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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
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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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.
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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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.
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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