Question

give a rational number between root 5 and root 6

Answer 1

Answer:

There are infinitely many such numbers — rationals between 5 and 6.

One way to show that these are infinite in number is to start by splitting the difference, getting 5.5.

A side effect is to create two “halves,” from 5 to 5.5, and from 5.5 to 6.

Split the difference again, in each half, producing 5.25 and 5.75.

Repeat the process again, getting another four numbers, each time splitting the difference. The process can be repeated indefinitely, producing infinitely many rational numbers.

However, even if this process is repeated forever, it only ends up finding an infinitesimal subset of all rationals between 5 and 6. There are ways of comprehensively listing ALL such rationals — because after all it is a “countable” set — but surely you can at least see that the number of rationals between 5 and 6 is infinite.