## Archive for the ‘Everything and More’ Category

### Is there some number that’s larger than .999… but smaller than 1.0?

October 4, 2007

We can show that the repeating decimal .999… is equal to 1.0 with only a couple of legal moves:

If x = .999…, then 10x = 9.999…

Now subtract x from 10x:

9.999999….

– 0.999999…

You get 9x = 9.0 and thus x = 1

So the repeating decimalĀ  .999… equals 1.0!

Is there some number that’s larger thanĀ  .999… but smaller than 1.0?

Such a number would involve an infinitesimal, meaning a literally infinitely small mathematical entity.

— From Everything and More by David Foster Wallace

### Ascending the staircase of abstraction …

October 3, 2007

Abstract thinking tends most often to strike during moments of quiet repose. As for example the early morning, especially if you wake up slightly before your alarm goes off, when it can suddenly and for no reason occur to you that you’ve been getting out of bed every morning without the slightest doubt that the floor would support you.

Lying there your mind engages in this train of thought:

It appears at least theoretically possible that some flaws in the floor’s construction could make it buckle.

Am I truly justified in my confidence about the floor? Yes, I have gotten out of bed in the morning thousands of times so far, and each time the floor supported me. It’s the same way I feel justified in believing that the sun will come up. Because they’ve happened over and over before.

What principle am I using? The principle involved is really the only way I can predict any of the phenomena I just automatically count on without having to think about them. And the vast bulk of daily life is composed of these sorts of phenomena; and without this confidence based on past experience I’d be unable to function because I’d have to stop and deliberate about every little last thing. It’s a fact: life as I know it would be impossible without this confidence.

Still, though: Is this confidence actually justified, or just highly convenient?

Notice in the above description the staircase that I am ascending in my thoughts. With each step I am going to higher levels of abstraction:

• I started by considering whether the floor would hold me, then
• … step … I moved up to thinking about how I assume things will occur in the future because they have occured in the past, then
• … step … I moved up to thinking about the principle being used in this assumption, and then
• … step … I moved up still another step in my thinking by asking whether this principle was justified.

— Extracted from Everything and More by David Foster Wallace