# Doubts about 0.999 being equal to 1

• Dragonfall
In summary, the conversation discusses the topic of whether 0.999... is equal to 1. Participants share their opinions and arguments, with some arguing that the two are not equal while others provide mathematical proofs to show their equality. The conversation also touches on the concept of limits and the use of infinite sequences in arithmetic operations. Ultimately, the conversation ends with a reminder to accept that 0.999...= 1 and the thread is closed.
Dragonfall
If anyone has any further doubts about 0.999... being equal to 1, please direct your attention to today's http://en.wikipedia.org/wiki/0.999..." .

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Dragonfall said:
If anyone has any further doubts about 0.999... being equal to 1, please direct your attention to today's http://en.wikipedia.org/wiki/0.999..." .
Eek! Aargh! Grumble, shriek, not another thread on this!

Hung, drawn and quartered is too good for you!

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Stevedye56 said:

Really? It became featured article only yesterday.

500 hundred threads *here*, all by people arguing the point.

I did think the following quote from the linked article was interesting though. :)
The equality has long been taught in textbooks, and in the last few decades, researchers of mathematics education have studied the reception of this equation among students, who often reject the equality

i honestly don't see how anybody can reject the equality.

All that is required is a firm belief that a "number" is exactly the same as the "numeral" used to represent it. Since "1" and "0.9999..." are different it follows immediately that they are different numbers! Just like 0.5 and 1/2are different numbers!

Actually not too many people will deny that 0.5= 1/2 but people who deny 1= 0.9999... are likely to deny 1/3= 0.3333...!

It's too bad everyone here is too well-informed, and there's no one left to argue the other side. It's kind of fun seeing what they can come up with when backed into a corner. Just for fun, here's how I would lay out the argument:

1. $$0.999... = \sum_{n=1}^\infty \frac{9}{10^n}$$
(by definition)

2.$$\sum_{n=1}^\infty \frac{9}{10^n} = \lim_{N \rightarrow \infty} \sum_{n=1}^N \frac{9}{10^n}$$
(by definition)

3. $$\sum_{n=1}^N \frac{9}{10^n} = \frac{(9/10) - 9/(10^{N+1}) }{1-(1/10)} = 1-{\left( \frac{1}{10} \right) }^N$$
(easy to show with algebra, and not many people would deny it because there aren't any infinities)

4. Let $\epsilon>0$. Then there is some N with $(1/10)^n<\epsilon$ for all n>N.
(again, pretty intuitive, doesn't involve infinites yet)

5. $$\lim_{N \rightarrow \infty} {\left( \frac{1}{10} \right) }^N =0$$
(by (4) and the definition of a limit)

6. $$\lim_{N \rightarrow \infty}1- {\left( \frac{1}{10} \right) }^N =1$$
(by (5) and the continuity of subtraction)

7. Therefore, by (1),(2),(3),(6), and the transitivity of equality
$$0.999... = 1$$

Now, if they want to deny the conclusion, they have to pick a premise to deny. (1) or (2) would just be disagreeing with definitions everyone else uses, and there's no point in arguing about something like that. (3) and (7) are pretty undeniable. (4) is a little tricky, but again, there are no infinities, so I don't think it would be too hard to convince people of. (6) is actually the most technical line, but I think it jives with intuition, so I don't think it would be a problem.

That leaves us with (5). Again, debating this would just be disagreeing with a definition. The essence of the problem is that people have a preconceived notion of what a limit is, as some sort of process, but this doesn't agree with the epsilon delta definition (in fact, it doesn't really make sense at all). If they can understand this definition, then I don't see how they could both accept all the definitions used above and still deny that 0.999...=1.

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The usual "objection" is that 0.999... is not the LIMIT of the sum, but the sum itelf, which the typical objector does not realize IS a limit itself.

The usual "objection" is that 0.999... is not the LIMIT of the sum, but the sum itelf, which the typical objector does not realize IS a limit itself. That's what I meant when I said it is a confusion of "number" with "numeral".

why don't people disagree with 1.000...=1. seems almost the same as .999...=1 to me.

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Because "zeroes don't count".

Okay, guys. A lack of people arguing $0.\bar{9} \neq 1$ isn't a bad thing, and we don't need to be picking up the slack.

Let c = 0.9...

10c = 9.9...
10c - c = 9
9c = 9
c = 9/9

"If you don't know where you are going, any road will get you there."
-- Lewis Carroll

To accept that you have to accept that ordinary arithmetic operations work on infinite sequences as well as finite. That's true but no more obvious than that 0.999...= 1 to begin with.

## What is the evidence that 0.999 is equal to 1?

There are several ways to prove that 0.999 is equal to 1. One way is to use the fact that every real number has a decimal representation. In this case, 0.999 is simply another way of writing the number 1.

## How can 0.999 be equal to 1 if they have different decimal representations?

While 0.999 and 1 may look different, they are actually just different ways of writing the same number. Just like 1/2 and 0.5 are different representations of the same fraction, 0.999 and 1 are different representations of the same real number.

## Why is it hard for people to accept that 0.999 is equal to 1?

Some people may have a hard time accepting that 0.999 is equal to 1 because it goes against their intuition. We are used to thinking of 0.999 as almost 1, but in reality, it is exactly 1. Additionally, the concept of infinity can be difficult for our minds to grasp, making it hard to accept that 0.999 is equal to 1.

## Are there any real-life examples that demonstrate the equality of 0.999 and 1?

Yes, there are many real-life examples that demonstrate the equality of 0.999 and 1. One example is the measurement of time. If we divide an hour into 60 minutes, each minute can be represented as 0.016666... hours, which can also be written as 0.999 minutes. This shows that 0.999 and 1 are two ways of representing the same amount of time.

## Can you explain the concept of limits and how it relates to the equality of 0.999 and 1?

Limits are used in mathematics to describe the behavior of a function as the input approaches a certain value. In the case of 0.999 and 1, we can think of 0.999 as approaching 1 as we add more 9s to the decimal representation. In the limit, 0.999 becomes equal to 1. This demonstrates the equality of the two numbers.

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