Quote:
|
What you have designed above is a limit---not a ratio of two integers, but a rational expression that approaches a real number. An infinitely long integer then is not an integer proper---it's infinity---and hence not in the set of integers.
|
I agree, as long as you specify that the numerator never starts repeating. If it starts repeating, then you do indeed have a rational number, because the unique sequence of digits is finite (even though the total number of digits in the numerator is infinite).
In fact you can express the number as the ratio of the unique sequence of digits (numerator) to a long string of 9s (denominator). So, for example, 1/7 can be expressed as .142856 repeated infinitely, or it can be expressed as 142856/999999. As soon as you have a repeat, you have two integers whose ratio is the number.
Or to put it another way, if your sequence doesn't repeat, then the numerator has to be infinite. If the numerator is infinite, then that number is not in the set of integers.