Show that there is a rational number and an irrational

Rapsinincke 2022-07-01 Answered
Show that there is a rational number and an irrational number between any two real numbers.
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Answers (1)

fugprurgeil
Answered 2022-07-02 Author has 12 answers
We know that Q is dense in R . This means for any x R and for any r > 0 , Q B ( x ; r ) ϕ where B ( x ; r ) = { y R : | x y | < r }.
Let a , b R be such that a < b. Then we have Q B ( a + b 2 ; b a 3 ) ϕ by above statement. This simply means there is at least one rational number between a and b.
Similarly, R Q is also dense in R . By the similar process as above , one can easily derive that there is at least one irrational number between any two distinct real numbers.

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