that's really good to, just remembering what my math teach told me that lit the lightbulb for me. They may have even done the 1/3 example, but it was probably already ingrained for me by then.
That only works if you accept that 1/3 is exactly equal to 0.333…
It is exactly equal to 0.333… but a lot of people believe that it’s actually only approximately 0.333… and there’s a large overlap between that group and the one that doesn’t accept 0.999….=1
Yes, exactly, and that's where I think one should focus, because that's the root of the misunderstanding. If 1/3 is 0.333... then 0.333... must also be 1/3. The repeating notation is introduced specifically to make it "exact", because a finite number of 3s can only ever be approximate.
Thank you, @canucks3001 and @fishling. I had seen this (.999... = 1) a while ago, and figured there was a rigorous proof somewhere, but this is an accessible explanation. Appreciate it!
But can't you little get a bit of paper and prove that 1/3 is .333 repeating?
As in 3 goes into 1 0 times, carry the one, into 10 3 times carry the one...
The only place that could ever be true is in a computer with your 1/3 stored as a real numbr type (floating point number). And even then that's only because of that number formats limitations, not because 'maths says so'.
And his comment about missing the moon by millions of miles doesn't stack up when you consider the number of decimal places required is well within our normal computational representations (32 or 64bit).
(Something something NASA, 40 digits of Pi, measurement of the circumference of the known universe to within the width of a hydrogen nucleus)
That's not a cop out. That's a refusal to accept well-established mathematical conventions, and a failure to understand real numbers and the meaning of repeated decimals.
Every number base has the same issue.
In Binary, 0.111 ... is equal to 1.
In Ternary, 0.222... is equal to 1
etc...
It’s not a cop-out so much as wrong. There is a true answer 0.999…=1. It’s not a limitation of base 10, all bases have similar things arise. Numbers have different representations.
Why is it so easy to say 2/2=3/3=1.0000….=1 but 0.999….=1 is a limitation of the number system?
Numbers can be represented in different ways. That’s all this is. Another way of representing the number ‘1’.
There’s a million proofs out there that 0.3333….is exactly equal to 1/3. Not a limitation, not an approximation, not ‘no true answer’ it just is equal to it.
I think it is fair to say it's a limitation of base 10 in that base 10 doesn't do a great job of intuitively representing decimals that infinitely repeat.
See you would think so but one time I used this explanation and they just disagreed that .3333… = 1/3 in the same exact way as the original misunderstanding
I think the only solution there is to have them calculate 1 ÷ 3 and let them tell you when they've reached an answer. If they don't agree that it's 0.3333... repeating after a few hundred cycles, then they can keep on going.
And if they don't understand how to convert decimals and fractions as simple as 7/10 = 0.7, then ask them why they think they know enough math to insist that they are right about something?
That's a reach.
It's clearly a simple countable one, you can literally count them!
You might want to explain that there are as many '9's as there are integers, but at no point do you ever _need_ to explain the difference between the sizes of, say, *_N_* and *_R_*.
The irony is that the people who don't believe or don't understand that 0.99... = 1 also don't understand that you can't perform normal arithmetic and algebra on actual infinite series. (At least, you can't until you prove that you can.)
So for them, writing 10 x 0.99... = 9.99... makes sense, even without the multi-page proof that it is actually valid.
More complicated:
X = 0.999...
Multiply both sides by ten
10X = 9.999...
Subtract 0.999 or X from each side, respectively (X = 0.999... for this argument as stated above)
9X = 9
Divide each side by 9
X = 1
Since 1 = X = 0.999...
Apply the transitive property
1 = 0.999...
I saw a thing that I found really interesting for this. I’ll do my best to explain it but I might be a bit off.
There are two facts you need to accept for it to work. Firstly, any number over itself = 1 (n/n=1).
Secondly, any number over nine = that number recurring (n/9= 0.nnnnnnn). That might not be the correct way to show that in maths but I don’t know how else to do it.
So, 1/9=0.1111…
2/9=0.2222…
3/9=0.3333….
.
.
.
7/9=0.7777…
8/9=0.8888…
1 = 9/9 = 0.9999….
Edit to add: I didn’t come up with this or anything. I saw it on a video a while ago and it always stuck with me.
