The number 1 needs no explanation. The number .999…, however, requires more thought. Starting down the road of formalization, it represents 9/10 + 9/100 + 9/1000 + … on to infinity. But formally we can’t carry out an infinite summation. Recognizing this, mathematicians have long relied on the notion of a limit. We can take enough terms in the summation 9/10 + 9/100 + 9/1000 + … to make the partial sum as close as we want to 1. In mathematical parlance, “in the limit as we take evermore terms in the summation 9/10 + 9/100 + 9/1000 + … we can get as close as we want to 1.”
The genius of this approach is that we sidestep the notion of performing an infinite summation. We’re always working in the land of the finite. The drawback is that we can’t say that 1 = .999…
It’s important to realize this isn’t a mere technicality. Admittedly, mathematicians will frequently write “1 = .999…” as shorthand, but infinite summations can lead to fundamental problems, which is why the notion of a limit was adopted in the first place.
With the expression .999… looked at through the lens of the limit, we start to see a version of one of Zeno’s famous paradoxes that goes as follows. A man reaches out to a wall. To do so, his hand must reach 9/10 of the way to the wall, then 9/10 of the remaining distance, then 9/10 of the remaining distance, and so on. Looked at in this way, his hand must cover an infinite number of distances, each of which individually has finite length. How, then, can his hand actually reach the wall?
The common response is “That’s what the notion of a limit was introduced for. If you understand limits the paradox disappears.” But that’s absolutely not the case. Translating the notion of a limit to the case of a man reaching toward a wall, a mathematician would say “yes, in reaching toward the wall in the way described, the hand can get as close as desired by passing through a large, though finite, number of distances.” As a mathematician, however, the individual would have to remain silent about actually reaching the wall. Zeno’s paradoxes are alive and well.
So does 1 = .999…? Working with the formal definitions provided by limits, we know the expressions can be used interchangeably. But are they actually equal? If we could carry out an infinite summation we’d find out.