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

Jul 16, 2015

The average $3 \frac{1}{2}$ 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: $\frac{{x}_{1} + {x}_{2}}{2}$ is a real number strictly between them.

${x}_{1} < \frac{{x}_{1} + {x}_{2}}{2} < {x}_{2}$

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:

$\left\lceil x \right\rceil = n$ such that $n - 1 < x \le n$

Let $q = 2 \cdot \left\lceil \frac{1}{{x}_{2} - {x}_{1}} \right\rceil$

and $p = \left\lceil q {x}_{1} \right\rceil + 1$

Then

${x}_{1} < \frac{p}{q} < {x}_{2}$