I'm talking about integer numbers. 2^53 = 9007199254740992. You can do any arithmetic operations with any integer number from -9007199254740992 to 9007199254740992 and results will be correct. E.g. 9007199254740991 + 1 = 9007199254740992. But outside of that range there will be errors, e.g. 9007199254740992 + 1 = 9007199254740992
You are are describing only one of the types of numerical error that can occur, and it is not commonly a problem: this is only an edge case that occurs at the significand limit where the exponent alone must be used to approximate larger magnitudes at which point integers become non-contiguous.
The types of errors being discussed by others are all in the realm of non-integer rationals where limitations in either precision or representation introduce error and then compound in operations no matter the order of magnitude... and btw _real_ life tends to contain _real_ numbers, that commonly includes rationals in use of IEEE 754.
