This is about the legitimacy of a possible operation. Take 0.999999... The operation is defined like this: 1) identify the insertion position with respect to the decimal. in this example we choose "3" as in the third position, 0.999999... 2) from the insertion position, inclusive, to the right, indefinitely, divide each digit by 10. in this example the result is 0.990999... 3) add some value in this example, we add 0.009 so the result is 0.999999... The operation might be specified by the particulars of the three steps. In this example, the thing might look like this: [3, 10, 0.009]->(0.999999...)=0.999999... First question is about the validity of this operation. Second question concerns applying the operation to numbers like this: [2, 10, 0]->(0.001000...)=0.000100... 1) identify the insertion position in this example we choose "2" as in the second position, 0.001000... 2) from the insertion position, inclusive, to the right, indefinitely, divide each digit by ten in this example the result is 0.000100... 3) add some value in this example, we add 0 so the result is 0.000100... Third question is the big question... in this second example the result of repeated applications of the operation [2,10,0] to 0.001000... moves the "1" digit progressively to the right by basically inserting zeros in front of it. In thinking about things like whether 0.999...=1, is it OK to imagine operations that behave like this? Is there a formal name for this? Does it abrogate any canonical math?