Introduction:
A square number is a number that is obtained by multiplying a number by itself. In simple terms, it is the product of a number multiplied by itself. For example, 3 * 3 = 9, so 9 is a square number. In this response, we will explore the concept of the greatest 4-digit square number.

Finding the range of 4-digit numbers:
To find the greatest 4-digit square number, we need to determine the range of 4-digit numbers. A 4-digit number has four places, ranging from 1000 to 9999. So, the range of 4-digit numbers is from 1000 to 9999.

Determining the square root range:
To find the greatest 4-digit square number, we need to determine the range of square roots within the 4-digit number range. Taking the square root of a number helps us determine the value that, when multiplied by itself, gives the original number.

Calculating the square roots:
To calculate the square roots, we need to find the square roots of the smallest and largest 4-digit numbers. The smallest 4-digit number is 1000, and its square root is approximately 31.62. The largest 4-digit number is 9999, and its square root is approximately 99.99.

Finding the greatest square number:
Now that we have the range of square roots, we need to find the square number closest to the higher end of the range, which is 99.99. Since the square root of 100 is 10, the square number closest to 99.99 is 10 * 10 = 100.

Conclusion:
Therefore, the greatest 4-digit square number is 100, as it is the highest square number within the range of 4-digit numbers.