Question

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

Answer 1

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

Explanation:

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`