Theorem 4.8.1 Irrationality of V2 V2 is 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
Related questions
Question
Suggest you refer to Theorem 4.8.1.
Please help me prove this using Theorem 4.8.1
![Theorem 4.8.1 Irrationality of V2
V2 is irrational.
Proof (by contradiction): [We take the negation and suppose it to be true.] Suppose
not. That is, suppose V2 is rational. Then there are integers m and n with no com-
mon factors such that
m
4.8.1
n
[by dividing m and n by any common factors if necessary]. [We must derive a contradic-
tion.] Squaring both sides of equation (4.8.1) gives
bonsh
m2
2 =
Or, equivalently,
m2 = 2n².
4.8.2
Note that equation (4.8.2) implies that m is even (by definition of even). It follows
that m is even (by Proposition 4.7.4). We file this fact away for future reference and
also deduce (by definition of even) that
2k for some integer k.
4.8.3
m =
Substituting equation (4.8.3) into equation (4.8.2), we see that
² = (2k) = 4k = 2n².
m
Dividing both sides of the right-most equation by 2 gives
n? = 2k?.
Consequently, n is even, and so n is even (by Proposition 4.7.4). But we also know
that m is even. [This is the fact we filed away.] Hence both m andn have a common
factor of 2. But this contradicts the supposition that m and n have no common fac-
tors. [Hence the supposition is false and so the theorem is true.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd34ec7b4-6ecf-40a8-b01d-5afafc11be89%2Fb1463257-e324-4b5a-a6ba-8103ce49f224%2Fzzjbgi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Theorem 4.8.1 Irrationality of V2
V2 is irrational.
Proof (by contradiction): [We take the negation and suppose it to be true.] Suppose
not. That is, suppose V2 is rational. Then there are integers m and n with no com-
mon factors such that
m
4.8.1
n
[by dividing m and n by any common factors if necessary]. [We must derive a contradic-
tion.] Squaring both sides of equation (4.8.1) gives
bonsh
m2
2 =
Or, equivalently,
m2 = 2n².
4.8.2
Note that equation (4.8.2) implies that m is even (by definition of even). It follows
that m is even (by Proposition 4.7.4). We file this fact away for future reference and
also deduce (by definition of even) that
2k for some integer k.
4.8.3
m =
Substituting equation (4.8.3) into equation (4.8.2), we see that
² = (2k) = 4k = 2n².
m
Dividing both sides of the right-most equation by 2 gives
n? = 2k?.
Consequently, n is even, and so n is even (by Proposition 4.7.4). But we also know
that m is even. [This is the fact we filed away.] Hence both m andn have a common
factor of 2. But this contradicts the supposition that m and n have no common fac-
tors. [Hence the supposition is false and so the theorem is true.]
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![Algebra: Structure And Method, Book 1](https://www.bartleby.com/isbn_cover_images/9780395977224/9780395977224_smallCoverImage.gif)
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell
![Elements Of Modern Algebra](https://www.bartleby.com/isbn_cover_images/9781285463230/9781285463230_smallCoverImage.gif)
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![Algebra: Structure And Method, Book 1](https://www.bartleby.com/isbn_cover_images/9780395977224/9780395977224_smallCoverImage.gif)
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell
![Elements Of Modern Algebra](https://www.bartleby.com/isbn_cover_images/9781285463230/9781285463230_smallCoverImage.gif)
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
![Algebra for College Students](https://www.bartleby.com/isbn_cover_images/9781285195780/9781285195780_smallCoverImage.gif)
Algebra for College Students
Algebra
ISBN:
9781285195780
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
![College Algebra (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305652231/9781305652231_smallCoverImage.gif)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
![College Algebra](https://www.bartleby.com/isbn_cover_images/9781337282291/9781337282291_smallCoverImage.gif)