Item 8. Constructing the Square Area of Two Proving No Irrationals. 6 Total Pages
|
|
- Shona Carter
- 5 years ago
- Views:
Transcription
1 Item 8 Constructing the Square Area of Two Proving No Irrationals 6 Total Pages 1
2 2
3 We want to start with Pi. How Geometry Proves No Irrations They call Pi the ratio of the circumference of a circle to its diameter, even though they have no ratio. Someone had proved that there could not be one. Is that really true? Our claim is that the correct Pi has a ratio 201/64, as a decimal Geometry gives a way to prove that this is true. We are going to use one as our diameter, so that one times Pi becomes our circumference length, so that the number for Pi and our circumference length are one and the same. If our diameter is one, our radius is 1/2, that can construct a hexagon within our circumference putting 6 points on our circumference giving us six equal lengths. This would mean that the number for Pi that is the same as our circumference length would have to be dividable by 1, 2, 3, 4, 5, and 6 and it is. This is true for , yet impossible for the present day calculation for Pi. You can t even list it, yet alone divide it into 6 equal divisions. You can also use two as our diameter length so that our circumference length is here again our radius of one can construct the six lengths of our straight line hexagon, where again, we put 6 points on our circumference line, so that is also dividable by 1,2 3, 4, 5, and 6, and it is. Thanks to geometry we have a correct ratio for Pi. Applying Quantum Math First we apply quantum math of reducing our ratio to a single integer by addition to show how it is related to our geometry so that 201/64 = 3/10 = 3/1 = 3 our hexagon length. Then, for our decimal part = 21 = 3, again our hexagon, thanks to quantum math. If we use our listing of countable numbers starting with one as our diameter lengths, our ever increasing multiplication by Pi where we apply quantum math of reducing to a single integer by addition we will get this infinite repeating set 3, 6, 9, 3, 6, 9... infinite. 3
4 It appears that 3, 6, and 9 represent infinity. If you look at item 15 we show our three columns of integers for our twin primes, the only 3 integers is our central column between our twin primes is 3, 6, and 9, that total 18. For our first twin prime we also had only 3 integers 2, 5, and 8 that totals 15, column one. Then for our second prime column, column three, our three integers were 1, 4, and 7, that totaled 12. These 3 columns account for all of our single quantum integers one through nine, that total 45 our 34 curved lines plus our 11 time lines of our Poincare one geometry. When we form our quantum fraction from our dual number 45 we have 4/5 the positive parts of 31/32 that reduces to 4/5 from our probability model (A) M-Theory that produced our probability models (B), (C), (D), (E), and (F). thanks to the relativity of numbers. Later we will show how our 3 infinite sets of integers 2, 5, and 8, then 3, 6, and 9 plus 1, 4, and 7 proves that our Poincare Geometry are true. In these 3 sets our primes total 17, the 17 curved lines for each half of our Poincare one geometry. Then our single time integers 1, 2, 4, 5, 7, and 8 total 27 that represents the lowest state for bosons that produced the mass structure of our universe, protons, and neutrons. Whole Number Squares with Square Roots While we can form whole number and fraction squares that have square roots, we want to just stay with our whole number applications, as it was here that false applications began. By using our countable numbers we can establish what squares we can establish that have whole number square roots. We can do so by squaring our whole numbers so that one squared equals one that has a square root of one. Next we square two so that two squared equals 4, so that the square root of 4 = 2. We can square 3 so that 3 squared = 9 as the square root of 9 is 3. Then we have 4 squared = 16 with its square root of 4. For 5 we have 5 squared equals 25 with its square root of 5. 4
5 Then, we have 6 squared and so on. We can then make a list of squares that have square roots that can exist 1, 2, 4, 9, 16, 25, 36, 49, 64, 81, and so on where all of the dual numbers have application of our two geometries and our probability models. We will show, there exist no other squares between these squares that have a square root. What happened with mathematics when they thought they proved that the square root of two was irrational. They began to say that the square root of three was irrational, also the square root of 5, 6, 7, and so on, were irrational when the truth of this matter was that these squares did not exist. Their lived a man who once said every house is constructed by someone, however, he who constructed all things is God. How Geometry Proves No Irrational