## Do you think 0.99|9 is equal to 1?

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## Do you think .99|9 is equal to 1?

Yes
9
28%
No
23
72%

### Do you think 0.99|9 is equal to 1?

...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.

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.

Catalyst
Apostle of Peace

Posts: 2016
Joined: Fri Jul 15, 2005 9:38 pm

I see no dispute. Or a flaw. Decimals are horribly inaccurate. Fractions, are not. So really, all your arguments fall down. I'll just jab at a few things that amused me.

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

He wasn't subtracting from either side of the equality, he was subtracting one equation from another. There is nothing messed up here.

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

Nice try, but if we forget mathematics, of course you'll be right.

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.

You've not made any real mathematical case that 0.99 recurring does not equal one. Who are these mathematicians? Be interested to see how they can show me it doesn't.

I don't think it equals one, I know it does!
Last edited by Ranbir on Wed Jun 28, 2006 10:40 am, edited 1 time in total.
"The imaginary number is a fine and wonderful resource of the human spirit, almost an amphibian between being and not being." - Gottfried Leibniz
Science snobbery.

Ranbir
For Queen and Country

Posts: 8515
Joined: Mon Aug 18, 2003 9:27 am

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...

Firstly, we can't really multiply recurring decimals. Anytime we do, (while remaining in decimals) will result in an approximation.

Your little look at the following is merely multiplying 99/100, not 0.99 recurring. Not even a calculator can do that.

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.

No, it wouldn't exist. And to see you actually think you know what is at the 'end' just makes me want to squeeze something until it's eyes bulge out. You're abusing infinity, stop it. 0.99 recurring. That means the 9 never stops. NEVER stops. Although, you seem content you know the mysteries of infinite, so maybe you can tell me the ending of Pi. Would you also like to give me a case that 0.46464646 recurring does not equal 46/99?
"The imaginary number is a fine and wonderful resource of the human spirit, almost an amphibian between being and not being." - Gottfried Leibniz
Science snobbery.

Ranbir
For Queen and Country

Posts: 8515
Joined: Mon Aug 18, 2003 9:27 am

Ranbir wrote:I see no dispute. Or a flaw. Decimals are horribly inaccurate. Fractions, are not. So really, all your arguments fall down. I'll just jab at a few things that amused me.

Ah, but then so are all of theirs, leading to the inevitable fact that (1/3)*3=1. They use decimals to find something that to me doesn't look right, and you seem to praise them, but when I do it, you mock me. Hmm... a double-standard, perhaps?

He wasn't subtracting from either side of the equality, he was subtracting one equation from another. There is nothing messed up here.

Erm, yes there is. Because when you do the math at the bottom by replacing the x with .99|9, you get the correct answer of 1=1. It's not something too hard to do, but you are supposed to do it whenever you already know what the value of x is.

Your little look at the following is merely multiplying 99/100, not 0.99 recurring. Not even a calculator can do that.

Of course. No one wants to push to buttons long enough.

No, it wouldn't exist. And to see you actually think you know what is at the 'end' just makes me want to squeeze something until it's eyes bulge out. You're abusing infinity, stop it. 0.99 recurring. That means the 9 never stops. NEVER stops.

Please, by the end of this, you'll want to kill every kitten in sight.

Now, you need to listen to this part, because I'm not debating whether that .99|9 is recurring or not. I'm saying that .99|9 shouldn't even have to be cared about.

However, once again, we return to part at hand. Anytime you use a finite version of the .99|9*3 (for instance, .9999*3, or .999999999999999999*3) you get a 7 at the end of the decimals. Now, when you look at the infinite, you see a continually recurring pattern of 7 and 9. Think about it as if, instead of just being a number, it's an infinitely increasing bunch of these things: .9999*3. So, the 7 will appear fleetingly, only to be instantly replaced by the new 9, as the next larger set of .99999 is put in. The only way to accept it completely is to admit that the 7 and the 9 have to exist simultaneously. It's all depending on how you look at that infinite. I'm choosing to look at it as the result of the multiplication. You and others look at it as simply the answer.

Although, you seem content you know the mysteries of infinite, so maybe you can tell me the ending of Pi. Would you also like to give me a case that 0.46464646 recurring does not equal 46/99?

Pi and this are two very different things. I have no math to base the numbers in Pi off of, and to me they're random, meaningless numbers.

Additionally, I'd be inclined to think that 46/99 doesn't equal .4646|46, using the same cases above. I'm sorry, it's an approximation.

Catalyst
Apostle of Peace

Posts: 2016
Joined: Fri Jul 15, 2005 9:38 pm

However, once again, we return to part at hand. Anytime you use a finite version of the .99|9*3 (for instance, .9999*3, or .999999999999999999*3) you get a 7 at the end of the decimals. Now, when you look at the infinite, you see a continually recurring pattern of 7 and 9. Think about it as if, instead of just being a number, it's an infinitely increasing bunch of these things: .9999*3. So, the 7 will appear fleetingly, only to be instantly replaced by the new 9, as the next larger set of .99999 is put in. The only way to accept it completely is to admit that the 7 and the 9 have to exist simultaneously. It's all depending on how you look at that infinite. I'm choosing to look at it as the result of the multiplication. You and others look at it as simply the answer.

