Number Shapes
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
9N1.5 Determine the Square Root of Positive Rational Numbers That Are Perfect Squares
9N1.5 Determine the square root of positive rational numbers that are perfect squares. Square Roots and Perfect Squares The square root of a given number is a value that when multiplied by itself results in the given number. For example, the square root of 9 is 3 because 3 × 3 = 9 . Example Use a diagram to determine the square root of 121. Solution Step 1 Draw a square with an area of 121 units. Step 2 Count the number of units along one side of the square. √121 = 11 units √121 = 11 units The square root of 121 is 11. This can also be written as √121 = 11. A number that has a whole number as its square root is called a perfect square. Perfect squares have the unique characteristic of having an odd number of factors. Example Given the numbers 81, 24, 102, 144, identify the perfect squares by ordering their factors from smallest to largest. Solution The square root of each perfect square is bolded. Factors of 81: 1, 3, 9, 27, 81 Since there are an odd number of factors, 81 is a perfect square. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Since there are an even number of factors, 24 is not a perfect square. Factors of 102: 1, 2, 3, 6, 17, 34, 51, 102 Since there are an even number of factors, 102 is not a perfect square. Factors of 144: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 Since there are an odd number of factors, 144 is a perfect square. -
Triangular Numbers /, 3,6, 10, 15, ", Tn,'" »*"
TRIANGULAR NUMBERS V.E. HOGGATT, JR., and IVIARJORIE BICKWELL San Jose State University, San Jose, California 9111112 1. INTRODUCTION To Fibonacci is attributed the arithmetic triangle of odd numbers, in which the nth row has n entries, the cen- ter element is n* for even /?, and the row sum is n3. (See Stanley Bezuszka [11].) FIBONACCI'S TRIANGLE SUMS / 1 =:1 3 3 5 8 = 2s 7 9 11 27 = 33 13 15 17 19 64 = 4$ 21 23 25 27 29 125 = 5s We wish to derive some results here concerning the triangular numbers /, 3,6, 10, 15, ", Tn,'" »*". If one o b - serves how they are defined geometrically, 1 3 6 10 • - one easily sees that (1.1) Tn - 1+2+3 + .- +n = n(n±M and (1.2) • Tn+1 = Tn+(n+1) . By noticing that two adjacent arrays form a square, such as 3 + 6 = 9 '.'.?. we are led to 2 (1.3) n = Tn + Tn„7 , which can be verified using (1.1). This also provides an identity for triangular numbers in terms of subscripts which are also triangular numbers, T =T + T (1-4) n Tn Tn-1 • Since every odd number is the difference of two consecutive squares, it is informative to rewrite Fibonacci's tri- angle of odd numbers: 221 222 TRIANGULAR NUMBERS [OCT. FIBONACCI'S TRIANGLE SUMS f^-O2) Tf-T* (2* -I2) (32-22) Ti-Tf (42-32) (52-42) (62-52) Ti-Tl•2 (72-62) (82-72) (9*-82) (Kp-92) Tl-Tl Upon comparing with the first array, it would appear that the difference of the squares of two consecutive tri- angular numbers is a perfect cube. -
An Heptagonal Numbers
© April 2021| IJIRT | Volume 7 Issue 11 | ISSN: 2349-6002 An Heptagonal Numbers Dr. S. Usha Assistant Professor in Mathematics, Bon Secours Arts and Science College for Women, Mannargudi, Affiliated by Bharathidasan University, Tiruchirappalli, Tamil Nadu Abstract - Two Results of interest (i)There exists an prime factor that divides n is one. First few square free infinite pairs of heptagonal numbers (푯풎 , 푯풌) such that number are 1,2,3,5,6,7,10,11,13,14,15,17…… their ratio is equal to a non –zero square-free integer and (ii)the general form of the rank of square heptagonal Definition(1.5): ퟑ ퟐ풓+ퟏ number (푯 ) is given by m= [(ퟏퟗ + ퟑ√ퟒퟎ) + 풎 ퟐퟎ Square free integer: ퟐ풓+ퟏ (ퟏퟗ − ퟑ√ퟒퟎ) +2], where r = 0,1,2…….relating to A Square – free Integer is an integer which is divisible heptagonal number are presented. A Few Relations by no perfect Square other than 1. That is, its prime among heptagonal and triangular number are given. factorization has exactly one factors for each prime that appears in it. For example Index Terms - Infinite pairs of heptagonal number, the 10 =2.5 is square free, rank of square heptagonal numbers, square-free integer. But 18=2.3.3 is not because 18 is divisible by 9=32 The smallest positive square free numbers are I. PRELIMINARIES 1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23……. Definition(1.1): A number is a count or measurement heptagon: Definition(1.6): A heptagon is a seven –sided polygon. -
Grade 7/8 Math Circles the Scale of Numbers Introduction
