Pythagorean Triples and Rational Points on the Unit Circle
|
|
- Jasmine Mitchell
- 7 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 left-hand side is two more than a multile of 4 while the right-hand 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. Double-click 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 non-zero 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 one-to-one corresondence between the Pythagorean triles and the rational oints on the unit circle in the first quadrant). 7
1 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationStanford Math Circle: Sunday, May 9, 2010 Square-Triangular Numbers, Pell s Equation, and Continued Fractions
Stanford Math Circle: Sunday, May 9, 00 Square-Triangular 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 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 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 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 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 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 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 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 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 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 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 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 informationLesson Plan. N.RN.3: Use properties of rational and irrational numbers.
N.RN.3: Use properties of rational irrational numbers. N.RN.3: Use Properties of Rational Irrational Numbers Use properties of rational irrational numbers. 3. Explain why the sum or product of two rational
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
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 informationBasic numerical skills: EQUATIONS AND HOW TO SOLVE THEM. x + 5 = 7 2 + 5-2 = 7-2 5 + (2-2) = 7-2 5 = 5. x + 5-5 = 7-5. 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 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 informationThe Cubic Formula. The quadratic formula tells us the roots of a quadratic polynomial, a polynomial of the form ax 2 + bx + c. The roots (if b 2 b+
The Cubic Formula The quadratic formula tells us the roots of a quadratic olynomial, a olynomial of the form ax + bx + c. The roots (if b b+ 4ac 0) are b 4ac a and b b 4ac a. The cubic formula tells us
More informationUnit 6 Trigonometric Identities, Equations, and Applications
Accelerated Mathematics III Frameworks Student Edition Unit 6 Trigonometric Identities, Equations, and Applications nd Edition Unit 6: Page of 3 Table of Contents Introduction:... 3 Discovering the Pythagorean
More informationMATH10040 Chapter 2: Prime and relatively prime numbers
MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
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 informationDigitalCommons@University of Nebraska - Lincoln
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-1-007 Pythagorean Triples Diane Swartzlander University
More informationLINEAR INEQUALITIES. Mathematics is the art of saying many things in many different ways. MAXWELL
Chapter 6 LINEAR INEQUALITIES 6.1 Introduction Mathematics is the art of saying many things in many different ways. MAXWELL In earlier classes, we have studied equations in one variable and two variables
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 information1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.
More information3.3 Real Zeros of Polynomials
3.3 Real Zeros of Polynomials 69 3.3 Real Zeros of Polynomials In Section 3., we found that we can use synthetic division to determine if a given real number is a zero of a polynomial function. This section
More informationUnit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.
Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material
More informationSystems of Equations Involving Circles and Lines
Name: Systems of Equations Involving Circles and Lines Date: In this lesson, we will be solving two new types of Systems of Equations. Systems of Equations Involving a Circle and a Line Solving a system
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 informationAs we have seen, there is a close connection between Legendre symbols of the form
Gauss Sums As we have seen, there is a close connection between Legendre symbols of the form 3 and cube roots of unity. Secifically, if is a rimitive cube root of unity, then 2 ± i 3 and hence 2 2 3 In
More informationChapter 9. Systems of Linear Equations
Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables
More informationChapter 11 Number Theory
Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications
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 information1.6 The Order of Operations
1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative
More information4. How many integers between 2004 and 4002 are perfect squares?
5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started
More informationIntegrals of Rational Functions
Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t
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 informationDomain of a Composition
Domain of a Composition Definition Given the function f and g, the composition of f with g is a function defined as (f g)() f(g()). The domain of f g is the set of all real numbers in the domain of g such
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 informationPigeonhole Principle Solutions
Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such
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 one-dimensional random walks. In
More information2 When is a 2-Digit 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 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 informationMath 4310 Handout - Quotient Vector Spaces
Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More informationCongruent Number Problem
University of Waterloo October 28th, 2015 Number Theory Number theory, can be described as the mathematics of discovering and explaining patterns in numbers. There is nothing in the world which pleases
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 informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column
More informationVieta s Formulas and the Identity Theorem
Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion
More informationSAT Math Hard Practice Quiz. 5. How many integers between 10 and 500 begin and end in 3?