Numbers Look at our first page, where we show how to construct the square area of two, by two square root triangles of one half base times height. For illustration two we show our square area of two within our two by two square. Here it is easy to see that the square area of two is exactly ½ of our two by two square, as the parts that lie outside of our area of two square can represent our vacuum space (equal representation) of our Poincare one geometry another part of P = NP. If you cut out our square area of two and separate our two square root triangle by cutting along our base line we have two separated square root triangles. Now, cut our height line of each that will give 4 total parts where you can form two squares of area one, with two sets of square roots where the square root of one is one. Next, we want to show importance of listing the squares that exist. You can look up the golden ratio that they feel is associated with the Fibonacci Sequence. They start with a line segment and label its two end points (A) and (C). then, they added a point (B) between (A) and (C). as a result they claimed a ratio equation of AB to BC. 5
6 I think that they forgot that they constructed (A) to (C) by the motion of forming that line of (A) to (C). we can also use motion to form point (B) by going backwards from (C) then forward again to our end point (C). What does our motion (no dought related to time) produce, one line length from (A) to (B) then 3 lengths of lines from (B) to (C) showing that they can t have the ratio that they claim. Next, they claim that they have an exact value for their claim of the golden ratio. They start with one (our additional one of Fibonacci Sequence) so that we have one plus the square root of five over two. Here is the important part. There exist no square of area five, deleting the square root of five. The only thing that we have left is our additional one from our Fibonacci Sequence that when divided by two goes on forever without ever reaching one, as we have ½, ¼, 1/8, 1/16, and so on so that the golden ratio does not exist, fictitious. Thanks, Richard Eicholtz 6
A Brief Proof of the Riemann Hypothesis, Giving Infinite Results. Item Total Pages
A Brief Proof of the Riemann Hypothesis, Giving Infinite Results Item 42 We will also show a positive proof of Fermat s Last Theorem, also related to the construction of the universe. 9 Total Pages 1 A
More informationItem 6. Pi and the Universe. Gives Applications to the Geometry of our Universe. 5 Total Pages
Item 6 Pi and the Universe Gives Applications to the Geometry of our Universe 5 Total Pages 1 Pi and the Universe How Geometry Proves the Correct Ratio for Pi 201/64, as a decimal 3.140625 For our geometry
More informationItem 37. The End of Time for All Governments and Rulers. 18 Total Pages
Item 37 The End of Time for All Governments and Rulers 18 Total Pages With new information for the construction of the universe, we have learned that there are two possible outcomes for the destruction
More informationAddition to Daniel s Prophecy. Item Total Pages. Showing Two Possible Outcomes
Addition to Daniel s Prophecy Item 30 6 Total Pages Showing Two Possible Outcomes 1 Addition to Daniel s Prophecy Our First Possible Outcome In the history of the scriptures, their was two past times,
More informationQuestion 1: Is zero a rational number? Can you write it in the form p, where p and q are integers and q 0?
Class IX - NCERT Maths Exercise (.) Question : Is zero a rational number? Can you write it in the form p, where p and q are integers and q 0? q Solution : Consider the definition of a rational number.
More informationClass IX Chapter 1 Number Sustems Maths
Class IX Chapter 1 Number Sustems Maths Exercise 1.1 Question Is zero a rational number? Can you write it in the form 0? and q, where p and q are integers Yes. Zero is a rational number as it can be represented
More informationStepping stones for Number systems. 1) Concept of a number line : Marking using sticks on the floor. (1 stick length = 1 unit)
Quality for Equality Stepping stones for Number systems 1) Concept of a number line : Marking using sticks on the floor. (1 stick length = 1 unit) 2) Counting numbers: 1,2,3,... Natural numbers Represent
More informationThe GED math test gives you a page of math formulas that
Math Smart 643 The GED Math Formulas The GED math test gives you a page of math formulas that you can use on the test, but just seeing the formulas doesn t do you any good. The important thing is understanding
More informationWhat is proof? Lesson 1
What is proof? Lesson The topic for this Math Explorer Club is mathematical proof. In this post we will go over what was covered in the first session. The word proof is a normal English word that you might
More information8th Grade. The Number System and Mathematical Operations Part 2.
