it's a notation problem. people can't understand that .9... (repeating or bar or whatever u want to call it) is the same as writing the number 1.
what do you mean notation problem? i mean that the bar (-) above the nine (9), or the elipsis (...) after the nine is a mathematical operator, just like the division (/) or factorial (!) operator.
People understand that 1/3 can also be written as a never-ending series of 3s after a decimal because they understand that 1/3 means you have to perform an operation on the numbers 1 and 3.
Similarly, some operators only take a single input !4 means that you use the number 4 as the input for the factorial operator. Which gives !4 the numerical meaning of 24. (1 x 2 x 3 x4 = 24)
What confuses people is that the bar (-) above a decimal or the elipsis (...) after the decimal is an operator that takes one input. In the case of 0.9... the one input is the 9 and the operator is telling you to repeat the nine indefinitely. There is a slight difference here, in that the operators here are more lexigraphical operators than mathematic ones. The elipsis or the bar don't tell you to perform a mathematical operation, they are simply a short-hand to tell you that the rest of the number is implied but has not been written.
The good news is, that .9... is not the result of some measurement, but rather an abstract idea, so it's not going to cause us too much real-world trouble. But if people were to take some finite measurement of .9999999999999999999999999999999999999 no matter how many nines there were, as long as the list is finite, .99999 will never be equal to one.