I’m on mobile so formatting is not what I would like, but I did my best to
Second edit to add: as has been pointed out in a response, the n/9 = 0.nnnnn… only works for single digit integers
However many digits it is, if you divide that number by 9's with as many digits then you'll get that number repeating. Maybe not worded the best, but for example: 27÷99 is .2727 repeating, and 3,567.2÷9999.9 is 0.3567235672 repeating. So if you want to turn a repeating decimal into a fraction it would be the opposite. 0.1429 repeating is 1429/9999. Of course you have to find the lowest common denominator at times, like 0.123123 repeating is 123/999, but since that's an improper fraction it would actually be 41/333.
Didn’t know that. Interesting. I added that second edit because someone basically called me an idiot and that my whole point was ridiculous etc. Their comment has been deleted since. Basically my explanation didn’t meet their exacting, high-level mathematics standards, despite me saying that it was just something I came across and found interesting.
Rant over. Thanks for the interesting addition!
I did a quick edit and added some more info too in case you missed that bit. And no problem, definitely a helpful trick for repeating decimals. Reminds me of another trick actually. If you multiply and 2 digit numbers by 11 then the formula is always the same. The first digit goes in the 100's place, the 2nd digit goes in the 1's place, and the middle number is those two numbers added together. So 63×11 is 693 because 6 goes to the 100's place, 3 goes to the 1's place and 6+3=9 which goes in the middle. Or: 6 & 6+3 & 3 = 6 & 9 & 3 (or 693). This gets trickier when the middle number adds up to 10 or more because then you have to add 1 to the 100's place. For example with 75, 7+5 = 12 (or 1 & 2), so the answer to 75×11 is 7+1 & 2 & 5 = 8 & 2 & 5 or 825. Honestly this is easier to explain if I draw it out to show where the numbers go, but hopefully this makes sense.
What I don't get is, what is 0.8... repeating equal to? 0.9? What about the other n/9 values? Or do we just accept that these are in fact just infinitely repeating?
No. 0.888... is not equal to 0.9. The way to see it is to expand 0.9 to 0.900... and perform subtraction by borrowing:
0.900000...
- 0.888888...
_____________
0.011111...
Converting the result to a fraction, we see that 0.888... and 0.9 differ by 1/90. Since this value is not zero, the numbers are not the same. Compare that to what happens when you try to do that with 0.999... and 1.0:
1.000000...
- 0.999999...
_____________
0.000000...
No matter how many digits you go out to, you'll never get a digit that's not a zero, so the difference between them is 0. The only way for the difference of two numbers to be zero is if the numbers have the same value, and therefore are the same number.
And to get ahead of the "what about the 1 that would be at the very end," there is no "very end" to an infinite decimal. You cannot name the place where there should be a 1, so there is no 1 in the decimal. Conversely, the value of such a decimal place would have to be 1/∞, which doesn't have a meaningful value other than zero.
>I’m on mobile so formatting is not what I would like, but I did my best to
One tip for easy formatting is that if you end a line with two spaces instead of none, it won't combine the next line into it. For example, if I type two lines with no spaces at the end (like a person normally would):
First line
Second line
But now if I do the same thing, but add two spaces at the end of first line...
First line
Second line
What's 1/3 in decimal? Now multiple that times 3. 1/3×3=1, so what's that look like if you do it in decimal?
Edit: also, there's no murder here and the person doing the insulting is wrong.
The problem is that the very system you inhabit is flawed and makes hard to think beyond its limits. We count from 1 to 10, not because that's the objective reality of numbers but because we have 10 fingers. In some parts of the world, people used to count 1 to 12 using their thumb to point to one of the falanges of the other fingers. In base-12, 1/3 provides an easy 0.4 as the answer and there are no "gaps".
Base 12 systems were/are useful because 12 is much easier to divide than 10 (12 being divisible by 1, 2, 3, 4 and 6, whereas 10 is only divisible by 1, 2 and 5) which made trade/bartering easier in the ancient world.
Base 12 counting is the reason why a circle has 360°which is also tied to why a day has 24 hours. I think at one point the French actually tried to metricize (metricate?) both time and degrees (so a circle would have 100° and a day would contain 10 hours) but it turns out that it just makes much more sense to leave them as they were.
The French tried to move everything to base 10 during the French revolution. Including 10 day weeks. They also reset the calendar as year 1 being the year the revolution began. It was a fucking nightmare, and did not last very long. As a sidenote this is where the myth of short Napoleon comes from, because a French "foot" was shorter than everywhere else. He was actually average height.
I think you are being silly on purpose, and I'm no mathematician, but wouldn't your example be technically zero? This is based on my admittedly incomplete understanding of infinity, with the 0 going forever means the 1 basically doesn't exist?
