Confirm that if x is rational and x neq 0, then frac{1}{x} is rational.

iohanetc 2021-02-13 Answered
Confirm that if x is rational and x0, then 1x is rational.
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insonsipthinye
Answered 2021-02-14 Author has 83 answers
Given:
x is rational
x0
TO PROOF:1x is rational
DIRECT PROOF
Property rational numbers:
if a is a rational number, then there exists two integers y and z such that
a=yz(withz0).
x is an rational number and thus there exists integers y and z such that:
x=yz
Since x0, we then also know that the numerator cannot be zero
y0
We are interesred in 1x:
1x=1yz=z/y
Since z and y are integers(with y0), we then know that 1x is rational.
Result: if x is rational and x0, then 1x is rational.
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