Which pair of number does square root 105 fall between?

Answer 1

There is no unique pair of numbers that satisfies these requirements.

Suppose a and b is such a pair, and sqrt(105) = x

then you want a < x < b

But a < (a+x)/2 < x < (b+x)/2 < b

So that (a+x)/2 and (b+x)/2 are a closer pair.

and you can then find a closer pair still - ad infinitum.

The question can be answered (sort of) if it asked about "integers" rather than "numbers".

100 < 105 < 121

Taking square roots, this equation implies that

10 < sqrt(105) < 11

so the answer could be 10 and 11.

But (and this is the reason for the "sort of") the above equation also implies that

-11 < sqrt(105) < -10

giving -11 and -10 as a pair of consecutive integers.

So, an unambiguous answer is possible only if the question specifies positive integers.