Um, the seven is a result from the multiplication done from it is using rounded numbers, it isn't using the infinite recurring decimal. You can't terminate the recurring. I've never once seen the infinite and I'm rather surprised you have, care to show me it? Your result of the multiplication isn't a recurring decimal, thus whatever it shows, it does not show anything about 0.999 recurring not equalling one. Again, you have yet to provide a mathematical proof where 0.99 recurring does not equal 1.

Erm, yes there is. Because when you do the math at the bottom by replacing the x with .99|9, you get the correct answer of 1=1. It's not something too hard to do, but you are supposed to do it whenever you already know what the value of x is.

...
No, really...it's fine. 10x = 9.9999... Let's call this equation 1. Equation 2 is x = 0.99999...
Equation 1 minus equation 2 will get us 9x = 9. Thus x = 1. This means equation 1's x value is the same as equation 2's. What you did was assume from the get go. Which ultimately doesn't matter, given that 0.99... is 1.

Pi and this are two very different things. I have no math to base the numbers in Pi off of, and to me they're random, meaningless numbers.

Both are recurring decimals. And I've yet to see the math you have to base 0.99 recurring as not equalling 1. Pi is meaningless? I'll have you know sir, the ratio of the circumference to the diameter of a circle is very meaningful and important.

Additionally, I'd be inclined to think that 46/99 doesn't equal .4646|46, using the same cases above. I'm sorry, it's an approximation.

0.4646 recurring itself isn't an approximation, given it's infinite nature. When using it practically, then naturally you will have to approximate.

0.99 recurring itself does equal 1. A 0.99 approximation, naturally does not.
"The imaginary number is a fine and wonderful resource of the human spirit, almost an amphibian between being and not being." - Gottfried Leibniz
Science snobbery.

Ranbir
For Queen and Country

Posts: 8515
Joined: Mon Aug 18, 2003 9:27 am

Jaime had to point me in the direction of this thread, so I offer food for thought:

The argument is moot. The derivitive is and always will be 0.

Calculus solves all arguments!
Remember, children, "No two people are not on fire."

Lion's Mane
Lord of Annwn

Posts: 1274
Joined: Fri Jun 14, 2002 9:25 pm
Location: Challenging the wind...

We have a geometric and limit proofs. That'd cover the calculus toolage.
"The imaginary number is a fine and wonderful resource of the human spirit, almost an amphibian between being and not being." - Gottfried Leibniz
Science snobbery.

Ranbir
For Queen and Country

Posts: 8515
Joined: Mon Aug 18, 2003 9:27 am

From the point of view of a non mathemetician, (though my maths was OK I guess!) it doesn't seem logical to say that .99l9=1, IMO, 1=1, .99l9=.99l9. 1/3x3= 1 because it's 1/3, not .33l3, with decimals you can never get a third, therefore you cant get .99l9 with decimals either.

Sorry, it's a bit silly cos I dont understand this very well, but it just seems a falousy or a paradox to me- why bother even try to say .99l9=1?!

Shi Tong
Stupid Egg of Shen Zhou

Posts: 4034
Joined: Mon Apr 11, 2005 10:12 pm
Location: London, England

Well 1/3 is 0.33|3.

1/3 is a recurring decimal. Many fractions form recurring decimals.

There is no paradox. All the proofs have been formed from the very simple fundamentals of mathematics. It seems a bit silly to me how we can argue against that.

As to the point of it, there isn't one in terms of practicality. It's just a truth. Just as there are infinitenly many numbers between 1 and 2 as there are between 0.5 and 0.6. It might not mean anything to the average joe, but it's still a truth.

Since we do have a few niave. (gosh, I'm the only one that's voted yes!)

Find me a number between 0.99|9 and 1.00|0.

If they are two different numbers, you will find one. If you do not, then you must come to the conclusion they are the same.

I'll give you some help though; it's impossible to write a decimal expansion of a number between 0.99|9 and 1.00|0.
"The imaginary number is a fine and wonderful resource of the human spirit, almost an amphibian between being and not being." - Gottfried Leibniz
Science snobbery.

Ranbir
For Queen and Country

Posts: 8515
Joined: Mon Aug 18, 2003 9:27 am

All goofing off about calculus aside, I also say yes along the lines Ranbir has been stating. There is no number whatsoever between 0.99|9 and 1.00|0. In fact, with seriousness regarding calculus, as you approach infinity, the limit of 0.99|9 is always going to be 1.00|0. That means the more 9s at the end of the number, the more akin to 1 the number becomes.
Remember, children, "No two people are not on fire."

Lion's Mane
Lord of Annwn

Posts: 1274
Joined: Fri Jun 14, 2002 9:25 pm
Location: Challenging the wind...

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