Is 0.999... equal to one?

#1

Someone posted this question on my facebook group for math. The question is whether 0.999… (that is, the number represented by an infinite stream of the digit 9 after the decimal point) is exactly equal to one. Not close, not good enough for government work, but exact.

Let’s see if this board does better than facebook. The bar is not set very high.

I posted a spoof thread in the spam section, but a humourless moderator locked it immediately. No rule violation was cited, but moparscape does have the notorious Rule 0, which says the rules are whatever the mods say the rules are.

#2

x = 0.999999…

  • 10

10x = 9.99999…

  • 1x

9x = 9

/ 9

x = 1

The symbol ‘…’ refers to a limit. A limit means “approaches ___ at infinity” (in basic terms)… and 0.99999… approaches 1.

#3

[quote=“Davidi2, post:2, topic:555378”]x = 0.999999…

  • 10

10x = 9.99999…

  • 1x

9x = 9

/ 9

x = 1

The symbol ‘…’ refers to a limit. A limit means “approaches ___ at infinity” (in basic terms)… and 0.99999… approaches 1.[/quote]
Shut the fuck up, you couldn’t count to 1 if you used your fingers.

#4

I think the question then is, do you define the symbol 0.999… as the limit of the sequence, 0.9, 0.99, 0.999, 0.9999, 0.99999, and so on, or as something different from that? If something different, then what?

#5

there was this formula I learned in calculus to convert limits into fractions and the sequence 0.999999~ always converts to 1 as a fraction

that being said it’s always 0.000001~ off to me, but you could argue that is 0 the same way

#6

I would avoid this sort of construction. It can be shown that it is impossible to extend the real number system to include an “infinity” number, and still have the infinity number obey the same rules of arithmetic as the other numbers. You can have an “infinity” number and have it obey some of the rules of arithmetic, but then you have to be extra careful to make sure you are not performing an illegal operation on infinity.

If the idea is to define things like 0.000…1 as the limit of the sequence 0.1, 0.01, 0.001, 0.00001, etc., then it would be equal to zero. Which is certainly consistent with what you have said earlier. 0.000…1 just seems like very odd notation to me - how can there be a “1” after an infinite stream of “0” digits, since the infinite stream of “0” digits has no end? Many of the moparians seem to have some alternate arithmetic system with a number 0.000…1 that is not equal to zero. I am glad you do not do this

#7

The difference between 1 and 0.999… is 0, so they are the same.

#8

not really tho

#9

[quote=“Davidi2, post:2, topic:555378”]x = 0.999999…

  • 10

10x = 9.99999…

  • 1x

9x = 9

/ 9

x = 1

The symbol ‘…’ refers to a limit. A limit means “approaches ___ at infinity” (in basic terms)… and 0.99999… approaches 1.[/quote]

This is the best way to explain how it does equal 1.

#10

I once had a philosophy professor who was so smart that he took a run around the dog house so fast that he ran right smack into himself. After he revived he didn’t know who he had run into, but he did know that he was wrong. We were all forced to agree that he was right, he was wrong, but he never knew it. To this day he thinks he was wrong, he was right.

0.999… Is an open ended phrase, written in a self limiting notation. It has little if any meaning. What number when subtracted from 0,999… would equal 0.999… ending with the numeral 8 instead of 9? How would you write such a number using your notation? Would you contend that such a number does not exist?

0.999…9 - 0.000…1 = 0.999…8 Weeee, wasn’t that fun?

Rewrite that in your limited notation. 0.999… is a trick phrase. It is used to confuse and deceive. It eliminates infinite possibilities and generates nothing but circular arguments.

Does 0.111… = 0.112… or 0.110…? What is the limit of 0.111…?