1 8th Grade The Number System and Mathematical Operations Part 2 2015 11 20 www.njctl.org 2 Table of Contents Squares of Numbers Greater than 20 Simplifying Perfect Square Radical Expressions Approximating
More informationGrade 7/8 Math Circles November 21/22/23, The Scale of Numbers
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Introduction Grade 7/8 Math Circles November 21/22/23, 2017 The Scale of Numbers Centre for Education in Mathematics and Computing Last week we quickly
More informationClass IX Chapter 5 Introduction to Euclid's Geometry Maths
Class IX Chapter 5 Introduction to Euclid's Geometry Maths Exercise 5.1 Question 1: Which of the following statements are true and which are false? Give reasons for your answers. (i) Only one line can
More informationChapter 3: Graphs and Equations CHAPTER 3: GRAPHS AND EQUATIONS. Date: Lesson: Learning Log Title:
Chapter 3: Graphs and Equations CHAPTER 3: GRAPHS AND EQUATIONS Date: Lesson: Learning Log Title: Date: Lesson: Learning Log Title: Chapter 3: Graphs and Equations Date: Lesson: Learning Log Title: Notes:
More informationApril 28, 2017 Geometry 11.1 Circumference and Arc Length
11.1 Warmup April 28, 2017 Geometry 11.1 Circumference and Arc Length 1 Geometry 11.1 Circumference and Arc Length mbhaub@mpsaz.org 11.1 Essential Question How can you find the length of a circular arc?
More information8th Grade The Number System and Mathematical Operations Part
Slide 1 / 157 Slide 2 / 157 8th Grade The Number System and Mathematical Operations Part 2 2015-11-20 www.njctl.org Slide 3 / 157 Table of Contents Squares of Numbers Greater than 20 Simplifying Perfect
More information8th Grade The Number System and Mathematical Operations Part
Slide 1 / 157 Slide 2 / 157 8th Grade The Number System and Mathematical Operations Part 2 2015-11-20 www.njctl.org Slide 3 / 157 Table of Contents Squares of Numbers Greater than 20 Simplifying Perfect
More informationSolving and Graphing Inequalities
Solving and Graphing Inequalities Graphing Simple Inequalities: x > 3 When finding the solution for an equation we get one answer for x. (There is only one number that satisfies the equation.) For 3x 5
More informationMATH 521, WEEK 2: Rational and Real Numbers, Ordered Sets, Countable Sets
MATH 521, WEEK 2: Rational and Real Numbers, Ordered Sets, Countable Sets 1 Rational and Real Numbers Recall that a number is rational if it can be written in the form a/b where a, b Z and b 0, and a number
More informationA π day celebration! Everyone s favorite geometric constant!
A π day celebration! Everyone s favorite geometric constant! Math Circle March 10, 2019 The circumference of a circle is another word for its perimeter. A circle s circumference is proportional to its
More informationCMA Geometry Unit 1 Introduction Week 2 Notes
CMA Geometry Unit 1 Introduction Week 2 Notes Assignment: 9. Defined Terms: Definitions betweenness of points collinear points coplanar points space bisector of a segment length of a segment line segment
More informationLong division for integers
Feasting on Leftovers January 2011 Summary notes on decimal representation of rational numbers Long division for integers Contents 1. Terminology 2. Description of division algorithm for integers (optional
More information8th Grade. Slide 1 / 157. Slide 2 / 157. Slide 3 / 157. The Number System and Mathematical Operations Part 2. Table of Contents
Slide 1 / 157 Slide 2 / 157 8th Grade The Number System and Mathematical Operations Part 2 2015-11-20 www.njctl.org Table of Contents Slide 3 / 157 Squares of Numbers Greater than 20 Simplifying Perfect
More informationGrade 7/8 Math Circles Winter March 20/21/22 Types of Numbers
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Winter 2018 - March 20/21/22 Types of Numbers Introduction Today, we take our number
More informationPUTNAM TRAINING MATHEMATICAL INDUCTION. Exercises
PUTNAM TRAINING MATHEMATICAL INDUCTION (Last updated: December 11, 017) Remark. This is a list of exercises on mathematical induction. Miguel A. Lerma 1. Prove that n! > n for all n 4. Exercises. Prove
More informationWaves. If you are a bit "taken for geometry", the Cabalist Leon offers you these simple observations that you can explain to your grandmother too.
