...Also known as why Math is flawed.

About a week ago, an article appeared on digg. It was a link to a blog from a mathematics professor, who claimed that .999 (repeating the 9, also written as .99|9) is equal to 1. Here is the initial blog post. Currently, the main battlefield is here. Yes, that Catalyst is me.

The poster (Polymath), claimed this to be true, by using mathematical stuff that is generally intended to make you go "Wow, I don't get it, so I guess he's right." Additionally, he claims to have the backing of most of the mathematicians in the known world.

Upon reading the digg post, I was skeptical. Upon reading the blog, I was laughing. To me, this seemed to be the stupidest thing I've ever heard. So, jumping in to the fray for the side of the Not Equals, I put up a post or two. More recently, I've figured out a way to essentially prove them wrong.

So, before I start showing you how this is wrong, I ask you to think about whether or not you think he is right, based on his posts alone. Keep this thought in your head, or write it down if you need to.

Are you ready?

My first question is one this: Why should these two numbers which seem different when you look at them have to be proven to me to be equal? If they ARE equal, shouldn't it be VERY EASY to show that they're equal, by the simple fact that they look the same? But, since no answer is yet forthcoming, I have to do the rest of this.

1. His initial equation:

Let x=.99|9

10x=9.99|9

-x -.99|9

9x=9

Ok, so at first glance this looks correct, right? It seems to be the right math. However, he's successfully messed up one of the basic rules of algebra. What you do to one side, you must do the exact same to the other. So, you could only subtract an x from both sides. To be fair, since we know what x is, we should simply replace it in the initial equation, getting something like this:

10*(.99|9)=9.99|9

9.99|9=9.99|9

Simple solution to that, making that bit of "proof" debunked. Next proof.

2. He said:

1/3=.33|3, so when you multiply it by 3, you get 1=.99|9

Hmmm... on second thought, I'll come back to this, because I consider this the "coup de grace" of the entire argument.

3. He says:

What is 1- .99|9?

He claims that the answer would be an absurd thing that doesn't exist. He shows that the answer would be, in a sense, .00|0 with a 1 following the infinite 0's. I believe this is a perfectly possible occurence, and have seen many other numbers like it. And I can explain how they exist now, as well!

In order to understand it, forget basically anything you know about mathematics, especially in "notations".

This is what happens: because it is an infinite series of 0's, they will continue without end. However, because the two numbers you subtracted were not equal, then there must be something at the "end". This is the 1. In a sense, it exists. However, it can only exist at the end of an infinite chain. If you don't get it, don't worry. I have another bit about numbers with infinite repetition followed by other numbers in a moment.

4. He said:

Stuff. I really have no idea what he was talking about in his part with definitions, though there was a digg poster who showed he was wrong in some way. I'll get back to you when I find a more math-smart person.

Next, we have a bunch of stuff from other posters in the group. (These are things I've addressed most recently)

1. "Can you think of a number between .999... and 1 anyway?"

First, I pose the question to them "does there need to be a number in between?" Think about this for a second: on the number line, every number is technically in order next to another.

Soon, they post something about how there are infinite numbers between 1 and .99|9. Or, wait, no. He doesn't. Because they're equal. Or, wait, no, he does. Hmm... do I smell contradiction in the air?

2. "Let's take the equation 1 x 3 = 3. Let's replace 1 with 0.99|9 (which is another way of writing 0.999...).

Now we have 0.99|9 x 3, which should equal 2.99|9, right?

"Wait, stop right there. How the heck are you able to multiply if you can't even find an end to start from?" you might say. Well, look at the following:

0.99 x 3 = 2.97

0.999 x 3 = 2.997

0.9999 x 3 = 2.9997

and so on...

If you know that the number to the right is another 9 that gets multiplied by 3 to yield that 2 (of the 9x3=27 on the left) needed to change the 7 (of 9x3=27 on the right) into a 9, then you can see that you'll reliably get the answer of 2.99|9."

Now, for this, I ignore the part about .99|9 = 1. It's really not the meat of this issue. Instead, I look at the part about repeating decimals, followed by a number.

We see that there is a 7 at the end of all of these multiplications, yes? This will always be happening. I explain it like this: The 7 is always there, at the "end". But, since it is infinite, it doesn't have an "end". However, if you look at it as the result of billions of multiplications, you see that the 7 would be there. If at any time, you try to make the number even slightly finite and usable as a number, you'd have a 7 at the end. However, as soon as you look at that 7, it becomes a 9, with the 7 forming in the next space over. Infinitely. So, by doing this, you find that the 7 MUST exist, but when you find it, it disappears. Such is the way of many numbers of this sort. The "ending" number is there, but because of the bit about infinity, it's also not there. It exists in both states at once.

3. Ok, so this is the coup de grace. Many of you are taught this in your school careers. After all, it's a generally accepted piece of mathematics. What is it?

1/3=.33|3

The best example of approximation I've ever seen, and, once exposed, the perfect way to make all of this ".99|9=1" nonsense go away.

The claim here is that when you take 1/3, you get .33|3. And, when you take this times 3, you end up with .99|9. But, shouldn't it equal 1? Polymath and most mathematicians accept this to mean that .99|9=1. They take the lazy way out, because, technically, their math is correct.

But think about this: I was taught that when two numbers don't look the same, they're generally different. And that makes sense, doesn't it? One doesn't go around saying they have an apple when the object in their hand is clearly an orange, so why do the same thing with numbers?

So, then we look into the issue. The math is clearly correct (not sarcasm). So, then, maybe it's one of our numbers? Well, the first and best answer to look at is the first result we got. I challenge you now to deny anything you've learned in math class about the decimal equivalents of fractions.

I ask you: WHAT IF 1/3 DOES NOT EQUAL .33|3?

I hear your nerdy cries representative of Luke Skywalker. "That's not true! That's impossible!" And yet, you'd be just as willing to accept that two numbers which when you see them look different, and it has to be proven to you how they're the same, are equal. So, taking this view, we must inevitably come to the conclusion that there is some part of this math that's wrong.

1/3 doesn't equal .33|3. It equals another number, in a sense. It is .33|3, but in the end, you have 1/3, after all of the repetitions of 3. I've shown how things like these can be possible. So, it is logical to think this can be a number. Now let's plug it into the equation:

1/3= .33|3...1/3

.33|3...1/3 *3=1

That's right, if you take the number I have proposed as the actual replacement for 1/3, you can multiply it by 3 to get 1. Does that sound unreasonable in any way? Why, I do believe it's a very easy to grasp concept. And the math works, too! Geez, every part of this is perfect... so why do they continue to dispute this?

Apparently, it's "not a number". Well, first off, anything is a number, because numbers are infinite. Additionally, they claim that it has infinite repetition of the 1/3 in it. Well, I ask, is that so wrong? It's the same sort of idea as .33|3, isn't it? It's an infinite repetition of a number. The only difference between my number and theirs' is that mine gives you the correct answer when you multiply by 3

<hr>

So, given what I've shown, I ask you now, what do you think? Is .99|9=1, or not? If you side with me, choose "No". If you side with the guys I'm against, click "Yes". I will not think any less of you for it. However, for those who choose "Yes", I'd like to see a reasonable argument besides things like "That's how it's always been" and "All the mathematicians agree with it". First off, if you take up a "That's how it always was" side, then remember that people once believed the world was flat. If you take up an "All mathematicians agree" side, I'll show you that you're wrong, as I happen to know a mathematician or two that disagree with the theory.