Yes, and no. The idea is the limit(Calculus speak) of .9999.... approaches 1.
If you need to assign a value to the concept of the infinite sum of .9+.09+.009+.0009....,it would be 1. Without lost of generality, this touches deeply on the Epsilon-Delta concept of a limit(Google it).
However, this is the classical view, there are more erotic view like the hyperreal number system. The idea in the hyperreal is the difference between .99999... and 1 is an infinitesimal quantity. Hence, there is no real number that can be consider to be the difference between .99999... and 1.

The idea of this is actually quite profound and touches on the fundamental property of what we consider real numbers. This property is known as the completeness of the reals.

Hi Yoyoyozo,

Try to look for a mathematical definition of " = " and you will see it is hard to find one for which 0.999... is not " = " to 1.

Eg: two numbers are equal when the difference between them can be as small as you fix it ; ...

*slaps drink out of yoyo's hand*

I assume this post was a cry for help. I cast shame on those of you who indulged it.

Oh God, not another one.

1/3 = .333......
3 * 1/3 = .999...
3/3 = 1
.999... = 1

& if a = 0.9999999, then 10 × a = 9.99999, so 9 a is 9.999999 minus 0.999999 giving 9a = 9 so a = 1 as someone else posted in one of the many previous threads on this.
Keep drinking Yoyoyozo. and we can discuss how many angels can sit on the point of a pin ( and paraphrasing from the Paul Kelly song ) how many notes on a saxaphone & how many tears in a bottle of gin.

Oh My God superkeiko I love it when you talk dirty math to me. Tell me the erotic view again ;) DriLL Integrals into me.
On a more serious note I made an A in calculus and learned nothing!
YAY!

Glad to hear that Vash Approval got you through your exams, Daniel :)

LOL that problem was completely wrong Captainmeme xD. The whole class tore me apart

I was just about to make fun of the sad soul that revived this thread. But alas...

Hey, you asked a math question, and I answered as any mathematician would. The idea is the conventional system(the reals), and more erotic ones, like complex(what is the square root of -1?), and even more erotic ones like hyper real, a good primer is the Infinitesimal approach to Calculus(https://www.math.wisc.edu/~keisler/calc.html).

The idea is that .999999... is an infinite sum, and if it converges, there is a limit, and if there is a limit, there is a real number associated with it. The following sum also converges to 1, 1/2+1/4+1/8+1/16....=1. However, be wary of there algebraic argument like TrPrado stated, though it gives the correct result in this case, rearrangement of infinite sums can fail.

a = 0.999....
10a = 9.999....
10a = 9 + a
9a = 9
a = 1

ah poop. Mitchell beat me to it, but he drowned his proof in text so my eyes glossed over it.

The proof I used is the easiest one to cite if you want to use an algebraic one. It's straightforward, makes sense, and is correct. Anyone who tries to do something else there is trying too hard to come up with a result.

i used to question it but i met a mathematician who cleared it up for me a bit more in detail. however, he also said he had a friend who worked at a swiss university in quantum physics (or something related) and they didn't work with 1=.999 as a constant all the time, but that stuff was crazy complicated

Divergent series are the invention of the devil
- N. H. Abel Mathematician
I did not mean to say that TrPrado was "wrong" or anything like that. However, I did wish to point out the counter intuitive nature of infinity, and to inject some rigor into this math discussion.
For instance, consider the sum, S = 1 - 1 + 1 - 1...., then.
1. S = (1-1)+(1-1)+(1-1)...=0
2. S = 1 +(-1+1) +(-1+1)....=1, and
3. S = 1-1+1-1+1-1....
0+S = 0+1-1+1-1+1....
which implies 2S =1, or S =1/2
This is known as the Grandi's Series.

The sum .99999... is what we called well behaved, and Grandi's Series isn't. Like division by 0, you need to be careful about the rules.

Thank you Superkeiko, i appreciate a mathematician when i see one.

To others, thank you for being rigourous non-mathematicians who still hold onto your ability to use mathematcs.