Waves Waves explained to the grandmother. If you are a bit "taken for geometry", the Cabalist Leon offers you these simple observations that you can explain to your grandmother too. A wave is an oscillation
More informationExercise 5.1: Introduction To Euclid s Geometry
Exercise 5.1: Introduction To Euclid s Geometry Email: info@mywayteaching.com Q1. Which of the following statements are true and which are false? Give reasons for your answers. (i)only one line can pass
More informationCOT 2104 Homework Assignment 1 (Answers)
1) Classify true or false COT 2104 Homework Assignment 1 (Answers) a) 4 2 + 2 and 7 < 50. False because one of the two statements is false. b) 4 = 2 + 2 7 < 50. True because both statements are true. c)
More informationAREA Judo Math Inc.
AREA 2013 Judo Math Inc. 7 th grade Geometry Discipline: Blue Belt Training Order of Mastery: Area 1. Square units/area overview 2. Circle Vocab (7G4) 3. What is Pi? (7G4) 4. Circumference of a circle
More informationPre-Algebra Unit 2. Rational & Irrational Numbers. Name
Pre-Algebra Unit 2 Rational & Irrational Numbers Name Core Table 2 Pre-Algebra Name: Unit 2 Rational & Irrational Numbers Core: Table: 2.1.1 Define Rational Numbers Vocabulary: Real Numbers the set of
More informationCLASS-IX MATHEMATICS. For. Pre-Foundation Course CAREER POINT
CLASS-IX MATHEMATICS For Pre-Foundation Course CAREER POINT CONTENTS S. No. CHAPTERS PAGE NO. 0. Number System... 0 3 0. Polynomials... 39 53 03. Co-ordinate Geometry... 54 04. Introduction to Euclid's
More informationCantor and Infinite Sets
Cantor and Infinite Sets Galileo and the Infinite There are many whole numbers that are not perfect squares: 2, 3, 5, 6, 7, 8, 10, 11, and so it would seem that all numbers, including both squares and
More informationGlossary. Glossary 981. Hawkes Learning Systems. All rights reserved.
A Glossary Absolute value The distance a number is from 0 on a number line Acute angle An angle whose measure is between 0 and 90 Addends The numbers being added in an addition problem Addition principle
More informationSlide 1 / 178 Slide 2 / 178. Click on a topic to go to that section.
Slide / 78 Slide 2 / 78 Algebra I The Number System & Mathematical Operations 205--02 www.njctl.org Slide 3 / 78 Slide 4 / 78 Table of Contents Review of Natural Numbers, Whole Numbers, Integers and Rational
More informationAlgebra I. Slide 1 / 178. Slide 2 / 178. Slide 3 / 178. The Number System & Mathematical Operations. Table of Contents
Slide 1 / 178 Slide 2 / 178 Algebra I The Number System & Mathematical Operations 2015-11-02 www.njctl.org Table of Contents Slide 3 / 178 Review of Natural Numbers, Whole Numbers, Integers and Rational
More informationCorresponding parts of congruent triangles are congruent. (CPCTC)
Corresponding parts of congruent triangles are congruent. (CPCTC) Corresponding parts of congruent triangles are congruent. (CPCTC) Definition: Congruent triangles: Triangles that have all corresponding
More informationCurvaceous Circles BUT IT DOES WORK! Yep we can still relate the formula for the area of a circle to the formula for the area of a rectangle
Curvaceous Circles So our running theme on our worksheets has been that all the formulas for calculating the area of various shapes comes back to relating that shape to a rectangle. But how can that possibly
More informationWhat can you prove by induction?
MEI CONFERENCE 013 What can you prove by induction? Martyn Parker M.J.Parker@keele.ac.uk Contents Contents iii 1 Splitting Coins.................................................. 1 Convex Polygons................................................
More information1.2 REAL NUMBERS. 10 Chapter 1 Basic Concepts: Review and Preview
10 Chapter 1 Basic Concepts: Review and Preview (b) Segment of a circle of radius R, depth R 2: A 4 R 2 (c) Frustum of cone: V 1 h R2 Rr r 2 R r R R 2 Conversion between fluid ounces and cubic inches:
More informationJennifer Duong Daniel Szara October 9, 2009
Jennifer Duong Daniel Szara October 9, 2009 By around 2000 BC, Geometry was developed further by the Babylonians who conquered the Sumerians. By around 2000 BC, Rational and Irrational numbers were used
More informationCN#4 Biconditional Statements and Definitions
CN#4 s and Definitions OBJECTIVES: STUDENTS WILL BE ABLE TO WRITE AND ANALYZE BICONDITIONAL STATEMENTS. Vocabulary biconditional statement definition polygon triangle quadrilateral When you combine a conditional
More informationP1-763.PDF Why Proofs?