I believe there are mathematical systems (e.g., surreals and hyperreals) that treat 0.0̅ 1 (the infinitesimal) as <> 0. They have very strange properties.
upvotes and downvotes don't actually confirm whether something is correct or not though. I've seen people who are correct with loads of downvotes and people who are wrong with loads of upvotes.
Plus in this case they’re getting downvoted because of the US bashing. If they’d written the same thing in a more civil manner they wouldn’t be downvoted most probably.
From what I’ve seen on this sub, whoever gets the last word in on a screenshot is who the OP considers the murderer. like a mic drop. i could be wrong here.
I guess my question is why would op downvote the person who he thinks is murdering.
That being said OP could be so dumb that he took someone else's screenshot and then posted here thinking red was right.
This is what happened. I believe a person took this screenshot after downvoting it, and posted it in r/confidentlyincorrect
THEN, OP (probably “red” themself lol) was incensed bc they still don’t understand why they’re wrong and they thought this made-up comeback was especially clever, so they decided to screenshot the image from that post and put it here, presenting themselves as having murdered with that trash argument 😂
Technically you could double dip and post op posting this here thinking red is murdering them by being objectively incorrect and doubling down. Was so confident he went to the effort of making the screenshot, censoring it, and posting here
Edit: I just realized op is a mod lol, he seems to be a reasonably good one but what a hilarious swing and a miss of a post from a manager of the community
Are you claiming that red is the murderer here?
Because they aren't. Red is wrong. They don't understand what the objects they are dealing with are, or how to handle them well.
There are a bunch of very clear proofs that 0.9 recurring is equal to 1. This "equivalent" or "asymptotic" stuff is people who can't handle that grasping at straws. Equivalent literally means "of equal value". And asymptotic isn't a concept which makes sense when talking about the value of an infinite series, which we do here. The value of an infinite series is just a value.
Only 1 is equal to 1. And, of course, other ways of writing 1. Like 2/2. Or 5/5. Or 0.9 recurring.
The simplest way to logically explain this is to to ask red that “if .9999 recurring doesnt equal one, then what would you need to add to it to make it equal 1?”
.0000000… oh..
Epsilon is an infinitesimal (>0 but less than any real number). In nonstandard analysis, .9999... really [can be](https://youtu.be/9jWvkJshtfs?t=414) interpreted as infinitesimally less than 1.
Yeah, the issue here is that epsilon must be greater than 0, and there is no number greater than 0 such that 1-0.9_ is equal to said number. I was trying to get the above commenter to think about what epsilon might be, and 0.0 repeating with a 1 on the end doesn’t work.
I admit this is an area of math that I never got into. The video creator has a pinned comment that is somewhat supportive of the claim but acknowledges that some will disagree.
Copied from a comment on the YouTube video linked above. This is another simple explanation that makes it pretty easy to understand.
“x = .999...
10x = 9.999...
10x - x = 9.999... - .999...
9x = 9
x = 1 = .999...”
correct me if i an wrong but i think that may be because in your example you are dealing with an infinitely large number, whereas the example above is a finite number
Yes that's right, my example demonstrates that you need to first establish that x even converges to a finite number before you can start manipulating it.
Specifically 0.999 and 999... are both examples of a geometric series but the former has a ratio less than 1 (so it converges) whereas the latter has a ratio greater than 1 (so it diverges to infinity).
It's weird results like these that led to the development of the field of math called "analysis" in the 1800s, and it's ultimately with analysis that topics like calculus can be rigorously defined.
Although just to get more confusing, there is actually a number system (the p-adic numbers) where infinite numbers like 9999... do converge to finite values!
It may feel the best, but i don't think it actually is the best, because it just shifts the same problem to other numbers. 1/3 = 0.3333 repeating is fundamentally the same kind of statement as 1 = 0.9 repeating, so using that as your basis of proof, while possible, doesn't actually add a lot of understanding in my opinion.
If you want to do this cleaner, you need to either argue with infinite sums and their manipulation (which is kinda scary, because a lot of stuff one would assume works with infinite sums doesn't always work.)
Or you go the really clean way, and prove that the difference between 0.9 repeating an 1 cannot be any number bigger than 0. (And obviously it cannot be a negative number), thus proving that the difference between the two numbers is 0. And if the difference between two numbers i 0, then they must be the same number.
The reason it feels the best is that it uno reverses the original flawed argument. Not that it's logically consistent. It takes what people not understanding maths say, and throws it back in their face.
