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0.999...= 1

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Re: 0.999...= 1

#71

Post by saagarjha »

thdrdt wrote: post #55
russellbeattie wrote: post #41

I'll just chime in with my completely ignorant theory that 1 - 0.999... = the infinitely smallest number, but is still, in my mind, regardless of any logic, reason, or educated calculations, greater than 0. I understand and accept this is wrong. However, somewhere in my brain I still believe it. Sort of like +0 and -0, which are also different in my head.

Philosophically you are right. Mathematically you are wrong. It is indeed true that if there was an entity that could reason beyond infinity 1-0.999... would be greater than 0.

From the perspective of IEEE-754, they’re pretty close ;)

Re: 0.999...= 1

#72

Post by The_Colonel »

notRobot wrote: post #29
larschdk wrote: post #25

Earlier quoted context omitted.

I don't mean to troll you, but if you were doubtful that 0.999... = 1, then you should also be doubtful that 0.333.. = 1/3. Any argument that 0.999... is not quite 1 can also be used to argue that 0.333... is not quite 1/3. I think it's mostly a matter of definition, since mathematicians consider sums of infinite series equal to their limit (if it's finite), i guess for many practical reasons. If you accept this, the…

> if you were doubtful that 0.999... = 1, then you should also be doubtful that 0.333.. = 1/3 I disagree. Any middle school student can calculate 1/3 to be 0.33333... using long division, but there's no immediately obvious way to go from 1 (or 1/1) to 0.9999...

> Any middle school student can calculate 1/3 to be 0.33333...

I can just do it backwards - is 1/3 equal to 0.33333...?

1 / 3 = 0.33333... 3 * 0.33333... = 0.99999... and my child brain "knows" that 1 != 0.99999...

In my child brain this proves that 1 / 3 is not equal to 0.33333..., it's just an approximation.

So I agree with larschdk, those problems are equivalent and one can't be used to prove the other ...

Re: 0.999...= 1

#73

Post by saagarjha »

tomtomtom1 wrote: post #31
saagarjha wrote: post #13

Earlier quoted context omitted.

Arithmetic breaks, as multiplication is no longer the inverse of division. (For example, 1/3 * 3 = 0.999… would no longer work.)

why? 1/3 * 3 could still be equal to one. but 1/3 != 0.33333... that is, 1/3 is not representable in base 10. Which makes way more sense. I wonder if taking 0.9999.. != 1, that is 0.0000...1 exists would allow us to reslove, the fact that some possible events have probability 0?

> but 1/3 != 0.33333...

But the issue is that this is easy to verify experimentally via (in this case infinitely) long division that you can do by hand. So it’s hard to convince people of this.

Re: 0.999...= 1

#74

Post by ball_of_lint »

baddox wrote: post #49
captainmuon wrote: post #45

Earlier quoted context omitted.

Well, if I could choose I wouldn't personally accept 1/3 = 0.333... . But rather, I'd say it equals a limit: 1/3 = lim(N -> oo) 0.3{N} (3 is N times repeated) Especially I would distinguish between infinitely many threes, and N threes, where N goes to infinity. In the first case, you would still be missing an infinitisimal amount, in the latter case you have the usual situation and the sequence has the least upper bo…

But you absolutely can evaluate that limit as N goes to infinity and correctly conclude that 1/3 does equal 0.333 repeating.

Repeating decimals may be introduced in a mathematics education long before any other infinite series or the methods used to tame them, such ass limits.

Re: 0.999...= 1

#75

Post by cryptica »

larschdk wrote: post #25
ginko wrote: post #9

I remember being doubtful when being presented with this in middle school, but after being shown this as fractions makes it obvious: 1/3 = 0.333.. 3 * 1/3 = 3 * 0.333.. 3/3 = 0.999.. 1 = 0.999..

I don't mean to troll you, but if you were doubtful that 0.999... = 1, then you should also be doubtful that 0.333.. = 1/3. Any argument that 0.999... is not quite 1 can also be used to argue that 0.333... is not quite 1/3. I think it's mostly a matter of definition, since mathematicians consider sums of infinite series equal to their limit (if it's finite), i guess for many practical reasons. If you accept this, the…

I agree. The argument ignores the rule that infinity is a point which can never be reached; it can only be approached. So repeating 9s infinitely many times will still not reach 1, it will only approach it.

Re: 0.999...= 1

#76

Post by saagarjha »

ltbarcly3 wrote: post #18
rs23296008n1 wrote: post #6

I'm actually curious what impact it would have on various proofs if 0.999... wasn't accepted as 1. What gets broken? What consequences do we hit?

So, lets take .9, .99, .999 and so on. If a sequence of rational numbers converges, it converges to a real number. What number does .9, .99, .999, .9999 (and so on) converge to? Which is to say, is there a number that it gets closer and closer to at every step? Clearly it gets closer and closer to 1 at every step, so the sequence converges to 1. This is one of the many (equivalent) ways the real numbers are defined t…

I don’t think this reasoning is correct, because the limit of an expression does not have to be the same as the value at the point the limit is being taken. For example, if your sequence is “sin(1/x) * x“, then it’ll slowly appear to converge to one as x approaches infinity, but it cannot reach it. So there’s really no relevant conclusion you can make.

Re: 0.999...= 1

#77

Post by saagarjha »

agumonkey wrote: post #67
knzhou wrote: post #44

Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards. The point is that in elementary school arithmetic, you define addition, multiplication, subtraction, division, decimals, and equality, but you never define "...". Until you've defined "...", it's just a meaningless sequence of marks on paper. You can't prove anything about it using arithmetic, or otherwise. What the "a…

And I thought mathematicians would reject normal arithmetic operation over the domain of `N...` elements. They're always so ultra rigorous to classify what is or is not, what is defined, the domain .. but then they let an infinite sequence be treated as any finite number.

> but then they let an infinite sequence be treated as any finite number

Technically it’s an infinite series (sum of an infinite sequence), which is a finite number in certain cases like this one.

Re: 0.999...= 1

#78

Post by jhanschoo »

bytedude wrote: post #2

Flame wars over this used to be common on the internet. People intuitively have the notion that the left side approaches 1, but never actually equals it. They see it as a process instead of a fixed value. Maybe the notation is to blame.

The intuition is right, and the mathematical definition relies on the intuition. It's just that people haven't been exposed to the actual definition when it comes to real numbers.

Mathematically, mathematicians prove that there is a unique number that this process goes to, (and not, say, two distinct numbers), and define the notation to represent this unique number.

Re: 0.999...= 1

#79

Post by jhanschoo »

vfinn wrote: post #62

Sorry for my naivety, but why one couldn't prove by induction that adding 9s never close the gap, or let's say, that by definition the operation is such, that it never closes the gap. If you can always halve the pie, then you can continue eating forever. To me it would be much easier to accept that (1/3)*3 is not 1.

It's because the rigorous mathematical definition is more subtle. It's defined as the smallest real number such that repeating this process (putting more 9's in the end) can't result in a number that's greater than it. So as long as you have a number that's smaller than 1, there's a gap there, and repeating the process of adding 9's will eventually give you a number that lies in that gap.

Re: 0.999...= 1

#80

Post by jhanschoo »

saagarjha wrote: post #70
vfinn wrote: post #62

Sorry for my naivety, but why one couldn't prove by induction that adding 9s never close the gap, or let's say, that by definition the operation is such, that it never closes the gap. If you can always halve the pie, then you can continue eating forever. To me it would be much easier to accept that (1/3)*3 is not 1.

Induction can be non-intuitive to people who are troubled by this problem.

Induction has nothing to do with this. It's about a taking the limit of a process.
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