This erotic numbers caper is one thing, but what about a "real world" application ?.. For example, Is 0.99999(recurring) of a Brainbomb equal to one Brainbomb ? Might that undefined, unquantifiable part that is the difference contain the essential spark that makes the difference between genius artist and the banal imitator ?

@Mahor Mitchel
I love that your points are at 999

Pretty sure you mean "exotic" numbers there superkeiko. A bunch of us are giggling themselves silly over those "erotic" numbers. Some have said the most attractive waist-to-hip ratio in a woman is 0.7, so maybe that's one of those erotic numbers.

@MajorMitchell The real world application of infinite sums are numerous. First, infinite sums are either converging or not.
Let's consider the converging one(well behaved). If so, we can use a finite number of that sum to get reasonable accurate approximation of more complex computation accurately.
A clear example is computation method for sin(x), cos(x), tan(x) using Taylor series. The quick and efficient trigonometry is crucial for navigation on sea and air, and geospatial application(GPS) though we as a species has long master this type of application(there are question of physics here as well).
In earlier times, you had trigonometry table that require much more labor to chart sea lane and so on.
The idea of making sure that a infinite sum, or finding a infinite sum that are well behaved that suits your application is crucial you don't get a contradiction somewhere. Since if you are not careful, you could get 1=2 if you accidentally do a division by 0. The idea here is the same.
In fact, I think the cutting edge of this field is trying to tame divergent infinite sums in computing quantum physics-related questions(not so well behaved infinite sum).

Mods, please move the contents of this thread into the one called "Unanswerable Questions".

Thank you.

Does anybody really know what time it is? Does anybody really care? (to know what time) I can't imagine why.

The adventure of the journey is not in the arrival at the destination. Do we really want to arrive at one.

It is approximately now. Like RIGHT now, time measurements are isually relative, and absolutely depend on your frame of reference... And Einstien says it is even more complicated, that there is no absolute time (or space).

In fact the universe is bundled up into little self-contained areas disconnected from each other by time and space. You could theoretically connect them with a wormhole, and even go 'back' in time, but you'd end up so far away from your starting point there would be no way to send a message back to tell yourself not to go.

No plausible way for the past of 'far far away' to affect the present now. So in effect it is a different self-contained universe. (Unless you figure out how to build wormholes... which MIGHT allow you to connect two of these).


So When is it? Well on a local level it is meaningful to say when we are... But nothing Universal.

* usually and Einstein

Does anybody really know what time it is? Does anybody really care? (to know what time) I can't imagine why.

I care what time is

Well superkeiko clearly the contradictions ( and paradoxes ) of domesticity with the fire breathing MemSahib, Her Indoors lying on a sofa eating cake are an infinite sum without any practical limits. Personally I prefer a few "stiffening whiskies" before I try any "erotic numbers" on the MemSahib navigable parts. I am trying to keep up to speed with Orathaic's dissertation on "now" and "when". When is a way to link the past with the future in timespeak that references now. Any fool knows that. I get a little lost contemplating the "event horizons" that differentiate and delineate future when and past when from now. Stuff like that is beyond the limits of my clockwork brain, probably because it's timing is off and needs to be advanced by several degrees. Looking forward to when, and remembering when at the same time is indeed tricky.

Actually superkieko, my favourite imaginary (?) number is the square root of minus one. There's a number that definitely "punches well above its weight class" Before we had i, finding the square root of a negative number was a vexatious exercise without solution. Once I was introduced, life became so much easier... Square root of minus sixteen ? It no longer strkes fear into the heart of budding mathematicians... "Plus, or minus 4i " comes the carefree response of these mollycoddled young maths swots. When we were young we didn't have these gimcrack luxuries, useful as they are.

