Solution Found!
Find the mistakes in the “proofs” that the sum of any two
Chapter 4, Problem 39E(choose chapter or problem)
In 35–39 find the mistakes in the “proofs” that the sum of any two rational numbers is a rational number.
“Proof: Suppose r and s are rational numbers. If r + s is rational, then by definition of rational r + s = a/b for some integers a and b with b \(\neq\) 0. Also since r and s are rational, r = i/j and s = m/n for some integers i, j,m, and n with j \(\neq\) 0 and n \(\neq\) 0. It follows that
\(r+s=\frac{i}{j}+\frac{m}{n}=\frac{a}{b}\),
which is a quotient of two integers with a non zero denominator. Hence it is a rational number. This is what was to be shown.”
Text Transcription:
neq
r + s = i/j + m/n = a/b
Questions & Answers
QUESTION:
In 35–39 find the mistakes in the “proofs” that the sum of any two rational numbers is a rational number.
“Proof: Suppose r and s are rational numbers. If r + s is rational, then by definition of rational r + s = a/b for some integers a and b with b \(\neq\) 0. Also since r and s are rational, r = i/j and s = m/n for some integers i, j,m, and n with j \(\neq\) 0 and n \(\neq\) 0. It follows that
\(r+s=\frac{i}{j}+\frac{m}{n}=\frac{a}{b}\),
which is a quotient of two integers with a non zero denominator. Hence it is a rational number. This is what was to be shown.”
Text Transcription:
neq
r + s = i/j + m/n = a/b
ANSWER:Solution:
Step 1
In this problem we have to find the mistakes in “proofs’’ that the sum of any two rational numbers is a rational number.