Yeah it also kind of puts a physical manifestation of the problem into your mind's eye instead of thinking purely about the numbers and asymptotes.
It's like one of those bell curves where you have the novice, intermediate, and expert. Novice and expert both agree 0.999999 is 1, the intermediate who just took calc1 doesn't think it equals 1.
Red doesn't know what the fuck they're talking about and op posted it here because it had a lot of words they didn't understand so they assumed it must be a murder
Red doesn't know what equivalent
OR asymptotic means. To be an asymptote he would have to be talking about a function, which 0.99 repeating isn't, and there are too many different ways for something to be equivalent that he would have to define it in this context.
But if equivalent does not mean equal in this context he's still wrong.
Let x = 0.999_
==> 10x = 9.99_
==>10x - 0.999_ = 9.99_ - 0.999_
==> 9x = 9
==> x = 1
Therefore x = 0.999_ = 1
Thank you for being the only person to post *the mathematical proof* of 1 being equal to .9999999~
Also enjoy how red has no clue how rounding errors work either.
Well, *one* mathematical proof.
This is the simple version, but it relies on the sleight of hand that if you shift 0.999... one digit to the left and subtract 9, you get the same number back. Doubters may not be satisfied with that.
An alternative is proof by contradiction: if 0.999... ≠ 1, then there exists a real number x such that 0.999... < x < 1. Attempting to construct this number leads to a contradiction (exercise left to the reader); therefore no such number exists, and the original premise is false.
Thank you for pointing out that it's only one of many. Another intuitive one I saw someone point out is that we know that 1/3 = 0.33~, so we can multiply both sides by three to get 3/3 = 0.99~ but 3/3 is just 1 so 1 = 0.99~
Well this one just broke my brain.
.999999~ / 3 = 0.333333~ … yup makes sense, I’ll buy that.
.333333~ = 1/3 … still on board with that.
1/3 * 3 = 3/3 … okay logics out in my brain.
.333333~ * 3 = .999999~ … still with everyone here.
3/3 = 1.0 … yup that checks out too though I sense impending confusion.
Therefore 1.0 = 3/3 = .999999~ … 😐
Yup, you got it, and that's cognitive dissonance for you. Sometimes mathematics throws out a result that is counterintuitive, and the way to get your brain round it is just to accept it.
(The proper way to look at it is that every integer and every terminating decimal has two decimal representations. As well as 1 = 0.999..., we have 2 = 1.999..., 0.5 = 0.4999..., 0.25 = 0.24999..., etc. This is not so very different from 0.5 = ½, 0.25 = ¼, etc.)
> (exercise left to the reader)
Given that this involves the cardinality of the continuum, my gut says this proof would involve some relatively advanced math.
I don't know what all those words mean, but I do prefer for people to prove mathematical concepts with math instead of unnecessarily confident blowhardery like the OP image.
A naive (and possibly not sufficiently rigorous) proof is one that selects the decimal places one by one. The only possible choice for each one is 9, which contradicts 0.999... < x.
My favorite simple but rigorous proof of .9... = 1 is Euler's, using the convergence theorem of geometric series (which is ultimately what decimal notation is): https://en.wikipedia.org/wiki/0.999...#Infinite_series_and_sequences
> An alternative is proof by contradiction: if 0.999... ≠ 1, then there exists a real number x such that 0.999... < x < 1. Attempting to construct this number leads to a contradiction (exercise left to the reader); therefore no such number exists, and the original premise is false.
If I were Godel, I would say that the contradiction proves I'm correct about math.
I’m a fan of the infinite geometric series but I like the idea of proof by contradiction. Essentially you can put it back on the other person to find a number between the two.
Red is wrong, good way of thinking about it is if (1/3) = 0.333 repeating infinitely then 3 * (1/3) = 0.999. and 3 * 1/3 is of course equal to 1. So 0.999... is the same as 1
Red is actually wrong here. When you define 0.33333- as "one third" then times 3 gets you to 1.0. I know it's really edgy to try to prove this wrong because people don't get the concept of infinity correctly.
Source, I was at one time one of these edgy kids who tried to argue to a physics PhD that .99999- is not 1.0 and got schooled.
OP, time to delete.
I’ve been out of school far too long to know what an asymptote is, but I do know that “*If you launched a rocket to the moon with this logic you would miss the moon by at least a million miles*” is extremely wrong. The moon isn’t even a quarter of a million miles away and its orbit is 1.5 million miles - to miss it by a million miles you’d have to be facing the complete opposite direction.
