Why do decimal places "move" when multiplying or dividing decimals? Suppose we have the two decimal values, 0.2 and 0.3. Of course, multiplying these values yields the result: 0.06. Of course, this is simply the calculation 2*3 with the decimal place moved according to the sum of the number of significant digits after the decimal of our two original values.
Similarly, many math tutorial websites offer the following explanations:
http://www.math.com/school/subject1/lessons/S1U1L5GL.html

Place the decimal point in the answer by starting at the right and moving a number of places equal to the sum of the decimal places in both numbers multiplied.

https://www.mathsisfun.com/multiplying-decimals.html

...put the decimal point in the answer - it will have as many decimal places as the two original numbers combined.

...etc.
However, why is this done? What is the mathematical reasoning for why these decimal points "magically" move when multiplying or dividing decimal values?
 A: It's simply a matter of counting how many factors of 10 appear in the denominator after the multiplication. Each factor of 10 in the denominator moves the decimal point one place to the left.
So you may ask, why do we have to count the factors of 10 and move the decimal point? The reason is that when you replace "0.2" by "2", you really multiplied 0.2 by 10 -- so you have to divide by 10 again in the end to have a net result of no change. Same for your conversion of "0.3" into "3"--that produces another factor of 10 you introduced, so you have to divide by 10 to undo that one as well.
The same reasoning shows that if you are multiplying $0.002\times 0.0003$, you can multiply $2\times 3$ to get $6$, and then move the decimal point 7 places to the left. That's because converting "$0.002$" to "$2$" introduced 3 factors of 10, and converting "$0.0003$" to "$3$" introduced 4 factors of 10, so you must eventually divide by 10 a total of $3+4=7$ times -- that is, move the decimal point to the left 7 times: $6 \to 0.0000006$ .
