Because I haven't done it yet: 0.999... = ?

[packbat]

Point nines recurring^[1] equals one.

I understand that this claim is intuitively displeasing to many intelligent people. "Point nine recurring doesn't equal one!", they say, and give various reasonable accounts of why the two should be distinguished. However, as an engineer, I submit to the reader that we should equate them, however displeasing it should appear, on the following practical grounds.

Decimal notation is an incredibly effective way of describing numbers. (Or, more accurately, positional notation is. We use decimal specifically because we have ten fingers. I think senary or quarternary notation would be a superior substitute, but I'm not in charge.) Using a small set of symbols and a few characters, we can represent in a very close-grained fashion numbers of a huge span of orders of magnitude - and even larger if we permit exponential notation (e.g. 6.022 * 10²³). Further, in decimal notation, comparison of the magnitudes of numbers is dead simple. (Is 14/25 larger than 9/16? 14/25 = .56 and 9/16 = .5625, so no. Compare that with performing the fraction subtraction!) However, it pays for these advantages in the corresponding flaw: there are many^[2] numbers which cannot be written in standard decimal notation. Included among these are the majority of fractions. A good example of this is 1/3.

1/3 = 10/30
    = 9/30 + 1/30
    = 9/3 * 1/10 + 1/3 * 1/10
    = 3 * 1/10 + 1/3 * 1/10
    = 3 * 1/10 + 3 * 1/10 * 1/10 + 1/3 * 1/10 * 1/10
    = 3 * 1/10 + 3 * 1/102 + 3 * 1/103 + ...

As is obvious, there's no way to write this completely out as a decimal, like you could with 14/25 and 9/16. So, what do you do? You make up a new notation. Instead of trying to hit the end, you go until you've got the pattern, and then...

1/3 = 3 * 1/10 + 3 * 1/102 + 3 * 1/103 + ...
    = .3 + .03 + .003 + ...
    = .333...

Now, recall, there's no requirement that this notation exist. If, for example, you were programming a digital computer, you might choose not to bother with this notation, as you would never store an infinite decimal. However, it is convenient, defined this way. One especially convenient thing about it, for example, is that one can perform the regular arithmetic operations without trouble:

2/3 = 2 * 1/3
    = 2 * .333...
    = .666...

4/3 = 4 * 1/3
    = 4 * .333...
    = 1.2
     + .12
      +.012
       +...
    = 1.333...

Et cetera.

However, the other consequence is the following.

3/3 = 3 * 1/3
    = 3 * .333...
    = .999...
 and
3/3 = 1
 thus
 1  = .999...

So, one is left with the dilemma. Preserve this useful notation, or throw it out to avoid this (to some) distasteful equality? And, looking at it like that, is it really any sort of choice?

Thus, point nine recurring equals one. Q.E.D.

1. Some may say, for example, "repeating", rather than "recurring". (Google suggests the terms are approximately equal in popularity.) Naturally, the specific term is irrelevant. ^
2. "Many", in this case, meaning "an infinite number of" - specifically, 2 raised to the (infinite) number of counting numbers. What aleph infinity this is depends on whether you accept the continuum hypothesis. ^

Comments

[roaminrob.livejournal.com]

Or, to put it another way:

A mathematician walks in to Euclid's Machining, Inc., and speaks to the head machinist. "Sir," he says, "I'm working on a project, and I need a cylinder and a donut, CNC machined out of stock aluminum. The donut must have an inside diameter of 1.000 inches, and the cylinder must have a diameter of .999 inches."

"No problem," says the machinist, "but it'll cost you a fair bit for precision to thousandths of an inch." The mathematician agrees to the cost, and decides to return the following week for the completed pieces.

When he gets returns, he tries desperately to fit the two pieces together, and furiously approaches the machinist. "I want my money back! These parts aren't within tolerances!"

"Hey," shrugs the machinist, "you're the one that thinks .9999 is the same as 1 inch."