Poll: Does point 9 recurring equal one?
This is something I'm interested in getting people's opinions on.
15 comments
Why are you interested in getting people's opinions on a matter of fact? "Everyone is entitled to their own opinions, but they are not entitled to their own facts."
Some people regard it as "fact" that the Moon landings were faked. Some people regard it as "fact" that they've been abducted by aliens. Some people regard it as "fact" that they can power their car with water. Some people regard it as "fact" that there is only one size of infinity. Some people regard it as "fact" that you can choose one element from each of infinitely many non-empty sets. Some people regard it as "fact" that 0.99999... is less than one.
One of my colleagues at work recently told me that point nine recurring is less than one. I'm interested to hear what people here say.
One of my colleagues at work recently told me that point nine recurring is less than one. I'm interested to hear what people here say.
Nice work including the axiom of choice in there.
FWIW it's hard to imagine the mind that wouldn't believe in the axiom of choice but would believe in .999 == 1 being a good phrase to use; that person would probably say
- 'I have a procedure for constructing a # arbitrarily close to 1, and if you tell me how close to 1 you want it i'll tell you how many steps of my procedure you need to execute'
FWIW it's hard to imagine the mind that wouldn't believe in the axiom of choice but would believe in .999 == 1 being a good phrase to use; that person would probably say
- 'I have a procedure for constructing a # arbitrarily close to 1, and if you tell me how close to 1 you want it i'll tell you how many steps of my procedure you need to execute'
One of my colleagues at work recently told me that point nine recurring is less than one.
I've meanwhile looked at some of your other recent comments, and it appears that you have rather stronger math background than I, and could well explain why your colleague's opinion is indeed not a fact. Or just refer your colleague to the excellent Wikipedia article on the subject linked to by another participant here. Is there a work issue that turns on how people at your workplace interpret 0.99999 . . . ?
I've meanwhile looked at some of your other recent comments, and it appears that you have rather stronger math background than I, and could well explain why your colleague's opinion is indeed not a fact. Or just refer your colleague to the excellent Wikipedia article on the subject linked to by another participant here. Is there a work issue that turns on how people at your workplace interpret 0.99999 . . . ?
I've found that this question is one where people are often not persuadable, even by arguments that have been accepted for centuries. Somehow they "know" that 0.999... has to be less than 1. It "never gets there", or something. Articles on Wikipedia, MathWorld, etc., are regarded as simply wrong or irrelevant. They know better.
There are users of our systems who have similarly strong beliefs that I simply can't fathom. I'm interested in seeing what positions people take, and how they argue them. I'm learning a lot from this forum, specifically because it is not a random sample of ordinary people.
Linguists learn the most from pathological examples of maltreatment. Likewise, I'm learning a lot by probing the corners of the beliefs of non-typical people.
And there is an issue at work that uses similar analysis as that required by this question. More detailed, more complex, less obvious, but it boils down to a similar point. As his boss I can pull rank and tell him to do it as I say, but I'd rather engender understanding.
There are users of our systems who have similarly strong beliefs that I simply can't fathom. I'm interested in seeing what positions people take, and how they argue them. I'm learning a lot from this forum, specifically because it is not a random sample of ordinary people.
Linguists learn the most from pathological examples of maltreatment. Likewise, I'm learning a lot by probing the corners of the beliefs of non-typical people.
And there is an issue at work that uses similar analysis as that required by this question. More detailed, more complex, less obvious, but it boils down to a similar point. As his boss I can pull rank and tell him to do it as I say, but I'd rather engender understanding.
your colleague is 99.9...% wrong
[deleted]
Glad to see that the poll is nearly unanimous here.
Btw, the naive perspective on this issue is because people intuitively perceive the hyperreal numbers. In the hyperreal number system, .9999999999999999999999... and 1 are different. But in almost all phrasings of this question it is implied to be talking about the real number system.
Specifying it gets rid of all ambiguity. The real numbers .999... and 1 are equal. The hyperreal numbers .999... and 1 are not.
http://en.wikipedia.org/wiki/Hyperreal_number
Btw, the naive perspective on this issue is because people intuitively perceive the hyperreal numbers. In the hyperreal number system, .9999999999999999999999... and 1 are different. But in almost all phrasings of this question it is implied to be talking about the real number system.
Specifying it gets rid of all ambiguity. The real numbers .999... and 1 are equal. The hyperreal numbers .999... and 1 are not.
http://en.wikipedia.org/wiki/Hyperreal_number
At first I was thrown off by the way you wrote the value.
If "point 9 recurring" means the same thing as:
the limit of the sequence 0.9, 0.99, 0.999, etc.
then I know it equals 1 because that's what I was told.
Math in engineering school was like studying hammers. You don't take them apart, and you don't ask why. You just grab one and start hitting things. For five years.