The thing is real numbers themselves are much more mysterious and elusive than you can imagine in everyday and were taught in school.
Most if not all numbers we used everyday or are taught in school are rational:
a. 1 dollar for that bread.
b. 0.12 dollars for the tax on that bread
c. 1/3 price per earnings for that stock you like, (or .333333....)
d. 1.3 cm for some measurement some scientist is doing.
Only sometimes does irrationals pop up:
a. pi, the ratio of diameter/radius to the circumference of a circle(we did fixed the convention of using diameter, but really, either ratios are irrational)
b. golden ratio(yeah, some designers like it)
c. square roots(whenever a triangle is present, or even navigation)
d. The values of sin(x) are generally irrational as well.
Now, here is the crazy part. All but a countable set of real numbers are irrational. And countable set is the "smallest" type of infinity we know. What does that mean? It means that most of us have been looking at sands on a infinite beach while ignoring much more massive infinite ocean.
Now, here is the crazier part, most real numbers we know are algebraic. More precisely, can be written as

roots of non zero polynomial with rational coefficients. Those that are not are known as transcendental numbers. And we only know a few, pi, and Euler's number(e) come to mind.
And All but a countable set of real numbers are transcendental. So, we have a really massive unexplored ocean that we as a species have never ever set foot on even in the real numbers.
Imagine, do you think the universe is a transcendental place where we as humans are doomed to measure it with only algebraic rulers. Never being given that chance to taste all but a fraction of what our universe has to offer, and only being able to gleams it ever so elusively in our own thoughts, if ever.

Pure mathematics is, in its way, the poetry of logical ideas.
-Albert Einstein

Is arithmetic consistent?

Unstupid, you mean complete? I guess you could ask Kurt Friedrich Gödel.

Luxury, what a luxurious mathematics you young fighters take for granted. When I were a lad we had to struggle, work all day and night in coal pit, lick road clean with tongue, live with eighteen point 99999 recuring other families in old paper bag caught on barb wire fence, just to get a glimpse of a logarithm, and if we looked too long at it, why Dad would beat us to death with a broken beer bottle... The good old days. Aye, and nowadays, you tell young folk of it, and they just don't believe you.

Typos.young blighters..not young fighters.

Bump. As for erotic numbers, google sexy prime. =)

I did. According to Wikipedia such things exist. Prime numbers that differ by six. Plus a blurb about prime triplets. Who could doubt the veracity of information at Wikipedia. LoL

@Orathaic, no. I mean "Are the basic rules of set theory consistent with each other"

@Unstupid, and Orathaic
We need a formal definition of consistent, and completeness in logical systems before proceeding. (They are different but related concepts).

Let us consider only logic system has statements, which can either be true or false.
(So we are doing the 2-valued/binary logic systems only.)
Some statements are axioms[we generally want as little axioms as possible], which are statement that are the "foundation" and need not be proven.
Other true statements are theorem that follow from the axioms using a logical process(aka proofs)
A logic system is consistent if using the axiom(s), for all statement P which is true, then ~P(the negation of P) is false. Now, we generally want consistent systems since proof by contradiction is so fundamental to so many proofs. Note here that formulation of "good" axioms that don't contradict each other somewhere along the way is the key to a "consistent" system.

A logic system is complete, if using the axiom(s), all possible "well formed" statements are decidable. Decidable means that we can, even if we don't know the actual proof, know that the proof actually exist within the logic system.
Now, you will need to specific which system are you talking about? ZFC? Or Peano Arithmetic?
The reality is that for we don't know if ZFC(the conventional set theory) is consistent or not. However, no one has found any inconsistent yet, and mathematician generally think ZFC is consistent, and all proof in ZFC have an implied "assuming ZFC is consistent, then...." However, if ZFC is consistent, then it is definitely not complete(incompleteness theorem, hence the reference to Godel).

The amazing thing is how the completeness theorem leads to turing machines and the stopping problem.

a turing machines mechanism is equivalent to a set of axioms, and so incompletness maps onto not knowing whether a particular input will stop. Thus resolve as true or false whatever question you may have asked (the input).

This means any computer program (taken as input) can be in a state of running forever, if you're not careful.

And several computer scientists/marhematicians have contrasted this with human creativity and problem solving - to basically say we are not turing machines.

@orathaic, I think it might be a good thing that humans exhibition contradiction then. I personally think that historically mathematics and set theory has been focusing too much on building a consistent system.(AKA the program of axiomizing all of math, and Godel put a monkey wrench in it, imagine those scholars that devotes lifetimes to it to be told it is impossible) If you can keep the contradiction into a finite and known set, perhaps a complete but finitely inconsistent system would be preferable.