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

1 Answer
Jul 16, 2015

Answer:

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#