#11

[quote=“drubrkletern, post:10, topic:555378”]0.999…9 - 0.000…1 = 0.999…8 Weeee, wasn’t that fun?[/quote]That is not correct, because the second number doesn’t exist. You can’t have infinite zeros and then a 1, so what you’re actually saying is 0.9999… - 0.000… = 0.999… which is correct. It also happens to equal 1 :slight_smile:

So yes, a number subtracted from 0.999… that equals 0.999…8 does not exist, because 0.999…8 does not exist. The fact that you are attempting to use them for your argument shows that you don’t understand what 0.999… means

The limit of 0.111… is 1/9 (which is equal to 0.111…; limits do not ‘create’ new numbers every time, the limit of x=2 is 2 for example)

And just for giggles:

x = 0.111…
10x = 1.111…
9x = 1 (notice that 9x would be 0.999… ? which we just showed equals 1, funny coincidence)
x = 1/9

#12

I would say that it definitely doesn’t matter for some things - for example, the ancient Romans didn’t have the modern understanding of the real number system, but they used numbers, performed calculations, and built buildings, bridges, and aqueducts based on those calculations, and those things didn’t fall down because they didn’t have the modern understanding.

But in other areas, I would say it is important to get these things right. Calculus is absolutely fundamental to understanding, for example, Newton’s laws of motion (presumably a reason Newton felt the need to develop calculus!), and in any treatment (either standard analysis, using the real number system, where there is no such thing as an infinitely small number not equal to zero, or some other systems, in which there is), it’s important to get these things right. If we say, these two things are really close together, maybe they are equal, or maybe they’re not but the difference is incredibly small, so we won’t worry about the problem too much - do that, and we may well get the wrong answer.

So I would say the understanding of questions like this one have been important in the development of branches of mathematics that have proved incredibly useful in modelling very real-world phenomena. Maybe it’s not the only intellectual path the world couldn’t have taken to understand things like Newton’s laws of motion, but it’s the path it did take.

#13

0.999… is always equal to the next number, no matter what. If you say 24.999…, you’re really just saying 25.

0.999… is another way of writing 0.333… * 3, 0.333… of course being equal to 1/3 (and 3/3 being equal to 1)

If you don’t believe that 1/3 = 0.333… (or any infinite number of decimals can occur), then this won’t work. In which case, David’s explanation is a better fitting.

x = 0.999…

  • 10
    [hr]
    10x = 9.999… (Note: You’ve just shifted the infinite number of decimals, but the amount of decimals still remains the same while still having a new 9 in the ones place.)
  • x (0.999…)
    [hr]
    9x = 9
    / 9
    [hr]
    x = 1
#14

[quote=“asshole_rule, post:8, topic:555378”][quote author=Bowser jr link=topic=674569.msg4508595#msg4508595 date=1463352497]
The difference between 1 and 0.999… is 0, so they are the same.
[/quote]

not really tho[/quote]
What number is between 1 and 0.999…?
If there is no, the numbers are the same.

#15

[quote=“Bowser jr, post:14, topic:555378”][quote author=asshole_rule link=topic=674569.msg4508607#msg4508607 date=1463370454]

not really tho
[/quote]
What number is between 1 and 0.999…?
If there is no, the numbers are the same.[/quote]You rephrased the correct answer with no explanation hoping it would help. (hint: it didnt)

#16

[quote=“Pure_, post:15, topic:555378”][quote author=Bowser jr link=topic=674569.msg4508629#msg4508629 date=1463407655]

What number is between 1 and 0.999…?
If there is no, the numbers are the same.
[/quote]You rephrased the correct answer with no explanation hoping it would help. (hint: it didnt)[/quote]
I asked him to give me a number between 1 and 0.999…, since he opposed to my statement.

#17

[quote=“Bowser jr, post:16, topic:555378”][quote author=Pure_ link=topic=674569.msg4508632#msg4508632 date=1463413977]

I asked him to give me a number between 1 and 0.999…, since he opposed to my statement.[/quote]0.9999

You clearly said,
The difference between 1 and 0.999… is 0, so they are the same.
I don’t see a question here, do you?

#18

Bunch of kids just round it up to 1

stop pretending to be smart you orphans

#19

The ability to reiterate a proof does not imply understanding.

It is almost impossible to find a college educated person who understands mathematics well enough to know when or how to apply it.

I have surveyed a cross section of college educated subjects on a simple common assertion.

Is 100 exactly equal to 100?
resp.: yes, 1000; no, 0.

Is this always true?
resp.: yes, 1000; no, 0

Can there be there any exceptions in the real world?
resp.: yes, 0; no, 976; 14 refused to answer.

Are there any examples that you can think of where it is not true?
resp.: yes, 0; no, 964; 36 refused to answer.

