There are two facts concerning the square of an integer that are useful in the inverse process of finding the square root. The first concerns the number of digits. In short:
If 0 < a < 10 then 0 < a2 < 100,
if 10 < a < 100 then 100 < a2 < 10 000,
if 100 < a < 1000 then 10 000 < a2 < 1 000 000,
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The point to see here is that the square of an integer has either twice as many digits as the integer itself, or one less than twice as many. So, since 9,409 has 4 digits its square root has 2 digits, and the square root of the 13 digit number 3,871,696,594,290 has 7 digits.
The second fact concerns the "geometry" of squaring a number. Consider, for example, the square of 249 so the geometric object to consider is a square with side of length 249 units. Write 249 as 200 + 40 + 9 then the square can be seen as
Now to find the square root of an integer you need first to determine the number of digits there will be. Let us find the square root of 64 009. Since 64 009 has 5 digits, the square root of 6 | 4 0 | 09 will have 3 digits. First recall some of the geometric detail of a 3 digit number.
A square of a 3 digit number is divided into 7 parts as shown in the diagram
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