Is one really the Loneliest Number?

Some time ago I was listening to a famous song of some sort. That song that sort of rambles on and on about how lonely the number one is. And I thought about it in my usual way, namely, far too seriously.

It came to me that when you tally instead of using an alphanumeric system, one is indeed quite lonely. It's merely a | all by itself. There aren't any other tally marks to keep it company. Once you get to two || or three ||| the loneliness really breaks down. No one is really lonely at all at four |||| and at five |||| everyone is bound together quite nicely.

So it seems that one is indeed the loneliest number.

However, I noticed something odd once I reached the number six |||| | . While the sixth tally mark can not be said to be truly alone, it is certainly lonelier than five. The sixth mark is not a part of the group, and sits on the sidelines. Still, one remains the loneliest number. While lonely, six has the company of other marks to prevent it from going insane from isolation.

One is the loneliest number.

My mind was not at rest yet. I moved on through to ten |||| |||| to eleven |||| |||| | . It struck me that perhaps, eleven was lonelier than six. While there were more marks to possibly keep the eleventh mark company, there were now two groups of marks that excluded an outcast. Anyone can tell you, having two groups of people shun you is worse than having only one. But, while eleven was certainly lonelier than six, it couldn't be said to be lonelier than one, who doesn't even have contact with other marks to keep it's mind from madness.

Lonelier than thou one is.

I wasn't done yet. If eleven was lonelier than six because eleven was excluded by more groups than six, is the same true for sixteen and eleven? Sixteen |||| |||| |||| | certainly has more groups, and I would say that it certainly must be lonelier than eleven. Despite the deep loneliness that must accompany being so unfortunate as to have to have three groups shun you, one still trumps that.

What a lone ranger that one.

However, if sixteen is lonelier than eleven, then we can also say that twenty-one is lonelier than sixteen, and twenty-six is lonelier than twenty-one and so on and so forth. This gives us an equation: For any number that fits the equation 1 + 5n there is a number 1 + 5(n + 1) which is lonelier (where n is equal to the number of groups the outcast mark is shunned by, but not equal to zero).

But one is still lonelier, right?

We'll see. If we take the limit of the equation 1 + 5n as n goes to infinite, we have a number shunned by an inconceivable number of groups. A number with an extremely profound case of loneliness, which is irrefutably lonelier than any other number fitting the equation 1 + 5n.

One is still the loneliest number.

You could argue that because the theoretical number of the limit of 1 + 5n (as n goes to infinite) is still in the midst of other marks that it could never be as lonely as one is. However, I would argue the point that it seems to be unfathomably more lonely to be surrounded by close knit groups of friends and people and yet be so completely excluded from every last one. To be completely surrounded by something you desperately want, and yet completely unable to acquire is far more sanity destroying than simply being alone. Any school outcast will attest to that.

So the limit of 1 + 5n (as n goes to infinite) is actually the loneliest number.

Unfortunately a song with such lyrics would not likely do very well, and it is additionally impossible to prove empirically that one is the loneliest number or the limit of 1 + 5n as n goes to infinite is the loneliest number. It's impossible because we don't have just one person in the world to monitor for insanity, we have four billion, but four billion is also a far cry short of an infinite number and it's be both difficult and cruel to arrange for the complete shunning of someone with that many people anyway.

So which is it? One or the limit of 1 + 5n as n goes to infinite?

I don't know, take your pick. Obviously someone decided it was worthwhile enough to make their choice into a song. However, despite my preference I don't think I'll write a song about the limit of 1 + 5n as n goes to infinite.