Out of one thousand surveyed all answered yes to the first two questions and most no to the last two questions. Most went so far as to claim that the questions themselves where ridiculous.

The correct answers are, of course: no, no, yes, yes; in that order.

A simple example from everyday life:

100 dollars, for instance, is almost never equal to 100 dollars. Only under very special and specific conditions can it be said to be true, that 100 dollars is exactly equal to 100 dollars. E.g. Joe and Bil both were ticketed for seat belt violations. The fines were set at 100 dollars each. Joe has to work 10 hours to produce 100 dollars. Bill ones a software company and makes 100 dollars every second. To claim that Joe’s 100 dollars was exactly equal to Bill’s 100 dollars we would have to also claim that 36,000 seconds is exactly equal to 1 second.

A person would have to be out of their mind to make such a claim. Yet, 99% + of the population believes those two fines to be equal and fair. We have seen what the college educated population thinks. It would appear that it is reasonably safe to conclude that the entire population is out of their minds.

Every school age child should know that; 100 is exactly equal to 100, only exists in the mind, it looks good on paper, but it is almost never true off paper, in the real world.

This is not just a failure of the educational system but of the culture as a whole.

My point all along is that in real terms the math does not always fit the observation or situation. On paper 0.9999… can be shown and proven in fact to = 1, but can we really apply this to reality? I just can’t see how something that has no definitive ending can ever truly equal something that does? I understand the math and see the logic in it but my mind remains open to what we actually understand in real terms.

#20

[quote=“drubrkletern, post:19, topic:555378”]The ability to reiterate a proof does not imply understanding.

It is almost impossible to find a college educated person who understands mathematics well enough to know when or how to apply it.

I have surveyed a cross section of college educated subjects on a simple common assertion.

Is 100 exactly equal to 100?
resp.: yes, 1000; no, 0.

Is this always true?
resp.: yes, 1000; no, 0

Can there be there any exceptions in the real world?
resp.: yes, 0; no, 976; 14 refused to answer.

Are there any examples that you can think of where it is not true?
resp.: yes, 0; no, 964; 36 refused to answer.

Out of one thousand surveyed all answered yes to the first two questions and most no to the last two questions. Most went so far as to claim that the questions themselves where ridiculous.

The correct answers are, of course: no, no, yes, yes; in that order.

A simple example from everyday life:

100 dollars, for instance, is almost never equal to 100 dollars. Only under very special and specific conditions can it be said to be true, that 100 dollars is exactly equal to 100 dollars. E.g. Joe and Bil both were ticketed for seat belt violations. The fines were set at 100 dollars each. Joe has to work 10 hours to produce 100 dollars. Bill ones a software company and makes 100 dollars every second. To claim that Joe’s 100 dollars was exactly equal to Bill’s 100 dollars we would have to also claim that 36,000 seconds is exactly equal to 1 second.

A person would have to be out of their mind to make such a claim. Yet, 99% + of the population believes those two fines to be equal and fair. We have seen what the college educated population thinks. It would appear that it is reasonably safe to conclude that the entire population is out of their minds.

Every school age child should know that; 100 is exactly equal to 100, only exists in the mind, it looks good on paper, but it is almost never true off paper, in the real world.

This is not just a failure of the educational system but of the culture as a whole.

My point all along is that in real terms the math does not always fit the observation or situation. On paper 0.9999… can be shown and proven in fact to = 1, but can we really apply this to reality? I just can’t see how something that has no definitive ending can ever truly equal something that does? I understand the math and see the logic in it but my mind remains open to what we actually understand in real terms.[/quote]

I take it you assert mathematics to be invented rather than discovered (which is perfectly valid) by questioning if it can be shown in paper could it be shown in reality. :slight_smile:

Anyway, I think the issue here is the syntax looks different, but are the values really the same? I guess it depends on what concept of infinity you believe in. For example, with a real number such as 0.999…, I believe there is no number in between 0.999… and 1 (which has been posted earlier). In that case, I think if you reject the idea that 0.999… = 1 then you also reject 0.333… is 1/3; but don’t 3 thirds make a whole? According to one particular syntax it can: 3*(1/3) = 3/3 = 1. Turns out this problem also has a proof in this way:

Anyway, I guess my post is just philosophical rambling but I’m interested to see more discussion here.