Given the properties of the natural numbers N and integers N (i) m, n € Z➡➡m+n, m-n, mn € Z (ii) If m € Z, then m EN m≥1 (iii) There is no m € Z that satisfies 0 np for n < 0,q> 0. (d) Show that the sum a rational number and an irrational number is always irrational.
Given the properties of the natural numbers N and integers N (i) m, n € Z➡➡m+n, m-n, mn € Z (ii) If m € Z, then m EN m≥1 (iii) There is no m € Z that satisfies 0 np for n < 0,q> 0. (d) Show that the sum a rational number and an irrational number is always irrational.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 92E
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Question 1
![Given the properties of the natural numbers N and integers N
(i) m, n € Z ⇒ m+n,m-n, mn € Z
(ii) If mEZ, then meN
m≥1
(iii) There is no m€ Z that satisfies 0 <m < 1.
For integers m, n, p, q E Z, n, q0, use this and the field axioms
(a) Provem + P
mg+np and.
mp
9
ng
9
nq
(b) Show that the field axioms also hold for rational numbers Q.
(c) Prove <
2
mq> np for n < 0,q> 0.
(d) Show that the sum a rational number and an irrational number is always irrational.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4cb4d097-3492-4cc6-a3d0-40439978c013%2F0b2dbd40-76cd-410e-a731-95107f7c252e%2Fsd1cr0a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Given the properties of the natural numbers N and integers N
(i) m, n € Z ⇒ m+n,m-n, mn € Z
(ii) If mEZ, then meN
m≥1
(iii) There is no m€ Z that satisfies 0 <m < 1.
For integers m, n, p, q E Z, n, q0, use this and the field axioms
(a) Provem + P
mg+np and.
mp
9
ng
9
nq
(b) Show that the field axioms also hold for rational numbers Q.
(c) Prove <
2
mq> np for n < 0,q> 0.
(d) Show that the sum a rational number and an irrational number is always irrational.
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Assistance with (a) to (d)
![Given the properties of the natural numbers N and integers N
(i) m, n € Z ⇒ m+n,m-n, mn € Z
(ii) If mEZ, then meN
m≥1
(iii) There is no m€ Z that satisfies 0 <m < 1.
For integers m, n, p, q E Z, n, q0, use this and the field axioms
(a) Provem + P
mg+np and.
mp
9
ng
9
nq
(b) Show that the field axioms also hold for rational numbers Q.
(c) Prove <
2
mq> np for n < 0,q> 0.
(d) Show that the sum a rational number and an irrational number is always irrational.](https://content.bartleby.com/qna-images/question/4cb4d097-3492-4cc6-a3d0-40439978c013/ed2d3661-d2ec-43fe-944b-739e43b85668/w4cfjzo_thumbnail.jpeg)
Transcribed Image Text:Given the properties of the natural numbers N and integers N
(i) m, n € Z ⇒ m+n,m-n, mn € Z
(ii) If mEZ, then meN
m≥1
(iii) There is no m€ Z that satisfies 0 <m < 1.
For integers m, n, p, q E Z, n, q0, use this and the field axioms
(a) Provem + P
mg+np and.
mp
9
ng
9
nq
(b) Show that the field axioms also hold for rational numbers Q.
(c) Prove <
2
mq> np for n < 0,q> 0.
(d) Show that the sum a rational number and an irrational number is always irrational.
Solution
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