Prove the following statement directly from the definition of rational number. The difference of any two rational numbers is a rational number. Proof: Supposer and s are any two rational numbers. By definition of rational, r = and s= for some --Select--- b Writers in terms of a, b, c, and d as a quotient of two integers whose numerator and denominator are simplified as much as possible. The result is the following. r-s= a, b, c, and d with -Select-- Both the numerator and the denominator are integers because ---Select--- In addition, bd 0 by the -Select- Hence r-s is a -Select- of two integers with a nonzero denominator, and so by definition of rational, r-s is rational.
Prove the following statement directly from the definition of rational number. The difference of any two rational numbers is a rational number. Proof: Supposer and s are any two rational numbers. By definition of rational, r = and s= for some --Select--- b Writers in terms of a, b, c, and d as a quotient of two integers whose numerator and denominator are simplified as much as possible. The result is the following. r-s= a, b, c, and d with -Select-- Both the numerator and the denominator are integers because ---Select--- In addition, bd 0 by the -Select- Hence r-s is a -Select- of two integers with a nonzero denominator, and so by definition of rational, r-s is rational.
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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Transcribed Image Text:Prove the following statement directly from the definition of rational number.
The difference of any two rational numbers is a rational number.
Proof: Suppose r and s are any two rational numbers. By definition of rational, r = a
b
and s= for some ---Select---
d
Writer-s in terms of a, b, c,, and d as a quotient of two integers whose numerator and denominator are simplified as much as possible. The result is the following.
r-s=
Both the numerator and the denominator are integers because ---Select---
In addition, bd # 0 by the--Select--
Hencer - s is a ---Select--of two integers with a nonzero denominator, and so by definition of rational, r s is rational.
a, b, c, and d with ---Select--
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