Faculty of Mathematics Centre for Education in Waterloo, Ontario N2L 3G1 Mathematics and Computing Grade 7/8 Math Circles November 21/22/23, 2017 The Scale of Numbers Introduction Last week we quickly took a look at scientific notation, which is one way we can write down really big numbers. We can also use scientific notation to write very small numbers. 1 × 103 = 1; 000 1 × 102 = 100 1 × 101 = 10 1 × 100 = 1 1 × 10−1 = 0:1 1 × 10−2 = 0:01 1 × 10−3 = 0:001 As you can see above, every time the value of the exponent decreases, the number gets smaller by a factor of 10. This pattern continues even into negative exponent values! Another way of picturing negative exponents is as a division by a positive exponent. 1 10−6 = = 0:000001 106 In this lesson we will be looking at some famous, interesting, or important small numbers, and begin slowly working our way up to the biggest numbers ever used in mathematics! Obviously we can come up with any arbitrary number that is either extremely small or extremely large, but the purpose of this lesson is to only look at numbers with some kind of mathematical or scientific significance. 1 Extremely Small Numbers 1. Zero • Zero or `0' is the number that represents nothingness. It is the number with the smallest magnitude. • Zero only began being used as a number around the year 500. Before this, ancient mathematicians struggled with the concept of `nothing' being `something'. 2. Planck's Constant This is the smallest number that we will be looking at today other than zero. -
Induction for Tetrahedral Numbers
Induction for Tetrahedral Numbers Teacher Notes and Answers 7 8 9 10 11 12 TI-84PlusCE™ Investigation Student 50 min Teacher Notes: “Inductive Tetrahedrals” is part two of this three-part series. [Refer to Inductive Whole Numbers for Part One]. Part two follows a similar format to part one but is slightly more challenging from an algebraic perspective. Teachers can use part two in this series as a more independent environment for students focusing on providing feedback rather than instruction. A set of Power Point slides is provided that can be used to enhance the visual aspect of this lesson, the ability to visualise solutions can give problems more meaning. “Cubes and Squares” is part three of this three-part series. This section is designed as an assessment tool, marks have been allocated to each question. The questions are more reflective of the types of questions that students might experience in the short answer section of an examination where calculators may be used to derive answers or to simply verify by-hand solutions. A set of Power Point slides is provided that can be used to introduce students to this task. Introduction The purpose of this activity is to use exploration and observation to establish a rule for the sum of the first n tetrahedral numbers then use proof by mathematical induction to show that the rule is true for all whole numbers. What are the Tetrahedral numbers? The prefix ‘tetra’ refers to the quantity four, so it is not surprising that a tetrahedron consists of four faces, each face is a triangle. -
Input for Carnival of Math: Number 115, October 2014
Input for Carnival of Math: Number 115, October 2014 I visited Singapore in 1996 and the people were very kind to me. So I though this might be a little payback for their kindness. Good Luck. David Brooks The “Mathematical Association of America” (http://maanumberaday.blogspot.com/2009/11/115.html ) notes that: 115 = 5 x 23. 115 = 23 x (2 + 3). 115 has a unique representation as a sum of three squares: 3 2 + 5 2 + 9 2 = 115. 115 is the smallest three-digit integer, abc , such that ( abc )/( a*b*c) is prime : 115/5 = 23. STS-115 was a space shuttle mission to the International Space Station flown by the space shuttle Atlantis on Sept. 9, 2006. The “Online Encyclopedia of Integer Sequences” (http://www.oeis.org) notes that 115 is a tridecagonal (or 13-gonal) number. Also, 115 is the number of rooted trees with 8 vertices (or nodes). If you do a search for 115 on the OEIS website you will find out that there are 7,041 integer sequences that contain the number 115. The website “Positive Integers” (http://www.positiveintegers.org/115) notes that 115 is a palindromic and repdigit number when written in base 22 (5522). The website “Number Gossip” (http://www.numbergossip.com) notes that: 115 is the smallest three-digit integer, abc, such that (abc)/(a*b*c) is prime. It also notes that 115 is a composite, deficient, lucky, odd odious and square-free number. The website “Numbers Aplenty” (http://www.numbersaplenty.com/115) notes that: It has 4 divisors, whose sum is σ = 144. -
Tetrahedral Numbers