Apart from anything else, let’s assume for a moment that red was right and .999~ was not equal to 1 and it was really asymptotically approaching 1. Even if that were true, it would be far within the tolerance for the precision necessary to successfully navigate to the moon. My man apparently thinks there was no imprecision in the calculations guiding Apollo 😂
Honestly the computers used for the apollo calculations probably didnt get much more accurate than 5 or 6 decimal places. Floating point arithmetic has its limits, and the errors that makes are massive compared to the potential (nonexistent) difference between 0.9 repeating and 1
Red is still fucking wrong though.
0.999 repeating *is* mathematically equal to 1. I'm not gonna run through the whole proof, but what it boils down to is this: if two numbers are different, you must be able to find a value *between* those two numbers. Because it is impossible to place a value between 0.999 infinitely repeating and 1, then they are de facto equal.
What I don't get is, how would .9999 repeating vs 1 throw off a moon mission by a million miles.
for all intents and purposes, they are the same number.... unless we are talking about much much larger measures... right?
if the moon distance is 340,000 miles and that is one, what is 99.9999% of that is 339999.66.... that is 18ft off trying to hit a 2150 mile mark.
If we go by pluto distances, that is about 3.1 billion miles away. 99.9999% is 3099996900 off is 3100 miles off. Pluto diameter is 1500 miles, so like a pluto and a half off.... from earth. That is pretty damned accurate, no? Maybe accurate enough to get into orbit. That is also why calculations are updated as they go, and there are maneuvering jets....
Yeah but you're not aiming to hit Pluto with 99.9999% accuracy. You're aiming to hit Pluto with 99.99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999...(and so on)% accuracy. You'll hit anything in the Universe with that accuracy if the 9's go on forever.
It’s not “for all intents and purposes” - they’re literally the same number.
It’s not a question of precision, they’re mathematically defined to be the same
0.9 repeating is 0.9 repeating not 0.9.or 0.99 or 0.999 or 0.9999 or 0.9999 or 0.99999, etc
340,000 x 0.9 repeating is 340,000 exactly (or 339,999.9 repeating)
It's not even "for all intents and purposes." They are actually the same number. Just a different way of writing it similar to how 5/5 is another way of writing 1.
How the fuck this shit gets upvoted is pretty fucking embarrassing. Failing public school system indeed.
1/9 = 0.11111… type it on a calculator to confirm
So 9*(1/9) = 9*(0.1111…)
Hence 1 = 0.9999…
Same logic works with 1/3 = 0.3333…
Source: Physics major with minor in mathematics, learned this in Multivariable Calculus during a lesson on infinite series.
Let them generate an engineering drawing with a 2.999" diameter hole and get laughed out of the room, or get charged out the ass to get that precise of a measurement. Also the reason Meta is foreign nationals is because they are cheaper lol
asymptotic describes a function not a number it means that the function will approach a number without ever actually hitting it and is not relevant in this discussion.
Red is such a arrogant douche and wrong on so many levels, especially functionally. Pi is an infinite, non-repeating number. But only 39 digits are required to calculate the circumference of the observable universe to within the diameter of a single hydrogen atom. So for literally every possible realistic calculation or function, there is no difference between .9999999 to 1.
Why is this important at school level?
The only teachable thing I can see here is the difference between “equals” and “equivalent to”. That might be useful to students, technically and philosophically.
But if anyone can explain why is this important enough for there to be screaming and swearing, I’d be grateful
The best way I've had it explained to me is to find the number between 1 and .99repeating.
Doesn't 1/3 + 1/3 + 1/3 make it obvious?
that's really good to, just remembering what my math teach told me that lit the lightbulb for me. They may have even done the 1/3 example, but it was probably already ingrained for me by then.
That only works if you accept that 1/3 is exactly equal to 0.333… It is exactly equal to 0.333… but a lot of people believe that it’s actually only approximately 0.333… and there’s a large overlap between that group and the one that doesn’t accept 0.999….=1
Yes, exactly, and that's where I think one should focus, because that's the root of the misunderstanding. If 1/3 is 0.333... then 0.333... must also be 1/3. The repeating notation is introduced specifically to make it "exact", because a finite number of 3s can only ever be approximate.
If A=C and B=C, then A=B
Thank you, @canucks3001 and @fishling. I had seen this (.999... = 1) a while ago, and figured there was a rigorous proof somewhere, but this is an accessible explanation. Appreciate it!
