Roger
Retired SS teacher

The question is mine, which I wish to share with my 9th grade granddaughter Does two (2) have a square root or do the numbers just keep going? Are there any other numbers that behave like two when it comes to extracting the square root?

2 does have a square root. Take a square that is one unit by one unit. The length of the diagonal is the square root of 2. A difficulty arises when you try to express this length as a decimal. You get approximately 1.41421... and, as you said, "the numbers just keep going", that is the decimal expansion does not stop after a finite number of steps.

Many numbers have a decimal expansion that does not stop after a finite number of steps. For example  ¹/₇ = 0.142857142857142857... In this case however there is a pattern, the sequence 142857 repeats indefinitely. In the case of the square root of 2 there is no repeating pattern.

As you suggested, 2 is not the only integer whose square root exhibits this behavour. For any integer, if it is a perfect square, like 4, 9, or 64 then its square root is an integer. If it is not a perfect square, like 2, 3 or 12 then its square root does not have a finite or repeating decimal expansion.