Prove the Following Statement Directly from the Deﬁnitions of the Terms

Question

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The Answer is Proof: Suppose

r

and

s

, are any rational numbers with

s eq 0

.

/

We must show that

\frac { 2 r } { 5 s }

is a rational number./By definition of rational, there exist integers

a , b , c

, and

d

with

b eq 0

and

d eq 0

such that

r = \frac { a } { b } \quad ext { and } \quad s = \frac { c } { d }

In addition, since

c = s \cdot d

and since neither

s

nor

d

equals 0 , then

c eq 0

by the zero product property. Then

\frac { 2 r } { 5 s } = \frac { 2 \cdot \frac { a } { b } } { 5 \cdot \frac { c } { d } } = \frac { 2 a d } { 5 b c }

Now both

2 a d

and

5 b c

are integers because they are products of integers, and

5 b c eq 0

by the zero product property. Hence

\frac { 2 r } { 5 s }

is a ratio of integers with a nonzero denominator, and so

\frac { 2 r } { 5 s }

is a rational number by definition of rational [as was to be shown].

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