Q:

Counting Squares In a Range

I've been working on a series of programming challenges to work on my math skills, and I came across a solution that I don't know how to explain.
The problem: 

"For 1 <= A <= B <= 10^9, find the number of perfect squares between
  $A$ and $B$ (inclusive).

My initial solution was to loop through all the numbers in a range and count the perfect squares that way. However, this was obviously a very slow solution (~ 14 seconds for range 1 -> 10^9).
Then, I came across the following:

"The number of squares between $A$ and $B$ = sqrt(B) - sqrt(A - 1) rounded
  down". 

NOTE: From an answer to this question, rounded down is incorrect in this case, but this formula holds for finding the perfect squares in a given range. The code for this question actually converts the value to an int, which performs rounding to the nearest integer, not necessarily down.
This is cool and offered a vastly quicker solution, but I don't understand why.
Can anyone help me understand why this simple equation actually works?

A:

I presume $A$ and $B$ are positive integers.
The number of squares $\le B$ is $\lfloor \sqrt{B} \rfloor$. The number of squares $< A$ is the number of squares $\le A-1$, and thus $\lfloor \sqrt{A-1}\rfloor$.  The number between $A$ and $B$ (inclusive) is the difference between these. 
EDIT:
 But this is not the same as " sqrt(B) - sqrt(A - 1) rounded down".  For example, take $A = 3$, $B = 4$.  There is one square ($4$) between $A$ and $B$, inclusive, but $\lfloor \sqrt{4} - \sqrt{2}\rfloor = 0$.

