Arbtirary thoughts on nearly everything from a modernist poet, structural mathematician and functional programmer.

Monday, October 15, 2007

On .(9); that is: On 1

I was driving today (That is, when I originally wrote this) and I realized a flaw in a contradiction of a specific proof that 0.999...=1

A whole argument came flashing back to me. I was part of an argument on an online forum on the topic. There were actually mathematicians trying to argue that .99... does not equal 1, which, to me, sounds kind of like a biologist claiming that dolphins are not mammals because they live under water.
I assure you: they are.

The claim that the number in question is actually two numbers generally comes from a misunderstanding of infinity. Firstly, (.9...thly?) we must make sure we make a clarification. This argument is happening in the real number system. Hyperreal, superreal, surreal, and any other class/field of numbers have no place in this argument, because they different systems.

Next: infinity is not "the largest numebr ever." It is NOT a member of the real numbers. When invoking infinity within the reals, it is a concept: "greater than any real number". This is not to be mistaken for "the greatest number". Infinity+1=infinity. Period. Infinity-1=infinity.
It does not equal some number less than infinity, because infinity cannot relate to other numbers: it is like trying to walk from Hawaii to the Continent. As further explanation, infinity is not actually included in the set we shall call R (the set of all reals); it is, however, included in the affinitely extended real number line. That is {negative infinity, {R}, positive infinity} Arithmetic with infinity (which will be w) is defined in this (which I will call R, if I reference it again in this post) as follows:

x + w = w
- x = w
x*w = w Unless x = 0, in which case.. things go funny.
x/w = 0 Unless x=0, in which case, see above.
There are a few more, and these all work with -w as well, but, well... you get the point.

So, one last attempt to concretely define a purely abstract concept. Infinity never ends. There is no "last term" in an infinite sequence.

So... on to proofs:
.99... can be thought of as the sum of an infinite series:
9*(.1^n). So, there is a number with an INFINITE number of 9's. Now, the difference between 1 and this series can be thought of as a complementary sequence: 1*(.1^n). Notice, not the sum (that would be a series), but a sequence. So, when there is one 9 in the sequence (.9), the difference has no 0s followed by a one(1-.9=.1), when there are two 9s, there is one 0 followed by a 1 (1-.99=.01). So, logically, when there are infinite 9s, there are how many 0s? w-1, which we've stated earlier is infinity.
Now, we have here a problem. An infinite sequence is non-terminal, so the 1 at the end cannot be there: there IS NO end. thus, the difference between 1 and .99... is 0.0000 or, 0. If there is 0 difference between two numbers they are equal, thus 1=.99...

Next, a quicker proof:

An objection to this argument was made: when you make this argument, you invoke the "w +1"-th term in the series. There is no number after infinity, so the resulting ".99..." actually ends.

This is not true: w +1=infinity. Also, the length of the new series of 9s is w-1, which is infinity.
So, either way, we have an infinite series of 9s, which means the argument still stands.

Or we could go straight back to the series, and do the math to find what number it converges to. 9*(10^n) converges to 1. I don't feel like doing the math. If you are determined to try to prove me wrong, you can do the math... and see I'm right.

One more objection is "but in the real world there is no .99... It HAS to end."

True. In the real world .999... does not exist. In fact, in the real world, one does not exist. It is an abstract concept. There is no Platonic one that all real-world ones strive toward. There is no perfect, actual one. It is a concept used to understand the world around us. Numbers are abstract. You cannot go into nature to find an object called "one" or "two." There are only things. which can be counted. Using numbers. Which are concepts. The fact that they so accurately represent the world around us is a testament to the beautiful design of nature, and the the ingenuity of the human mind.

What this note is showing is that .99...=1. I don't mean "They are equal for all practical purposes". They are equal. For ALL purposes, practical or impractical. The pronoun "they" does not make sense, because there is no "they" there is it. It is one number. Not two. Not .9...; One.

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