In chapter 1.1 of “Elementary Number & Its Applications”, Kenneth H. Rosen defines greatest integer function:The greatest integer function in a real number x, denoted by [x], is the largest integer less than or equal to x. That is [x] is an integer satisfying:[x] ≤ x < [x]+1Kenneth H. Rosen also defines below:The fractional part of a real number x, denoted by {x}, is the difference between x and the largest integer less than or equal to x, namely [x]. That is, {x}=x-[x].I want to use these definitions to answer the following:Chapter 1.1, Exercise 13: Show that [x+y] ≥ [x] + [y] for all real numbers x and y. *c) The product of a rational number and an irrational number is irrational.d) The product of two irrational numbers is irrational.5. Use the well-ordering property to show that √3 is irrational.6. Show that every nonempty set of negative integers has a greatest element.7.Find the following values of the greatest integer function.a) [1/4]c) [22/7]ooe) [[1/2] + [1/2]]f) [-3+ [-1/2]]b) [-3/4]d) [-2]1.1 Numbers and Sequences8. Find the following values of the greatest integer function.c) [5/4]d) [[1/2]]a) [-1/4]b) [-22/7]e) [[3/2] + [-3/2]]f) [3- [1/2]]9. Find the fractional part of each of these numbers:a) 8/5b) 1/7c)-11/410. Find the fractional part of each of these numbers:a) -8/5b) 22/7c) -1d) 7d) -1/311. What is the value of [x] + [-x] where x is a real number?12. Show that [x] + [x + 1/2] = [2x] whenever x is a real number.13. Show that [x+y]≥ [x] + [y] for all real numbers x and y.14. Show that [2x] + [2y] ≥ [x] + [y] + [x+y] whenever x and y are real numbers.15. Show that if x and y are positive real numbers, then [xy] ≥ [x][y]. What is the situation whenboth x and y are negative? When one of x and y is negative and the other positive?16. Show that -[-x] is the least integer greater than or equal to x when x is a real number.17. Show that [x + 1/2] is the integer nearest to x (when there are two integers equidistant fromx, it is the larger of the two).a) 3, 11, 19, 27, 35, 43, 51, 59, 67, 75b) 5, 7, 11, 19, 35, 67, 131, 259, 515, 10271318. Show that if m and n are integers, then [(x + n)/m] = [([x]+n)/m] whenever x is a realnumber.predisco* 19. Show that [√[x]]= [√x] whenever x is a nonnegative real number.* 20. Show that if m is a positive integer, then[mx] = [x] + [x+ (1/m)] + [x+ (2/m)] ++ [x+ (m - 1)/m]whenever x is a real number.21. Conjecture a formula for the nth term of {a} if the first ten terms of this sequence are asfollows. hollal oh Judong toToc) 1, 0, 0, 1, 0, 0, 0, 0, 1, 0d) 1, 3, 4, 7, 11, 18, 29, 47, 76, 12322. Conjecture a formula for the nth term of {a} if the first ten terms of this sequence are asfollows.a) 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366b) 1, 1, 0, 1, 1, 0, 1, 1, 0, 1 1.1 Numbers and SequencesWe have s√2-s=s√2-1√√2=(5-1)√2. Because s√√2 = 2t and s are bothintegers, s√2-s=s√√2-1√√2 = (st)√2 must also be an integer. Furthermore, itis positive, because s√√2-s=s(√2-1) and 21. It is less than 's, because √2<2so that √2-141. This contradicts the choice of s as the smallest positive integer in S.It follows that √2 is irrational.■7The sets of integers, positive integers, rational numbers, and real numbers aretraditionally denoted by Z, Z+, Q, and R, respectively. Also, we write x € S to indicatethat x belongs to the set S. Such notation will be used occasionally in this book.We briefly mention several other types of numbers here, though we do not return tothem until Chapter 12.Definition. A number a is algebraic if it is the root of a polynomial with integercoefficients; that is, & is algebraic if there exist integers ao, a₁,..., a, such that ana" +++ao= 0. The number & is called transcendental if it is not algebraic.an-10-1Example 1.2. The irrational number √2 is algebraic, because it is a root of thepolynomial x² - 2.Note that every rational number is algebraic. This follows from the fact that the numbera/b, where a and b are integers and b0, is the root of bx-a. In Chapter 12,we will give an example of a transcendental number. The numbers e and are alsotranscendental, but the proofs of these facts (which can be found in [HaWr08]) are beyondthe scope of this book.The Greatest Integer FunctionIn number theory, a special notation is used for the largest integer that is less than orequal to a particular real number.Definition. The greatest integer in a real number x, denoted by [x], is the largest integerless than or equal to x. That is, [x] is the integer satisfying[x] ≤ x < [x] + 1.Example 1.3. We have [5/2] =2, [-5/2] =-3, [T]= 3, [-2]= -2, and [0] = 0.Remark. The greatest integer function is also known as the floor function. Instead ofusing the notation [x] for this function, computer scientists usually use the notation [x].The ceiling function is a related function often used by computer scientists. The ceilingfunction of a real number x, denoted by [x], is the smallest integer greater than or equalto x. For example, [5/21 = 3 and [-5/21 = -2.The greatest integer function arises in many contexts. Besides being important innumber theory, as we will see throughout this book, it plays an important role in theanalysis of algorithms, a branch of computer science. The following example establishes

By definition, [x] ≤ x and if k be any integer such that k ≤ x, then k ≤ [x]

Answered: Show that [x+y] ≥ [x] + [y] for all…

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