P1-763.PDF Why Proofs? During the Iron Age men finally started questioning mathematics which eventually lead to the creating of proofs. People wanted to know how and why is math true, rather than just
More informationMath 105A HW 1 Solutions
Sect. 1.1.3: # 2, 3 (Page 7-8 Math 105A HW 1 Solutions 2(a ( Statement: Each positive integers has a unique prime factorization. n N: n = 1 or ( R N, p 1,..., p R P such that n = p 1 p R and ( n, R, S
More informationCSE 1400 Applied Discrete Mathematics Proofs
CSE 1400 Applied Discrete Mathematics Proofs Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Axioms 1 Logical Axioms 2 Models 2 Number Theory 3 Graph Theory 4 Set Theory 4
More information2018 Entrance Examination for the BSc Programmes at CMI. Read the instructions on the front of the booklet carefully!
2018 Entrance Examination for the BSc Programmes at CMI Read the instructions on the front of the booklet carefully! Part A. Write your final answers on page 3. Part A is worth a total of (4 10 = 40) points.
More informationSolutions 2017 AB Exam
1. Solve for x : x 2 = 4 x. Solutions 2017 AB Exam Texas A&M High School Math Contest October 21, 2017 ANSWER: x = 3 Solution: x 2 = 4 x x 2 = 16 8x + x 2 x 2 9x + 18 = 0 (x 6)(x 3) = 0 x = 6, 3 but x
More information8th Grade Math Course Map 2013
Course Title: 8 th Grade Pre-Algebra 8th Grade Math Course Map 2013 Duration: 2 semesters Frequency: Daily 44-51 minutes Year Updated: 2013 Text: Prentice Hall Pre-Algebra Other materials: Kagan Cooperative
More informationD - E - F - G (1967 Jr.) Given that then find the number of real solutions ( ) of this equation.
D - E - F - G - 18 1. (1975 Jr.) Given and. Two circles, with centres and, touch each other and also the sides of the rectangle at and. If the radius of the smaller circle is 2, then find the radius of
More informationYes zero is a rational number as it can be represented in the
1 REAL NUMBERS EXERCISE 1.1 Q: 1 Is zero a rational number? Can you write it in the form 0?, where p and q are integers and q Yes zero is a rational number as it can be represented in the form, where p
More informationCONTENTS NUMBER SYSTEMS. Number Systems
NUMBER SYSTEMS CONTENTS Introduction Classification of Numbers Natural Numbers Whole Numbers Integers Rational Numbers Decimal expansion of rational numbers Terminating decimal Terminating and recurring
More informationIntroduction to Geometry
Introduction to Geometry What is Geometry Why do we use Geometry What is Geometry? Geometry is a branch of mathematics that concerns itself with the questions of shape, size, position of figures, and the
More informationMath-2A Lesson 2-1. Number Systems
Math-A Lesson -1 Number Systems Natural Numbers Whole Numbers Lesson 1-1 Vocabulary Integers Rational Numbers Irrational Numbers Real Numbers Imaginary Numbers Complex Numbers Closure Why do we need numbers?
More informationChapter 11 - Sequences and Series
Calculus and Analytic Geometry II Chapter - Sequences and Series. Sequences Definition. A sequence is a list of numbers written in a definite order, We call a n the general term of the sequence. {a, a
More informationEuler s Identity: why and how does e πi = 1?
Euler s Identity: why and how does e πi = 1? Abstract In this dissertation, I want to show how e " = 1, reviewing every element that makes this possible and explore various examples of explaining this
More informationMath-2 Section 1-1. Number Systems
Math- Section 1-1 Number Systems Natural Numbers Whole Numbers Lesson 1-1 Vocabulary Integers Rational Numbers Irrational Numbers Real Numbers Imaginary Numbers Complex Numbers Closure Why do we need numbers?
