[-x] where x is a real number? 12. Show that [x] + [x+ 1/2] = [2x] whenever x is a real number.
[-x] where x is a real number? 12. Show that [x] + [x+ 1/2] = [2x] whenever x is a real number.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
In chapter 1.1 of “Elementary Number Theory & It’s Applications”, Kenneth H. Rosen defines the greatest integer function:
The greatest integer in a real number x, denoted by [x], is the largest integer less than or equal to x. That is, [x] is the integer satisfying (“\lt” means “less than or equal to”)
[x] \lt x < [x]+1
I want to use this def. to answer the following:
Chapter 1.1, Exercise 12: Show that [x]+[x+1/2]=[2x] whenever x is a real number.
![1.1 Numbers and Sequences
7
We have s√2-s=s√√2-1√√2 = (S-1)√√2. Because s√2= 2t and s are both
integers, s√2-s =s√2-1√√2 = (st)√√2 must also be an integer. Furthermore, it
is positive, because s√2- s = s(√2-1) and √2 > 1.It is less than s, because √2 <2
so that √2-141. This contradicts the choice of s as the smallest positive integer in S.
It follows that V2 is irrational.
The sets of integers, positive integers, rational numbers, and real numbers are
traditionally denoted by Z, Zt, Q, and R, respectively. Also, we write x € S to indicate
that 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 to
them until Chapter 12.
Definition. A number a is algebraic if it is the root of a polynomial with integer
coefficients; that is, & is algebraic if there exist integers ao, a₁, ... , an such that ana" +
an-10-1
++ do 0. The number a is called transcendental if it is not algebraic.
Example 1.2. The irrational number √2 is algebraic, because it is a root of the
polynomial x2 -2.
Note that every rational number is algebraic. This follows from the fact that the number
a/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 andare also
transcendental, but the proofs of these facts (which can be found in [HaWr08]) are beyond
the scope of this book.
The Greatest Integer Function
In number theory, a special notation is used for the largest integer that is less than or
equal to a particular real number.
Definition. The greatest integer in a real number x, denoted by [x], is the largest integer
less 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 of
using 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 ceiling
function of a real number x, denoted by [x], is the smallest integer greater than or equal
to x. For example, [5/2] = 3 and [-5/2] = -2.
The greatest integer function arises in many contexts. Besides being important in
number theory, as we will see throughout this book, it plays an important role in the
analysis of algorithms, a branch of computer science. The following example establishes](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9acf228d-ec9e-49d8-a78a-8ceb13ba5105%2F8da5c926-9f5d-4ca6-a29a-acffef56241b%2Fb0ylis_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.1 Numbers and Sequences
7
We have s√2-s=s√√2-1√√2 = (S-1)√√2. Because s√2= 2t and s are both
integers, s√2-s =s√2-1√√2 = (st)√√2 must also be an integer. Furthermore, it
is positive, because s√2- s = s(√2-1) and √2 > 1.It is less than s, because √2 <2
so that √2-141. This contradicts the choice of s as the smallest positive integer in S.
It follows that V2 is irrational.
The sets of integers, positive integers, rational numbers, and real numbers are
traditionally denoted by Z, Zt, Q, and R, respectively. Also, we write x € S to indicate
that 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 to
them until Chapter 12.
Definition. A number a is algebraic if it is the root of a polynomial with integer
coefficients; that is, & is algebraic if there exist integers ao, a₁, ... , an such that ana" +
an-10-1
++ do 0. The number a is called transcendental if it is not algebraic.
Example 1.2. The irrational number √2 is algebraic, because it is a root of the
polynomial x2 -2.
Note that every rational number is algebraic. This follows from the fact that the number
a/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 andare also
transcendental, but the proofs of these facts (which can be found in [HaWr08]) are beyond
the scope of this book.
The Greatest Integer Function
In number theory, a special notation is used for the largest integer that is less than or
equal to a particular real number.
Definition. The greatest integer in a real number x, denoted by [x], is the largest integer
less 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 of
using 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 ceiling
function of a real number x, denoted by [x], is the smallest integer greater than or equal
to x. For example, [5/2] = 3 and [-5/2] = -2.
The greatest integer function arises in many contexts. Besides being important in
number theory, as we will see throughout this book, it plays an important role in the
analysis of algorithms, a branch of computer science. The following example establishes
![*
1.1 Numbers and Sequences
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.
c) [22/7]
d) [-2]
a) [1/4]
b) [-3/4]
00e) [[1/2] + [1/2]]
f) [-3+ [-1/2]]
8. Find the following values of the greatest integer function.
a) [-1/4]
c) [5/4]
b) [-22/7]
d) [[1/2]]
e) [[3/2]+[-3/2]]
f) [3 [1/2]]
9. Find the fractional part of each of these numbers:
a) 8/5
b) 1/7
c) - 11/4
10. Find the fractional part of each of these numbers:
a)-8/5
b) 22/7
c) - 1
d) 7
d) - 1/3
11. 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 when
both 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 from
x, it is the larger of the two).
18. Show that if m and n are integers, then [(x + n)/m] = [([x] + n)/m] whenever x is a real
number.
* 19. Show that [V[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]
13
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 as
follows.lvollok ah Indons
a) 3, 11, 19, 27, 35, 43, 51, 59, 67, 75
c) 1, 0, 0, 1, 0, 0, 0, 0, 1, 0
b) 5, 7, 11, 19, 35, 67, 131, 259, 515, 1027 d) 1, 3, 4, 7, 11, 18, 29, 47, 76, 123
a) 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366
b) 1, 1, 0, 1, 1, 0, 1, 1, 0, 1
22. Conjecture a formula for the nth term of {a} if the first ten terms of this sequence are as
follows.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9acf228d-ec9e-49d8-a78a-8ceb13ba5105%2F8da5c926-9f5d-4ca6-a29a-acffef56241b%2Fw8mmmds_processed.jpeg&w=3840&q=75)
Transcribed Image Text:*
1.1 Numbers and Sequences
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.
c) [22/7]
d) [-2]
a) [1/4]
b) [-3/4]
00e) [[1/2] + [1/2]]
f) [-3+ [-1/2]]
8. Find the following values of the greatest integer function.
a) [-1/4]
c) [5/4]
b) [-22/7]
d) [[1/2]]
e) [[3/2]+[-3/2]]
f) [3 [1/2]]
9. Find the fractional part of each of these numbers:
a) 8/5
b) 1/7
c) - 11/4
10. Find the fractional part of each of these numbers:
a)-8/5
b) 22/7
c) - 1
d) 7
d) - 1/3
11. 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 when
both 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 from
x, it is the larger of the two).
18. Show that if m and n are integers, then [(x + n)/m] = [([x] + n)/m] whenever x is a real
number.
* 19. Show that [V[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]
13
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 as
follows.lvollok ah Indons
a) 3, 11, 19, 27, 35, 43, 51, 59, 67, 75
c) 1, 0, 0, 1, 0, 0, 0, 0, 1, 0
b) 5, 7, 11, 19, 35, 67, 131, 259, 515, 1027 d) 1, 3, 4, 7, 11, 18, 29, 47, 76, 123
a) 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366
b) 1, 1, 0, 1, 1, 0, 1, 1, 0, 1
22. Conjecture a formula for the nth term of {a} if the first ten terms of this sequence are as
follows.
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