How do you find a "REAL NUMBER" between pairs of numbers 3 1/3 and 3 2/3?

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Answer 1 · 2015-07-16T08:31:41

The average $3 1/2$ is a real number between the two numbers.

If $x_1$ and $x_2$ are any pair of real numbers with $x_1 < x_2$ then their average: $(x_1 + x_2) / 2$ is a real number strictly between them.

$x_1 < (x_1 + x_2) / 2 < x_2$

In fact, you can always find a rational number strictly in between $x_1$ and $x_2$ :

In case you have not encountered it before, the ceiling function maps a number to the least integer that is greater or equal to it:

$ceil(x) = n$ such that $n - 1 < x <= n$

Let $q = 2*ceil(1/(x_2 - x_1))$

and $p = ceil(q x_1) + 1$

Then

$x_1 < p/q < x_2$