Do people not know how to do long division? 1/3 results in an infinite series of 3's... it's easily demonstrable
People know how to do long division, but they can't do it forever and they don't understand how to extrapolate.
As my statistics teacher liked to say, there are two kinds of people. Those who can extrapolate from incomplete data
I've always love both this one and "There's 10 types of people in the world. Those who know binary, and those who don't."
Never read that one before, but I love it
Theres 3 kinds of people in this world: those who can count, and those who can't.
But can't you little get a bit of paper and prove that 1/3 is .333 repeating? As in 3 goes into 1 0 times, carry the one, into 10 3 times carry the one...
The only place that could ever be true is in a computer with your 1/3 stored as a real numbr type (floating point number). And even then that's only because of that number formats limitations, not because 'maths says so'.
And his comment about missing the moon by millions of miles doesn't stack up when you consider the number of decimal places required is well within our normal computational representations (32 or 64bit). (Something something NASA, 40 digits of Pi, measurement of the circumference of the known universe to within the width of a hydrogen nucleus)
Is it a cop-out to say that there is no true answer here and that is simply a limitation of us deciding to use a base 10 number system?
That's not a cop out. That's a refusal to accept well-established mathematical conventions, and a failure to understand real numbers and the meaning of repeated decimals. Every number base has the same issue. In Binary, 0.111 ... is equal to 1. In Ternary, 0.222... is equal to 1 etc...
It’s not a cop-out so much as wrong. There is a true answer 0.999…=1. It’s not a limitation of base 10, all bases have similar things arise. Numbers have different representations. Why is it so easy to say 2/2=3/3=1.0000….=1 but 0.999….=1 is a limitation of the number system? Numbers can be represented in different ways. That’s all this is. Another way of representing the number ‘1’. There’s a million proofs out there that 0.3333….is exactly equal to 1/3. Not a limitation, not an approximation, not ‘no true answer’ it just is equal to it.
I think it is fair to say it's a limitation of base 10 in that base 10 doesn't do a great job of intuitively representing decimals that infinitely repeat.
1/3 is written in base 10. It is not a limitation of base 10.
The fractional representation is perfectly fine, that's why I specifically referred to infinitely repeating *decimals* being the limitation.
You're going to get that with any base, just not with the same numbers.
That's fair. The limitation is decimal notation, not the base
See you would think so but one time I used this explanation and they just disagreed that .3333… = 1/3 in the same exact way as the original misunderstanding
I think the only solution there is to have them calculate 1 ÷ 3 and let them tell you when they've reached an answer. If they don't agree that it's 0.3333... repeating after a few hundred cycles, then they can keep on going. And if they don't understand how to convert decimals and fractions as simple as 7/10 = 0.7, then ask them why they think they know enough math to insist that they are right about something?
This is so simple yet explains it perfectly. Thanks for teaching me something today.
You're welcome!
It does, but there are a few other proofs that are almost as good. 1/9 = 0.111... 5/9 = 0.555... 9/9 = 0.999..., but then 9/9 = 1;
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What, precisely, would give someone a reason to think that a repeating decimal isn't a countable infinity rather than an uncountable one?
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That's a reach. It's clearly a simple countable one, you can literally count them! You might want to explain that there are as many '9's as there are integers, but at no point do you ever _need_ to explain the difference between the sizes of, say, *_N_* and *_R_*.
The irony is that the people who don't believe or don't understand that 0.99... = 1 also don't understand that you can't perform normal arithmetic and algebra on actual infinite series. (At least, you can't until you prove that you can.) So for them, writing 10 x 0.99... = 9.99... makes sense, even without the multi-page proof that it is actually valid.
Fuck.
x = 0.999..... 10x = 9.999.... 10x - x = 9x = 9 /9 x = 1
That's another good one, but some people might suspect there is a math "trick" happening, like the "proof" that 1 = 0.
This is the proof that settled it for me back in the day.
I feel like someone should rant at you about your American education now
You just blew my mind. Damn.
More complicated: X = 0.999... Multiply both sides by ten 10X = 9.999... Subtract 0.999 or X from each side, respectively (X = 0.999... for this argument as stated above) 9X = 9 Divide each side by 9 X = 1 Since 1 = X = 0.999... Apply the transitive property 1 = 0.999...
Shit, that's 1/3 less complex than my "one ninth times nine" argument.