If "point 9 recurring" means the same thing as:
the limit of the sequence 0.9, 0.99, 0.999, etc.
then I know it equals 1 because that's what I was told.
Math in engineering school was like studying hammers. You don't take them apart, and you don't ask why. You just grab one and start hitting things. For five years.
1/3 = .333333....
1/3 + 1/3 + 1/3 = 1
So, yes
1/3 + 1/3 + 1/3 = 1
So, yes
I've always used this variation
1/9 = 0.111...
0.111... + 0.111... + ... = 0.999....
9 * 1/9 = 1
1/10 = 0.1
10 * 1/10 = 1i always ,,loved'' the variation that goes like: if 0.99... != 1, then there has to be an x that 0.99.... + x = 1.0 (or to say 1 - 0.99.. > 0 must hold), so you can start to find this x (as 0.01, 0.001, 0.0001, etc.) and fail of course. this is not so ,,simple'' as your examples (or the (10*0.99.. - 0.99..)/9) but lights the point where most people go wrong here (at least for me)
I stopped posting at Digg forever years ago after this very question was posed
None of the choices above work.
The LIMIT of .9999 is 1. Since .9 repeating is not actually the same number as .9 with an infinite number of nines at the end, you're always adding a nine out there to do something with it, right?
Sure .9 repeating infinitely is the same as one, but I'd argue that .9 with the repeating symbol is not, since the symbol indicates that the repeating has to take place manually. The repeating symbol is a mechanical one. And any manual, finite number of nines is not going to equal one, no matter how many times you manually add nines on to the end.
Plus I'm just feeling like arguing. Had a crappy day at work.
The LIMIT of .9999 is 1. Since .9 repeating is not actually the same number as .9 with an infinite number of nines at the end, you're always adding a nine out there to do something with it, right?
Sure .9 repeating infinitely is the same as one, but I'd argue that .9 with the repeating symbol is not, since the symbol indicates that the repeating has to take place manually. The repeating symbol is a mechanical one. And any manual, finite number of nines is not going to equal one, no matter how many times you manually add nines on to the end.
Plus I'm just feeling like arguing. Had a crappy day at work.
This downmodding is not making me any less cranky. (grin)
See also the discussion at http://news.ycombinator.com/item?id=692338
I'm sure someone can point out the situation for more, but I'm curious to the practical situation of where it truly matters.
I'd be surprised if many people here vote is doesnt equal one. Im a poor mathematician and I agree :D
It's happened ...
.9... = n
9.9... = 10n
9.9... - .9... = 10n - n
9 = 9n
1 = n
According to math folk, yes.
According to philosophy folk, no.
This crowd is more likely to be math folk. Both crowds just need to get out and relax more.
This crowd is more likely to be math folk. Both crowds just need to get out and relax more.
Many "philosophy folk" are very comfortable with the type of logical argument that leads us to the conclusion that 0.999... == 1, and most of the others will defer the question to mathematicians.
eventually
it approaches 1 but is not equal to 1.
(sorry to see you downvoted - because it is actually sometimes tought to grasp but...)
it doesnt approach one- your thinking of a nice graph where the line does approach but never touch, however 0.9... is NOT a function, cannot be plotted. It exists for infinite length.
Any distance it is from 1 is constant.
it doesnt approach one- your thinking of a nice graph where the line does approach but never touch, however 0.9... is NOT a function, cannot be plotted. It exists for infinite length.
Any distance it is from 1 is constant.
I interpreted repeating 9's as:
the function that writes some number of repeating 9's (replacing the zero in the last decimal place with a 9). This function approaches 1, does it not?
f(1) = 0.9
f(2) = 0.99
f(3) = 0.999
so f as defined above approaches 1
I interpreted the question in the context of numerical precision. Since we are usually not dealing with infinite precision, the two are not equal.
I am familiar with the obvious: 1/3 == 0.3333... and 3 * 1/3 == 1 so 0.9999... == 1
the function that writes some number of repeating 9's (replacing the zero in the last decimal place with a 9). This function approaches 1, does it not?
f(1) = 0.9
f(2) = 0.99
f(3) = 0.999
so f as defined above approaches 1
I interpreted the question in the context of numerical precision. Since we are usually not dealing with infinite precision, the two are not equal.
I am familiar with the obvious: 1/3 == 0.3333... and 3 * 1/3 == 1 so 0.9999... == 1
ahhh No the OP was referring to the value at the limit as x->inf of your function f.
yes and sometimes the integrals of those graphs converge on a value...
PG should be using this poll to disable user accounts.
that's harsh, bro :) I can smell the axe body spray all over your remark.
Read this for some supporting evidence. (sorry for being snarky thought this poll linked to the article.)
.999... === 1, but .999 != 1.
Equivalent, but not equal.
Equivalent, but not equal.