Tetrahedral Numbers Student Worksheet TI-30XPlus Activity Student 50 min 7 8 9 10 11 12 MathPrint™ Finding Patterns What are the Tetrahedral numbers? The prefix ‘tetra’ refers to the quantity four, so it is not surprising that a tetrahedron consists of four faces, each face is a triangle. This triangular formation can sometimes be found in stacks of objects. The series of diagrams below shows the progression from one layer to the next for a stack of spheres. Row Number 1 2 3 4 5 Items Added Complete Stack Question: 1. Create a table of values for the row number and the corresponding quantity of items in a complete stack. Question: 2. Create a table of values for the row number and the corresponding quantity of items that are added to the stack. Question: 3. The calculator screen shown here illustrates how to determine the fifth tetrahedral number. The same command could be used to determine any of the tetrahedral numbers. Explain how this command is working. Question: 4. Verify that the calculation shown opposite is the same as the one generated in Question 3. Question: 5. Enter the numbers 1, 2 … 10 in List 1 on the calculator. Enter the first 10 tetrahedral numbers in List 2. Once the values have been entered try the following: a) Quadratic regression using List 1 and List 2. Check the validity of the result via substitution. b) Cubic regression using List 1 and List 2. Check the validity of the result via substitution. Texas Instruments 2021. You may copy, communicate and modify this material for non-commercial educational purposes Author: P. -
The Observability of the Fibonacci and the Lucas Cubes
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 255 (2002) 55–63 www.elsevier.com/locate/disc The observability of the Fibonacci and the Lucas cubes Ernesto DedÃo∗;1, Damiano Torri1, Norma Zagaglia Salvi1 Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy Received 5 April 1999; received inrevised form 31 July 2000; accepted 8 January2001 Abstract The Fibonacci cube n is the graph whose vertices are binary strings of length n without two consecutive 1’s and two vertices are adjacent when their Hamming distance is exactly 1. If the binary strings do not contain two consecutive 1’s nora1intheÿrst and in the last position, we obtainthe Lucas cube Ln. We prove that the observability of n and Ln is n, where the observability of a graph G is the minimum number of colors to be assigned to the edges of G so that the coloring is proper and the vertices are distinguished by their color sets. c 2002 Elsevier Science B.V. All rights reserved. MSC: 05C15; 05A15 Keywords: Fibonacci cube; Fibonacci number; Lucas number; Observability 1. Introduction A Fibonacci string of order n is a binary string of length n without two con- secutive ones. Let and ÿ be binary strings; then ÿ denotes the string obtained by concatenating and ÿ. More generally, if S is a set of strings, then Sÿ = {ÿ: ∈ S}. If Cn denotes the set of the Fibonacci strings of order n, then Cn+2 =0Cn+1 +10Cn and |Cn| = Fn, where Fn is the nth Fibonacci number. -
Representing Square Numbers Copyright © 2009 Nelson Education Ltd
Representing Square 1.1 Numbers Student book page 4 You will need Use materials to represent square numbers. • counters A. Calculate the number of counters in this square array. ϭ • a calculator number of number of number of counters counters in counters in in a row a column the array 25 is called a square number because you can arrange 25 counters into a 5-by-5 square. B. Use counters and the grid below to make square arrays. Complete the table. Number of counters in: Each row or column Square array 525 4 9 4 1 Is the number of counters in each square array a square number? How do you know? 8 Lesson 1.1: Representing Square Numbers Copyright © 2009 Nelson Education Ltd. NNEL-MATANSWER-08-0702-001-L01.inddEL-MATANSWER-08-0702-001-L01.indd 8 99/15/08/15/08 55:06:27:06:27 PPMM C. What is the area of the shaded square on the grid? Area ϭ s ϫ s s units ϫ units square units s When you multiply a whole number by itself, the result is a square number. Is 6 a whole number? So, is 36 a square number? D. Determine whether 49 is a square number. Sketch a square with a side length of 7 units. Area ϭ units ϫ units square units Is 49 the product of a whole number multiplied by itself? So, is 49 a square number? The “square” of a number is that number times itself. For example, the square of 8 is 8 ϫ 8 ϭ . -
On Hardy's Apology Numbers
