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Return to Blackboard Irwin & Nelms, Basic Engineering Circult Analysis, 11e Help I System Announcements Assignment Gradebook Downloadable eTextbook ent Math Skills Assessment: System of Equations 3: 2 Variables in order, one decimal In the following equations i and v are variables that are real valued. Find the values of i and v: By accessing this Question Assistance, you will learn while you earn points based on the Point Potential Policy set Ali Rights Reserved. A Division of ohn Wiley & Sons. Inc Rights ley &.sens. Ins

Problem: Divisibility and Divisibility Rules There are many rules to determine if a number is divisible by other smaller numbers. In basic arithmetic courses, you learned divisibility and discussed a few common divisibility rules. Your task is to determine if the given number is divisible by 9 but is not divisible by 11 Given number: 7,176,825,942,116,027,211 Note: You want to use divisibility rules. Do not attempt to use long division and do not just use your calculator 1. Research the topic of divisibility 2. Research the divisibility rules for 9 and 11. 3. Using the divisibility rules found, determine how the given number is divisible by 9 and is not divisible by 11. "Note:I am telling you that the number IS divisible by 9 and IS NOT divisible by 11. Do not try to show the opposite of this statement. 4. Include the following information in a report, in this order: a. Explain (in a few sentences) the topic of divisibility, including any correct terms that apply b. Explain in your own words the divisibility rule of 9. C. Give an example using another number to help the explanation. d. Show how the given number is divisible by 9. Work out all your steps. e. Explain in your own words the divisibility rule of 11 f. Give an example using another number to help the explanation. g. Show how the given number is not divisible by 11. Work out all your steps. 5. You must locate a minimum of 2 sources to help you complete this project. More than two sources, though not required, can be used if it helps to give detail to the project. a. You must use at least one book or E-book (other than your textbook). Do not use the dictionary for your definition. b. You must use at least one website. Do not use Wikipedia c. Your sources should explain the topics of the problem (divisibility and divisibility rules) and have examples that help you with the problem. Your sources should be relevant to the project and not just be a math source because it has math in the context d. 6. Correctly cite, in MLA format, the sources that you located to complete the assignment. If you need help for citations in MLA format, you can try using the website EasyBih Be sure to do the entire project typed neatly 7. Papers will be compared for similarity Plagiarism is against the academic honesty policy and will receive a zera. Note: A detailed Grading Rubric is provided on Canvas and on the assignment submission link The due date for the project is listed on Canvas. The Math/Analytical Project must be submitted through Canvas. Directions for submitting can be found in the Course Syllabus and on Canvas.

42 Metric Spaces Proof. The proof is somewhat lengthy but straightforward. We subdivide it into four steps (a) to (d). We construct (a) X=(X, d) (b) an isometry T of X onto W, where W Then we prove: (c) completeness of X. (d) uniqueness of X, except for isometries. Roughly speaking, our task will be the assignment of suitable limits to Cauchy sequences in X that do not converge. However, we should not introduce "too many" limits, but take into account that certain se- quences "may want to converge with the same limit" since the terms o those sequences "ultimately come arbitrarily close to each other intuitive ide " This a can be expressed mathematically in terms of a suitable equivalence relation [see (1), below). This is not artificial but is suggested by the process of completion of the rational line mentioned at the beginning of the section. The details of the proof are as follows. X-(X, d). (%) and (x) be (a) Construction of Let Cauchy sequences in Х. Define (4) to be equivalente, to (x), written Let X be the set of all equivalence classes,of Cauchy sequences thus obtained wewrite.(%)eito mean that (x.) is a member of i (a representative of the class , we now set where (t)e i and (y)e y. We show that this limit exists. We have hence we obtain and a similar inequality with m and n interchanged. Together, 1e review of the concept of equivalence, see Al 4 in Appendix 1

1.6 Completion of Metric Spaces 43 Since (%) and (%) are Cauchy, we can make the right side as small as we please. This implies that the limit in (2) exists because Ris complete. We must also show that the limit in (2) is independent of the particular choice of representatives. In fact, if ( )-(x) and (%)-(y"), then by (1), as n-a, which implies the assertion We prove that d in (2) is a metric on X. Obviously, d satisfies (MI) in Sec. 1.1 as well as d(i,i)s0 and (M3). Furthermore, gives (M2), and (M4) for d follows from TU ) by letting n-00, with (b) Construction of an isometry T: X-Wc each bex we associate the class bex which contains the constant Cauchy sequence (b, b,). This defines a mapping T X_W onto the subspace W= TX) ,. The mapping T is given by b-→ b= T, where (b, b,.Jeb. We see that T is an isometry since (2) becomes simply メ here é is the class of (ya) where y- c for all n. Any isometry is injective, and T: X-→ W is surjective since T(X)=w. Hence W and X are isometric; cf. Def. 1.6-1(b). We show that W is dense in X. We consider any iex. Let ()e i. For every e>0 there is an N such that in> N)

Let (쯔스-k h. Then This shows that every s-neighborhood of the arbitrary i e X contains an element of W. Hence W is dense in X. (c) Completeness of X. Let () be any Cauchy sequence in X. Since W is dense in X, for every there is a &ne W such that Pl Hence by the triangle inequality and this is less than any given ε>0 for sufficiently large m and n because (辶) is Cauchy. Hence (2-) is Cauchy. Since T: X-w is isometric and i W, the sequence (z), where zm-Tim, is Cauchy in X. Let te X be the class to which (.) belongs. We show that i is the limit of () By (4) rt Since (m)e i (see right before) and i e W, so that (2 the inequality (5) becomes ei nm to and the right side is smaller than any given e>0 for sufficiently large n. Hence the arbitrary Cauchy sequence (L) in X has the limit ieX, and X is complete.

1.6 Completion of Metric Spaces 45 っn+ワ Fig. 11. Notations in part (d) of the proof of Theorem 1.6-2 (d) Uniqueness of X except for isometries. If (X, d) is another complete metric space with a subspace W dense in X and isometric with X, then for any i, fe X we have sequences ( ), (SJ in W such thatn and -y; hence follows from [the inequality being similar to (3)1. Since W is isometric with We X and W-X, the distances on X and X must be the same. Hence X and X are isometric. We shall see in the next two chapters (in particular in 2.3-2, 3.1- and 3.2-3) that this theorem has basic applications to individual incomplete spaces as well as to whole classes of such spaces.