded far beyond that. It is not an independent subject of mathematics. In fact, it depends on your knowledge
More informationVariations on the Gambler s Ruin Problem
Variations on the Gambler s Ruin Problem Mat Willmott December 6, 2002 Abstract. This aer covers the history and solution to the Gambler s Ruin Problem, and then exlores the odds for each layer to win
More informationPYTHAGOREAN TRIPLES PETE L. CLARK
PYTHAGOREAN TRIPLES PETE L. CLARK 1. Parameterization of Pythagorean Triples 1.1. Introduction to Pythagorean triples. By a Pythagorean triple we mean an ordered triple (x, y, z) Z 3 such that x + y =
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationSection 4.2: The Division Algorithm and Greatest Common Divisors
Section 4.2: The Division Algorithm and Greatest Common Divisors The Division Algorithm The Division Algorithm is merely long division restated as an equation. For example, the division 29 r. 20 32 948
More informationBasic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.
Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required
More informationQUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE
MODULE  1 Quadratic Equations 6 QUADRATIC EQUATIONS In this lesson, you will study aout quadratic equations. You will learn to identify quadratic equations from a collection of given equations and write
More informationRECURSIVE ENUMERATION OF PYTHAGOREAN TRIPLES
RECURSIVE ENUMERATION OF PYTHAGOREAN TRIPLES DARRYL MCCULLOUGH AND ELIZABETH WADE In [9], P. W. Wade and W. R. Wade (no relation to the second author gave a recursion formula that produces Pythagorean
More informationIt is time to prove some theorems. There are various strategies for doing
CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, it
More informationBasic numerical skills: EQUATIONS AND HOW TO SOLVE THEM. x + 5 = 7 2 + 52 = 72 5 + (22) = 72 5 = 5. x + 55 = 75. x + 0 = 20.
Basic numerical skills: EQUATIONS AND HOW TO SOLVE THEM 1. Introduction (really easy) An equation represents the equivalence between two quantities. The two sides of the equation are in balance, and solving
More informationInfinite Algebra 1 supports the teaching of the Common Core State Standards listed below.
Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School
More informationHomework from Section Find two positive numbers whose product is 100 and whose sum is a minimum.
Homework from Section 4.5 4.5.3. Find two positive numbers whose product is 100 and whose sum is a minimum. We want x and y so that xy = 100 and S = x + y is minimized. Since xy = 100, x = 0. Thus we have
More informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More informationPrice Elasticity of Demand MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W
Price Elasticity of Demand MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W The rice elasticity of demand (which is often shortened to demand elasticity) is defined to be the
More informationCS 173: Discrete Structures, Fall 2010 Homework 6 Solutions
CS 173: Discrete Structures, Fall 010 Homework 6 Solutions This homework was worth a total of 5 points. 1. Recursive definition [13 points] Give a simple closedform definition for each of the following
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More information2 When is a 2Digit Number the Sum of the Squares of its Digits?
When Does a Number Equal the Sum of the Squares or Cubes of its Digits? An Exposition and a Call for a More elegant Proof 1 Introduction We will look at theorems of the following form: by William Gasarch
More informationPythagorean Triples and Fermat s Last Theorem
Pythagorean Triples and Fermat s Last Theorem A. DeCelles Notes for talk at the Science Speakers Series at Goshen College Document created: 05/10/011 Last updated: 06/05/011 (graphics added) It is an amazing
More informationConfidence Intervals for CaptureRecapture Data With Matching
Confidence Intervals for CatureRecature Data With Matching Executive summary Caturerecature data is often used to estimate oulations The classical alication for animal oulations is to take two samles
More informationLINEAR EQUATIONS 7YEARS. A guide for teachers  Years 7 8 June The Improving Mathematics Education in Schools (TIMES) Project
LINEAR EQUATIONS NUMBER AND ALGEBRA Module 26 A guide for teachers  Years 7 8 June 2011 7YEARS 8 Linear Equations (Number and Algebra : Module 26) For teachers of Primary and Secondary Mathematics 510
More information6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks
6.042/8.062J Mathematics for Comuter Science December 2, 2006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Random Walks Gambler s Ruin Today we re going to talk about onedimensional random walks. In
More informationCongruent Numbers, the Rank of Elliptic Curves and the Birch and SwinnertonDyer Conjecture. Brad Groff
Congruent Numbers, the Rank of Elliptic Curves and the Birch and SwinnertonDyer Conjecture Brad Groff Contents 1 Congruent Numbers... 1.1 Basic Facts............................... and Elliptic Curves.1
More informationElementary Number Theory We begin with a bit of elementary number theory, which is concerned
CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,
More informationAlgebra 1 Chapter 3 Vocabulary. equivalent  Equations with the same solutions as the original equation are called.
Chapter 3 Vocabulary equivalent  Equations with the same solutions as the original equation are called. formula  An algebraic equation that relates two or more reallife quantities. unit rate  A rate
More information4.2 Euclid s Classification of Pythagorean Triples
178 4. Number Theory: Fermat s Last Theorem Exercise 4.7: A primitive Pythagorean triple is one in which any two of the three numbers are relatively prime. Show that every multiple of a Pythagorean triple
More informationx if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the
More information