ON HARDY’S APOLOGY NUMBERS HENK KOPPELAAR AND PEYMAN NASEHPOUR Abstract. Twelve well known ‘Recreational’ numbers are generalized and classified in three generalized types Hardy, Dudeney, and Wells. A novel proof method to limit the search for the numbers is exemplified for each of the types. Combinatorial operators are defined to ease programming the search. 0. Introduction “Recreational Mathematics” is a broad term that covers many different areas including games, puzzles, magic, art, and more [31]. Some may have the impres- sion that topics discussed in recreational mathematics in general and recreational number theory, in particular, are only for entertainment and may not have an ap- plication in mathematics, engineering, or science. As for the mathematics, even the simplest operation in this paper, i.e. the sum of digits function, has application outside number theory in the domain of combinatorics [13, 26, 27, 28, 34] and in a seemingly unrelated mathematical knowledge domain: topology [21, 23, 15]. Pa- pers about generalizations of the sum of digits function are discussed by Stolarsky [38]. It also is a surprise to see that another topic of this paper, i.e. Armstrong numbers, has applications in “data security” [16]. In number theory, functions are usually non-continuous. This inhibits solving equations, for instance, by application of the contraction mapping principle because the latter is normally for continuous functions. Based on this argument, questions about solving number-theoretic equations ramify to the following: (1) Are there any solutions to an equation? (2) If there are any solutions to an equation, then are finitely many solutions? (3) Can all solutions be found in theory? (4) Can one in practice compute a full list of solutions? arXiv:2008.08187v1 [math.NT] 18 Aug 2020 The main purpose of this paper is to investigate these constructive (or algorith- mic) problems by the fixed points of some special functions of the form f : N N. -
Newsletter 91
Newsletter 9 1: December 2010 Introduction This is the final nzmaths newsletter for 2010. It is also the 91 st we have produced for the website. You can have a look at some of the old newsletters on this page: http://nzmaths.co.nz/newsletter As you are no doubt aware, 91 is a very interesting and important number. A quick search on Wikipedia (http://en.wikipedia.org/wiki/91_%28number%29) will very quickly tell you that 91 is: • The atomic number of protactinium, an actinide. • The code for international direct dial phone calls to India • In cents of a U.S. dollar, the amount of money one has if one has one each of the coins of denominations less than a dollar (penny, nickel, dime, quarter and half dollar) • The ISBN Group Identifier for books published in Sweden. In more mathematically related trivia, 91 is: • the twenty-seventh distinct semiprime. • a triangular number and a hexagonal number, one of the few such numbers to also be a centered hexagonal number, and it is also a centered nonagonal number and a centered cube number. It is a square pyramidal number, being the sum of the squares of the first six integers. • the smallest positive integer expressible as a sum of two cubes in two different ways if negative roots are allowed (alternatively the sum of two cubes and the difference of two cubes): 91 = 6 3+(-5) 3 = 43+33. • the smallest positive integer expressible as a sum of six distinct squares: 91 = 1 2+2 2+3 2+4 2+5 2+6 2. -
Elementary School Numbers and Numeration
DOCUMENT RESUME ED 166 042 SE 026 555 TITLE Mathematics for Georgia Schools,' Volume II: Upper Elementary 'Grades. INSTITUTION Georgia State Dept. of Education, Atlanta. Office of Instructional Services. e, PUB DATE 78 NOTE 183p.; For related document, see .SE 026 554 EDRB, PRICE MF-$0.33 HC-$10.03 Plus Postage., DESCRIPTORS *Curriculm; Elementary Educatia; *Elementary School Mathematics; Geometry; *Instruction; Meadurement; Number Concepts; Probability; Problem Solving; Set Thory; Statistics; *Teaching. Guides ABSTRACT1 ' This guide is organized around six concepts: sets, numbers and numeration; operations, their properties and number theory; relations and functions; geometry; measurement; and probability and statistics. Objectives and sample activities are presented for.each concept. Separate sections deal with the processes of problem solving and computation. A section on updating curriculum includes discussion of continuing program improvement, evaluation of pupil progress, and utilization of media. (MP) ti #######*#####*########.#*###*######*****######*########*##########**#### * Reproductions supplied by EDRS are the best that can be made * * from the original document. * *********************************************************************** U S DEPARTMENT OF HEALTH, EDUCATION & WELFARE IS NATIONAL INSTITUTE OF.. EDUCATION THIS DOCUMENT HA4BEEN REPRO- DuCED EXACTLY AS- RECEIVEDFROM THE PERSON OR ORGANIZATIONORIGIN- ATING IT POINTS OF VIEWOR 01NIONS STATED DO NOT NECESSARILYEpRE SENT OFFICIAL NATIONAL INSTITUTEOF TO THE EDUCATION4