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Prove V3 is irrational using the well-ordering principle? Asked today Active today Viewed 1 time I am a begginer. State every step as clearly as possible. Question: Prove v3 is well-ordered using the well-ordering principle? Answer: Suppose /3 is rational. Then there exists integers a, b such that /3 = a/b. Now suppose the minimum integers a, b, which equal /3, are s, t. Then s = /3t. Hence s/3 – s = s/3 – t/3 = (s – t)/3. Since s/3 = (tv3)/3 = 3t is an integer and s is an integer, s/3 – s is an integer. Moreover, it is positive since s/3 > s hence s3 – s > 0 and V3 -1> 0. Therefore /3 > 1. Moreover, /3 < 3, but /3 – 2 < 1 which means s(/3 – 2) < s. This contradicts our hypothesis that s is the smallest integer. Hence there is no smallest integers a and b where 3 = a/b. Therefore, /3 is irrational. %3D The problem is /3 – 2 < 0. It is not the least positive integer. So is my proof correct? If not, what should I have actually done?

Prove V3 is irrational using the well-ordering principle? Asked today Active today Viewed 1 time I am a begginer. State every step as clearly as possible. Question: Prove v3 is well-ordered using the well-ordering principle? Answer: Suppose /3 is rational. Then there exists integers a, b such that /3 = a/b. Now suppose the minimum integers a, b, which equal /3, are s, t. Then s = /3t. Hence s/3 – s = s/3 – t/3 = (s – t)/3. Since s/3 = (tv3)/3 = 3t is an integer and s is an integer, s/3 – s is an integer. Moreover, it is positive since s/3 > s hence s3 – s > 0 and V3 -1> 0. Therefore /3 > 1. Moreover, /3 < 3, but /3 – 2 < 1 which means s(/3 – 2) < s. This contradicts our hypothesis that s is the smallest integer. Hence there is no smallest integers a and b where 3 = a/b. Therefore, /3 is irrational. %3D The problem is /3 – 2 < 0. It is not the least positive integer. So is my proof correct? If not, what should I have actually done?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Prove V3 is irrational using the well-ordering principle? Asked today Active today Viewed 1 time I am a begginer. State every step as clearly as possible. Question: Prove v3 is well-ordered using the well-ordering principle? Answer: Suppose v3 is rational. Then there exists integers a, b such that 3 = a/b. Now suppose the minimum integers a, b, which equal 3, are s, t. Then s = V3t. Hence s/3 – 8 = s/3 – t/3 = (s – t)/3. Since s/3 = (ty3)/3 = 3t is an integer and s is an integer, s3 – s is an integer. Moreover, it is positive since s/3 > s hence sy3 – s > 0 and V3 –1>0. Therefore V3 > 1. Moreover, /3 < 3, but 3 – 2 <1 which means s(V3 – 2) < s. This contradicts our hypothesis that s is the smallest integer. Hence there is no smallest integers a and b where 3 = a/b. Therefore, 3 is irrational. The problem is 3 – 2 < 0. It is not the least positive integer. So is my proof correct? If not, what should I have actually done? elementary-number-theory solution-verification alternative-proof Share Cite Edit Delete Flag asked just now Arbuja 68 03 111 A37
Transcribed Image Text:Prove V3 is irrational using the well-ordering principle? Asked today Active today Viewed 1 time I am a begginer. State every step as clearly as possible. Question: Prove v3 is well-ordered using the well-ordering principle? Answer: Suppose v3 is rational. Then there exists integers a, b such that 3 = a/b. Now suppose the minimum integers a, b, which equal 3, are s, t. Then s = V3t. Hence s/3 – 8 = s/3 – t/3 = (s – t)/3. Since s/3 = (ty3)/3 = 3t is an integer and s is an integer, s3 – s is an integer. Moreover, it is positive since s/3 > s hence sy3 – s > 0 and V3 –1>0. Therefore V3 > 1. Moreover, /3 < 3, but 3 – 2 <1 which means s(V3 – 2) < s. This contradicts our hypothesis that s is the smallest integer. Hence there is no smallest integers a and b where 3 = a/b. Therefore, 3 is irrational. The problem is 3 – 2 < 0. It is not the least positive integer. So is my proof correct? If not, what should I have actually done? elementary-number-theory solution-verification alternative-proof Share Cite Edit Delete Flag asked just now Arbuja 68 03 111 A37
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