More informationSummer Solutions Common Core Mathematics 8. Common Core. Mathematics. Help Pages
8 Common Core Mathematics 6 6 Vocabulary absolute value additive inverse property adjacent angles the distance between a number and zero on a number line. Example: the absolute value of negative seven
More informationMathematics Foundation for College. Lesson Number 1. Lesson Number 1 Page 1
Mathematics Foundation for College Lesson Number 1 Lesson Number 1 Page 1 Lesson Number 1 Topics to be Covered in this Lesson Sets, number systems, axioms, arithmetic operations, prime numbers and divisibility,
More informationGrade 8 Chapter 7: Rational and Irrational Numbers
Grade 8 Chapter 7: Rational and Irrational Numbers In this chapter we first review the real line model for numbers, as discussed in Chapter 2 of seventh grade, by recalling how the integers and then the
More informationIntroducing Proof 1. hsn.uk.net. Contents
Contents 1 1 Introduction 1 What is proof? 1 Statements, Definitions and Euler Diagrams 1 Statements 1 Definitions Our first proof Euler diagrams 4 3 Logical Connectives 5 Negation 6 Conjunction 7 Disjunction
More informationMath 016 Lessons Wimayra LUY
Math 016 Lessons Wimayra LUY wluy@ccp.edu MATH 016 Lessons LESSON 1 Natural Numbers The set of natural numbers is given by N = {0, 1, 2, 3, 4...}. Natural numbers are used for two main reasons: 1. counting,
More informationSolutions Best Student Exams Texas A&M High School Math Contest November 16, 2013
Solutions Best Student Exams Texas A&M High School Math Contest November 6, 20. How many zeros are there if you write out in full the number N = 0 (6 0000) = 0 (6 00) so there are 6 0 0 or N = (000000)
More informationNumbers and symbols WHOLE NUMBERS: 1, 2, 3, 4, 5, 6, 7, 8, 9... INTEGERS: -4, -3, -2, -1, 0, 1, 2, 3, 4...
Numbers and symbols The expression of numerical quantities is something we tend to take for granted. This is both a good and a bad thing in the study of electronics. It is good, in that we're accustomed
More informationFundamentals of Mathematics I
Fundamentals of Mathematics I Kent State Department of Mathematical Sciences Fall 2008 Available at: http://www.math.kent.edu/ebooks/10031/book.pdf August 4, 2008 Contents 1 Arithmetic 2 1.1 Real Numbers......................................................
More informationCoach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers
Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers CLASSIFICATIONS OF NUMBERS NATURAL NUMBERS = N = {1,2,3,4,...}
More informationThe University of Melbourne Department of Mathematics and Statistics School Mathematics Competition, 2016 INTERMEDIATE DIVISION: SOLUTIONS
The University of Melbourne Department of Mathematics and Statistics School Mathematics Competition, 2016 INTERMEDIATE DIVISION: SOLUTIONS (1) In the following sum substitute each letter for a different
More information=.55 = = 5.05
MAT1193 4c Definition of derivative With a better understanding of limits we return to idea of the instantaneous velocity or instantaneous rate of change. Remember that in the example of calculating the
More informationMath Tool: Dot Paper. Reproducible page, for classroom use only Triumph Learning, LLC
Math Tool: Dot Paper A Reproducible page, for classroom use only. 0 Triumph Learning, LLC CC_Mth_G_TM_PDF.indd /0/ : PM Math Tool: Coordinate Grid y 7 0 9 7 0 9 7 0 7 9 0 7 9 0 7 x Reproducible page, for
More informationThe Celsius temperature scale is based on the freezing point and the boiling point of water. 12 degrees Celsius below zero would be written as
Prealgebra, Chapter 2 - Integers, Introductory Algebra 2.1 Integers In the real world, numbers are used to represent real things, such as the height of a building, the cost of a car, the temperature of
More informationLecture 1: Axioms and Models
Lecture 1: Axioms and Models 1.1 Geometry Although the study of geometry dates back at least to the early Babylonian and Egyptian societies, our modern systematic approach to the subject originates in
More informationModular Arithmetic Instructor: Marizza Bailey Name:
Modular Arithmetic Instructor: Marizza Bailey Name: 1. Introduction to Modular Arithmetic If someone asks you what day it is 145 days from now, what would you answer? Would you count 145 days, or find
More information2. Two binary operations (addition, denoted + and multiplication, denoted
Chapter 2 The Structure of R The purpose of this chapter is to explain to the reader why the set of real numbers is so special. By the end of this chapter, the reader should understand the difference between
More informationWheels Radius / Distance Traveled