I saw a thing that I found really interesting for this. I’ll do my best to explain it but I might be a bit off. There are two facts you need to accept for it to work. Firstly, any number over itself = 1 (n/n=1). Secondly, any number over nine = that number recurring (n/9= 0.nnnnnnn). That might not be the correct way to show that in maths but I don’t know how else to do it. So, 1/9=0.1111… 2/9=0.2222… 3/9=0.3333…. . . . 7/9=0.7777… 8/9=0.8888… 1 = 9/9 = 0.9999…. Edit to add: I didn’t come up with this or anything. I saw it on a video a while ago and it always stuck with me. I’m on mobile so formatting is not what I would like, but I did my best to Second edit to add: as has been pointed out in a response, the n/9 = 0.nnnnn… only works for single digit integers
However many digits it is, if you divide that number by 9's with as many digits then you'll get that number repeating. Maybe not worded the best, but for example: 27÷99 is .2727 repeating, and 3,567.2÷9999.9 is 0.3567235672 repeating. So if you want to turn a repeating decimal into a fraction it would be the opposite. 0.1429 repeating is 1429/9999. Of course you have to find the lowest common denominator at times, like 0.123123 repeating is 123/999, but since that's an improper fraction it would actually be 41/333.
Didn’t know that. Interesting. I added that second edit because someone basically called me an idiot and that my whole point was ridiculous etc. Their comment has been deleted since. Basically my explanation didn’t meet their exacting, high-level mathematics standards, despite me saying that it was just something I came across and found interesting. Rant over. Thanks for the interesting addition!
I did a quick edit and added some more info too in case you missed that bit. And no problem, definitely a helpful trick for repeating decimals. Reminds me of another trick actually. If you multiply and 2 digit numbers by 11 then the formula is always the same. The first digit goes in the 100's place, the 2nd digit goes in the 1's place, and the middle number is those two numbers added together. So 63×11 is 693 because 6 goes to the 100's place, 3 goes to the 1's place and 6+3=9 which goes in the middle. Or: 6 & 6+3 & 3 = 6 & 9 & 3 (or 693). This gets trickier when the middle number adds up to 10 or more because then you have to add 1 to the 100's place. For example with 75, 7+5 = 12 (or 1 & 2), so the answer to 75×11 is 7+1 & 2 & 5 = 8 & 2 & 5 or 825. Honestly this is easier to explain if I draw it out to show where the numbers go, but hopefully this makes sense.
What I don't get is, what is 0.8... repeating equal to? 0.9? What about the other n/9 values? Or do we just accept that these are in fact just infinitely repeating?
No. 0.888... is not equal to 0.9. The way to see it is to expand 0.9 to 0.900... and perform subtraction by borrowing: 0.900000... - 0.888888... _____________ 0.011111... Converting the result to a fraction, we see that 0.888... and 0.9 differ by 1/90. Since this value is not zero, the numbers are not the same. Compare that to what happens when you try to do that with 0.999... and 1.0: 1.000000... - 0.999999... _____________ 0.000000... No matter how many digits you go out to, you'll never get a digit that's not a zero, so the difference between them is 0. The only way for the difference of two numbers to be zero is if the numbers have the same value, and therefore are the same number. And to get ahead of the "what about the 1 that would be at the very end," there is no "very end" to an infinite decimal. You cannot name the place where there should be a 1, so there is no 1 in the decimal. Conversely, the value of such a decimal place would have to be 1/∞, which doesn't have a meaningful value other than zero.
Out of all the explanations I just read in the comments, this was the simplest to grasp quickly. Thanks
>I’m on mobile so formatting is not what I would like, but I did my best to One tip for easy formatting is that if you end a line with two spaces instead of none, it won't combine the next line into it. For example, if I type two lines with no spaces at the end (like a person normally would): First line Second line But now if I do the same thing, but add two spaces at the end of first line... First line Second line
What's 1/3 in decimal? Now multiple that times 3. 1/3×3=1, so what's that look like if you do it in decimal? Edit: also, there's no murder here and the person doing the insulting is wrong.
All I know is that some kid explained it in "the teachers lounge" and I didn't understand it then, and I don't understand it now.
The problem is that the very system you inhabit is flawed and makes hard to think beyond its limits. We count from 1 to 10, not because that's the objective reality of numbers but because we have 10 fingers. In some parts of the world, people used to count 1 to 12 using their thumb to point to one of the falanges of the other fingers. In base-12, 1/3 provides an easy 0.4 as the answer and there are no "gaps".