SAT Math Hard Practice Quiz Numbers and Operations 5. How many integers between 10 and 500 begin and end in 3? 1. A bag contains tomatoes that are either green or red. The ratio of green tomatoes to red
More informationPolynomial and Rational Functions
Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving
More informationAlex, I will take congruent numbers for one million dollars please
Alex, I will take congruent numbers for one million dollars please Jim L. Brown The Ohio State University Columbus, OH 4310 jimlb@math.ohio-state.edu One of the most alluring aspectives of number theory
More informationContinued Fractions. Darren C. Collins
Continued Fractions Darren C Collins Abstract In this paper, we discuss continued fractions First, we discuss the definition and notation Second, we discuss the development of the subject throughout history
More informationis identically equal to x 2 +3x +2
Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3
More informationSAT Math Facts & Formulas Review Quiz
Test your knowledge of SAT math facts, formulas, and vocabulary with the following quiz. Some questions are more challenging, just like a few of the questions that you ll encounter on the SAT; these questions
More informationThis is a square root. The number under the radical is 9. (An asterisk * means multiply.)
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
More informationSouth Carolina College- and Career-Ready (SCCCR) Pre-Calculus
South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationMathematics 31 Pre-calculus and Limits
Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More information0.8 Rational Expressions and Equations
96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to
More informationBasic Proof Techniques
Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More informationSample Problems. 10. 1 2 cos 2 x = tan2 x 1. 11. tan 2 = csc 2 tan 2 1. 12. sec x + tan x = cos x 13. 14. sin 4 x cos 4 x = 1 2 cos 2 x
Lecture Notes Trigonometric Identities page Sample Problems Prove each of the following identities.. tan x x + sec x 2. tan x + tan x x 3. x x 3 x 4. 5. + + + x 6. 2 sec + x 2 tan x csc x tan x + cot x
More informationPlaying with Numbers
PLAYING WITH NUMBERS 249 Playing with Numbers CHAPTER 16 16.1 Introduction You have studied various types of numbers such as natural numbers, whole numbers, integers and rational numbers. You have also
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More information1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes
Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.
More informationLimits. Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as:
Limits Limits: Graphical Solutions Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as: The limits are defined as the value that the function approaches as it goes
More informationChapter 31 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M.
31 Geometric Series Motivation (I hope) Geometric series are a basic artifact of algebra that everyone should know. 1 I am teaching them here because they come up remarkably often with Markov chains. The
More informationTHE COMPLEX EXPONENTIAL FUNCTION
Math 307 THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following definition e iθ = cos θ + i sin θ. (1) This formula
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationCOMPLEX NUMBERS. a bi c di a c b d i. a bi c di a c b d i For instance, 1 i 4 7i 1 4 1 7 i 5 6i
COMPLEX NUMBERS _4+i _-i FIGURE Complex numbers as points in the Arg plane i _i +i -i A complex number can be represented by an expression of the form a bi, where a b are real numbers i is a symbol with
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More informationSection 1.1 Real Numbers
. Natural numbers (N):. Integer numbers (Z): Section. Real Numbers Types of Real Numbers,, 3, 4,,... 0, ±, ±, ±3, ±4, ±,... REMARK: Any natural number is an integer number, but not any integer number is
More informationPowerScore Test Preparation (800) 545-1750
Question 1 Test 1, Second QR Section (version 2) Two triangles QA: x QB: y Geometry: Triangles Answer: Quantity A is greater 1. The astute student might recognize the 0:60:90 and 45:45:90 triangle right
More information6.3 Conditional Probability and Independence
222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted
More informationThe last three chapters introduced three major proof techniques: direct,
CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements
More informationPythagorean Triples. becomes
Solution Commentary: Solution of Main Problems: Pythagorean Triples 1. If m = (n + 1), n = [m -1]/ and n+1 = [m +1]/. Then, by substitution, the equation n + (n + 1) = (n+1) m + 1 becomes + m =. Now, because
More information3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.
More informationStudent Outcomes. Lesson Notes. Classwork. Discussion (10 minutes)
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 Student Outcomes Students know the definition of a number raised to a negative exponent. Students simplify and write equivalent expressions that contain
More informationQuadratics - Build Quadratics From Roots
9.5 Quadratics - Build Quadratics From Roots Objective: Find a quadratic equation that has given roots using reverse factoring and reverse completing the square. Up to this point we have found the solutions
More information1.7 Graphs of Functions
64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most
More informationGeometry: Classifying, Identifying, and Constructing Triangles
Geometry: Classifying, Identifying, and Constructing Triangles Lesson Objectives Teacher's Notes Lesson Notes 1) Identify acute, right, and obtuse triangles. 2) Identify scalene, isosceles, equilateral
More information