To most of us, the number 1 (one) is the “basic number”. When learning basic arithmetic at the start of our school years, it was the beginning of any numerical chain of thought or rudimentary sequence (one to ten, for example). We are implied to say it was the first number any of us understood, and it’s so well-used that it has even been integrated in our everyday grammar. It’s not so much the word itself as it is the property it represents, that allows it such a firm place in our lives, but have you ever stopped and wondered: “When does a number become 1?”

This might seem to be a basic question at first glance, and indeed, it can be. It could logically be answered with a mathematical problem that totals to one. “Well, that’s easy. Just take 20, subtract 19, and there you go.” is a logical answer, as is “If I have 0 of something, and add 1 to it, I‘ll have one.” However, to understand what is technically correct, the mathematic involved have to be a little more complex.

Let’s look at the above examples, and modify them a bit. Suppose that, instead of subtracting 19 from 20, we decided that 19.00000000001 was to be taken. This would leave us with .9999999999, an ugly number to write (or type) out, but a number none the less. Would you consider this abomination equal to 1? Likewise, turning our attention to the second example, if instead of having 1 of an something, I have .9999999999 of a something, does that mean I have 1 (if the “something“ is a physical object, I‘d find it interesting to see what ,9999999999 of, say, a dollar, looks like. Is the corner of the bill ripped off, or is the decimal so small that only a few molecules have been removed? What if I want to buy something that costs a dollar? Does that mean I’d have to dig out a penny because the store won‘t accept my almost one dollar bill?)?

Physically, it is very difficult, if not impossible, to express a decimal result so close to the nearest real number. But in mathematics, the question is already solved, at least for the most part. Such numbers are usually rounded up to 1, unless if a given situation calls for extreme accuracy in calculations, then…science finds a way, I suppose. But this leads us back to our titular question. Where exactly does 1 begin and .9999 end?

One might say that point is whenever you can round up to one, such as at .5, but if the same rules still apply, that would also mean that, when it is possible to round up to .5, it is also possible to round all the way 1; 49999 would round up to .5, which rounds up to 1, but .49 = 1 doesn’t sit very well with me. That’s the same as saying that 49%=100%. Conversely, if we rule that .999999 is the only decimal that can be rounded to 1, we would have to be absolutely sure that, somewhere along the lines of infinite 9’s there’s not an 8 or 7 in sight, and in most of those results, that’s often the case. Even finding a middle ground between the two is rather gray.

This could be chalked up as another curiosity in the constructs we’ve created to manage the world we inhabit. It’s easy to find discontinuity in language, but examples in mathematics are somewhat more jarring, given the empirical correctness it’s so well known for. For now this oddity can stand alongside dividing by zero and -0 (the latter of which actually exists, believe it or not) as the subject of hopeless debate and amusement.

As the old joke goes:

How many mathematicians does it take to screw in a light bulb?

.99999999.…