Poll: Does point 9 recurring equal one?
This is something I'm interested in getting people's opinions on.
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.
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 . . . ?
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.
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'
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://polymathematics.typepad.com/polymath/2006/06/no_im_so...
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 + 1/3 + 1/3 = 1
So, yes
1/9 = 0.111...
0.111... + 0.111... + ... = 0.999....
9 * 1/9 = 1
1/10 = 0.1
10 * 1/10 = 1The 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 crowd is more likely to be math folk. Both crowds just need to get out and relax more.
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.
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
Were not at all dealing with f(very big numbers, increasing)
Equivalent, but not equal.