Between
two rational numbers there
are infinite
rational number
justify them?
Answers
Step-by-step explanation:
There are infinitely many rational numbers between 0 and 1 . At the very least, we have 1/n , where n∈{2,3,4,…} . More can be constructed explicitly, but this will do.
Consider any two distinct rational numbers a/b and c/d , where ad≠bc .
We can bijectively map a/b to 0 and c/d to 1 by f(x)=x−a/bc/d−a/b .
The inverse of f is f−1(y)=a/b+(c/d−a/b)y , which maps 0 to a/b and 1 to c/d .
Thus every distinct rational between 0 and 1 we can map to a distinct rational between a/b and c/d . Since there are infinitely many in the first interval, there are infinitely many in the second.
There are other ways to construct further rational numbers between pairs of distinct rational numbers, such as taking the mean of the two, then taking the mean of the new number and one of the end points, and so on.
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Answer:
There are infinitely many rational numbers between 0 and 1 . At the very least, we have 1/n , where n∈{2,3,4,…} . More can be constructed explicitly, but this will do.
Consider any two distinct rational numbers a/b and c/d , where ad≠bc .
We can bijectively map a/b to 0 and c/d to 1 by f(x)=x−a/bc/d−a/b .
The inverse of f is f−1(y)=a/b+(c/d−a/b)y , which maps 0 to a/b and 1 to c/d .
Thus every distinct rational between 0 and 1 we can map to a distinct rational between a/b and c/d . Since there are infinitely many in the first interval, there are infinitely many in the second.
There are other ways to construct further rational numbers between pairs of distinct rational numbers, such as taking the mean of the two, then taking the mean of the new number and one of the end points, and so on.