Mechanics Teacher Note to the teacher On these pages, students will learn about the relationships between wheel radius, diameter, circumference, revolutions and distance. Students will use formulas relating
More information35 Chapter CHAPTER 4: Mathematical Proof
35 Chapter 4 35 CHAPTER 4: Mathematical Proof Faith is different from proof; the one is human, the other is a gift of God. Justus ex fide vivit. It is this faith that God Himself puts into the heart. 21
More informationPaper 1H GCSE/A1H GCSE MATHEMATICS. Practice Set A (AQA Version) Non-Calculator Time allowed: 1 hour 30 minutes
Surname Other Names Candidate Signature Centre Number Candidate Number Examiner Comments Total Marks Paper 1H GCSE MATHEMATICS CM Practice Set A (AQA Version) Non-Calculator Time allowed: 1 hour 30 minutes
More informationAppendix: a brief history of numbers
Appendix: a brief history of numbers God created the natural numbers. Everything else is the work of man. Leopold Kronecker (1823 1891) Fundamentals of Computing 2017 18 (2, appendix) http://www.dcs.bbk.ac.uk/~michael/foc/foc.html
More informationSome Review Problems for Exam 1: Solutions
Math 3355 Fall 2018 Some Review Problems for Exam 1: Solutions Here is my quick review of proof techniques. I will focus exclusively on propositions of the form p q, or more properly, x P (x) Q(x) or x
More informationAlgebra I. abscissa the distance along the horizontal axis in a coordinate graph; graphs the domain.
Algebra I abscissa the distance along the horizontal axis in a coordinate graph; graphs the domain. absolute value the numerical [value] when direction or sign is not considered. (two words) additive inverse
More informationAdd Math (4047/02) Year t years $P
Add Math (4047/0) Requirement : Answer all questions Total marks : 100 Duration : hour 30 minutes 1. The price, $P, of a company share on 1 st January has been increasing each year from 1995 to 015. The
More informationCalculating methods. Addition. Multiplication. Th H T U Th H T U = Example
1 Addition Calculating methods Example 534 + 2678 Place the digits in the correct place value columns with the numbers under each other. Th H T U Begin adding in the units column. 5 3 4 + 12 16 17 8 4+8
More informationMechanics, Heat, Oscillations and Waves Prof. V. Balakrishnan Department of Physics Indian Institute of Technology, Madras
Mechanics, Heat, Oscillations and Waves Prof. V. Balakrishnan Department of Physics Indian Institute of Technology, Madras Lecture 08 Vectors in a Plane, Scalars & Pseudoscalers Let us continue today with
More informationHMMT February 2018 February 10, 2018
HMMT February 018 February 10, 018 Algebra and Number Theory 1. For some real number c, the graphs of the equation y = x 0 + x + 18 and the line y = x + c intersect at exactly one point. What is c? 18
More informationThe Pigeonhole Principle
The Pigeonhole Principle 2 2.1 The Pigeonhole Principle The pigeonhole principle is one of the most used tools in combinatorics, and one of the simplest ones. It is applied frequently in graph theory,
More informationMATH 201 Solutions: TEST 3-A (in class)
MATH 201 Solutions: TEST 3-A (in class) (revised) God created infinity, and man, unable to understand infinity, had to invent finite sets. - Gian Carlo Rota Part I [5 pts each] 1. Let X be a set. Define
More informationRising 7th Grade Math. Pre-Algebra Summer Review Packet
Rising 7th Grade Math Pre-Algebra Summer Review Packet Operations with Integers Adding Integers Negative + Negative: Add the absolute values of the two numbers and make the answer negative. ex: -5 + (-9)
More informationBasic Quantum Mechanics Prof. Ajoy Ghatak Department of Physics Indian Institute of Technology, Delhi
Basic Quantum Mechanics Prof. Ajoy Ghatak Department of Physics Indian Institute of Technology, Delhi Module No. # 05 The Angular Momentum I Lecture No. # 02 The Angular Momentum Problem (Contd.) In the
More information1 Sequences and Summation
1 Sequences and Summation A sequence is a function whose domain is either all the integers between two given integers or all the integers greater than or equal to a given integer. For example, a m, a m+1,...,
More informationCounting Out πr 2. Teacher Lab Discussion. Overview. Picture, Data Table, and Graph. Part I Middle Counting Length/Area Out πrinvestigation
5 6 7 Middle Counting Length/rea Out πrinvestigation, page 1 of 7 Counting Out πr Teacher Lab Discussion Figure 1 Overview In this experiment we study the relationship between the radius of a circle and
More informationSTA2112F99 ε δ Review
STA2112F99 ε δ Review 1. Sequences of real numbers Definition: Let a 1, a 2,... be a sequence of real numbers. We will write a n a, or lim a n = a, if for n all ε > 0, there exists a real number N such
More informationPHASE 9 Ali PERFECT ALI-PI.
