Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
expand_more
expand_more
format_list_bulleted
Related questions
Question
Transcribed Image Text:**Problem Statement:**
20. Prove: Let \( x \) and \( y \) be real numbers. If \( x \) is rational and \( y \) is irrational, then \( x + y \) is irrational.
**Explanation:**
This problem requires a proof in mathematical logic. The objective is to demonstrate that the sum of a rational number \( x \) and an irrational number \( y \) always results in an irrational number.
Key Points to Consider:
- A **rational number** can be expressed as the quotient of two integers (i.e., \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \)).
- An **irrational number** cannot be expressed as a simple fraction and has a non-repeating, non-terminating decimal expansion.
Steps for Proof:
1. Assume \( x \) is rational and can be written as \( \frac{m}{n} \) where \( m \) and \( n \) are integers, \( n \neq 0 \).
2. Assume \( y \) is irrational and cannot be expressed as a fraction.
3. Suppose \( x + y \) is rational. Then \( x + y = \frac{p}{q} \) for some integers \( p \) and \( q \).
4. Rearranging gives \( y = \frac{p}{q} - \frac{m}{n} \).
5. The right side of this equation is a difference of two rational numbers, which must be rational.
6. This contradicts the assumption that \( y \) is irrational.
Conclusion: The assumption that \( x + y \) is rational leads to a contradiction, hence \( x + y \) must be irrational. This completes the proof.
Expert Solution
arrow_forward
Step 1
Step by stepSolved in 2 steps with 2 images
Knowledge Booster
Similar questions
- 7. - Let I be the set of real numbers that are not rational; elements of I are called irrational numbers. Prove if a < b. then there exists x € I such that a< x < b. Hint: First show that {r+ √2; re Q} CI.arrow_forward6. Prove that x is a rational number if and only if 2x - 1 is a rational number.arrow_forward3. Prove or disprove: there exist rational numbers a and b such that a/b is irrational.arrow_forward
- use the contrapositive to prove: for all x in the positive real numbers, if x is irrational, then the square root of x is irrational. You will need to use the following consequence of the closure properties for the rational numbers: if x is rational, then x2 is irrational.arrow_forwardProve that if x is an irrational number, then √√x + 2 is also irrational number. State the type of the proof you are using and show all steps in all details. Your proof:arrow_forwardFind the matrix of the linear transformation T from R? R?, where T([1, 0]) = [1, -2], and T([2, 1]) = [2, 3] (Note: matrix A = [T([1,0]), T([0, 1])] ). 1 (a) 1 (b) 1. (c) -2 1 -2 3 -2 1 (d) 0. 3arrow_forward
- Show that if r is a rational number, then 2r2+3r+ 1 is rational as well.arrow_forwardLet Q be the set of all rational numbers. Which of these equations has no solution in Q? I: x? – 2 = 0 A. I and II only B. II only С. Ш only D. II and III only II: 2x2 +1 = 7 III: x3 – 4 = -31 E. None of the above.arrow_forwardHelp mearrow_forward
arrow_back_ios
arrow_forward_ios
Recommended textbooks for you
- Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
Numerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill Education
Introductory Mathematics for Engineering Applicat...Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEY 
Mathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,
Advanced Engineering Mathematics
Advanced Math
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:9780073397924
Author:Steven C. Chapra Dr., Raymond P. Canale
Publisher:McGraw-Hill Education
Introductory Mathematics for Engineering Applicat...
Advanced Math
ISBN:9781118141809
Author:Nathan Klingbeil
Publisher:WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:9781337798310
Author:Peterson, John.
Publisher:Cengage Learning,