PHASE 9 PERFECT ALI-PI Pi as a Fraction pi is written and expressed as definite fraction and ratio of two numbers: pi = 19 /6 = 3.16666666. pi = 3 + 1/6 Any rational number which cannot be expressed as
More information6. This sum can be rewritten as 4( ). We then recall the formula n =
. c = 9b = 3 b = 3 a 3 = a = = 6.. (3,, ) = 3 + + 3 = 9 + + 3 = 6 6. 3. We see that this is equal to. 3 = ( +.) 3. Using the fact that (x + ) 3 = x 3 + 3x + 3x + and replacing x with., we find that. 3
More informationExtended Essay - Mathematics
Extended Essay - Mathematics Creating a Model to Separate or Group Number Sets by their Cardinalities Pope John Paul II C.S.S. September 2009 Candidate Number: 001363-012 The conquest of the actual infinite
More informationBRITISH COLUMBIA SECONDARY SCHOOL MATHEMATICS CONTEST,
BRITISH COLUMBIA SECONDARY SCHOOL MATHEMATICS CONTEST, 014 Solutions Junior Preliminary 1. Rearrange the sum as (014 + 01 + 010 + + ) (013 + 011 + 009 + + 1) = (014 013) + (01 011) + + ( 1) = 1 + 1 + +
More informationA proof has to be rigorously checked before it is published, after which other mathematicians can use it to further develop the subject.
Proof in mathematics is crucial to its development. When an idea is formulated or an observation is noticed it becomes necessary to prove what has been discovered. Then again, the observation may prove
More informationWhy write proofs? Why not just test and repeat enough examples to confirm a theory?
P R E F A C E T O T H E S T U D E N T Welcome to the study of mathematical reasoning. The authors know that many students approach this material with some apprehension and uncertainty. Some students feel
More informationHow can you find decimal approximations of square roots that are not rational? ACTIVITY: Approximating Square Roots
. Approximating Square Roots How can you find decimal approximations of square roots that are not rational? ACTIVITY: Approximating Square Roots Work with a partner. Archimedes was a Greek mathematician,
More informationPAPER 1H GCSE/A1H GCSE MATHEMATICS. Practice Set A Non-Calculator Time allowed: 1 hour 30 minutes
Surname Other Names Candidate Signature Centre Number Candidate Number Examiner Comments Total Marks PAPER 1H GCSE MATHEMATICS CM Practice Set A Non-Calculator Time allowed: 1 hour 30 minutes Instructions
More informationADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS
ADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS 1. Cardinal number of a set The cardinal number (or simply cardinal) of a set is a generalization of the concept of the number of elements
More informationInspiring and enriching lessons at school. Gerry Leversha MA Conference, Oxford April 2016
Inspiring and enriching lessons at school Gerry Leversha MA Conference, Oxford April 2016 Inspiring lessons Until a few months ago, the buzzword was mastery The new national curriculum, having abolished
More informationSun Life Financial Canadian Open Mathematics Challenge Section A 4 marks each. Official Solutions
Sun Life Financial Canadian Open Mathematics Challenge 2015 Official Solutions COMC exams from other years, with or without the solutions included, are free to download online. Please visit http://comc.math.ca/2015/practice.html
More informationPre-Algebra Notes Unit Two: Solving Equations
Pre-Algebra Notes Unit Two: Solving Equations Properties of Real Numbers Syllabus Objective: (.1) The student will evaluate expressions using properties of addition and multiplication, and the distributive
More information