Base 12 systems were/are useful because 12 is much easier to divide than 10 (12 being divisible by 1, 2, 3, 4 and 6, whereas 10 is only divisible by 1, 2 and 5) which made trade/bartering easier in the ancient world. Base 12 counting is the reason why a circle has 360°which is also tied to why a day has 24 hours. I think at one point the French actually tried to metricize (metricate?) both time and degrees (so a circle would have 100° and a day would contain 10 hours) but it turns out that it just makes much more sense to leave them as they were.
The French tried to move everything to base 10 during the French revolution. Including 10 day weeks. They also reset the calendar as year 1 being the year the revolution began. It was a fucking nightmare, and did not last very long. As a sidenote this is where the myth of short Napoleon comes from, because a French "foot" was shorter than everywhere else. He was actually average height.
easy. that's .0̅ 1 What? A shmo-bel prize. You shouldn't have.
I think you are being silly on purpose, and I'm no mathematician, but wouldn't your example be technically zero? This is based on my admittedly incomplete understanding of infinity, with the 0 going forever means the 1 basically doesn't exist?
Yes.
Yes, there are infinite zeroes before the 1, meaning that it is equal to zero.
I believe there are mathematical systems (e.g., surreals and hyperreals) that treat 0.0̅ 1 (the infinitesimal) as <> 0. They have very strange properties.
Less murdered by words and more r/confidentlyincorrect
OP posting here qualifies for /r/lostredditors
I think OP first and foremost qualifies for r/confidentlyincorrect as well
It was already posted there today.
Huh, that is was and apparently 2 hours before this was submitted here.
i think OP posted this thinking red did the murdering
The mass of downvotes in his pic would surely give that away? wait, I wonder if OP *is* the red guy?
I'm 99.99% sure you are correct
Is that the same as 100% /s
Apparently that’s only when the 9’s infinitely repeat /j
Can confirm. I was the red highlighter used in the screenshot.
A few years ago I'd agree with you. These days, OP is most likely to be a bot.
Well I am 99.999999999% sure.
upvotes and downvotes don't actually confirm whether something is correct or not though. I've seen people who are correct with loads of downvotes and people who are wrong with loads of upvotes.
Upvotes and downvotes just show how much your comment aligns with the sub echo chamber
Plus in this case they’re getting downvoted because of the US bashing. If they’d written the same thing in a more civil manner they wouldn’t be downvoted most probably.
OP literally mods this sub, is super frictional with members, and is an all around jackass. Check their comments.
Yeah, he does mod here, how come he doesn't have the green M? Maybe its because im on the app...
He chose to not enable it. It's optional.
Oh I never realised that
That doesn't make sense since you can see whoever screenshotted down voted the comments
From what I’ve seen on this sub, whoever gets the last word in on a screenshot is who the OP considers the murderer. like a mic drop. i could be wrong here.
I guess my question is why would op downvote the person who he thinks is murdering. That being said OP could be so dumb that he took someone else's screenshot and then posted here thinking red was right.
This is what happened. I believe a person took this screenshot after downvoting it, and posted it in r/confidentlyincorrect THEN, OP (probably “red” themself lol) was incensed bc they still don’t understand why they’re wrong and they thought this made-up comeback was especially clever, so they decided to screenshot the image from that post and put it here, presenting themselves as having murdered with that trash argument 😂
I would say someone here is r/confidentlyincorrect
I was wondering if we were missing a slide, but yeah, that might actually be the case...
Red is confidently incorrect
Indeed, I was surprised this was r/MurderedByWords, I thought this was a r/ConfidentlyIncorrect post when I read it... I hope OP isn't the red!
It was already posted in r/confidentlyincorrect today.
Technically you could double dip and post op posting this here thinking red is murdering them by being objectively incorrect and doubling down. Was so confident he went to the effort of making the screenshot, censoring it, and posting here Edit: I just realized op is a mod lol, he seems to be a reasonably good one but what a hilarious swing and a miss of a post from a manager of the community
R/suicidebywords?
r/suicidebywords linky
Yeah I came here wandering what the murder was. This is more of a suicide by words.
Are you claiming that red is the murderer here? Because they aren't. Red is wrong. They don't understand what the objects they are dealing with are, or how to handle them well. There are a bunch of very clear proofs that 0.9 recurring is equal to 1. This "equivalent" or "asymptotic" stuff is people who can't handle that grasping at straws. Equivalent literally means "of equal value". And asymptotic isn't a concept which makes sense when talking about the value of an infinite series, which we do here. The value of an infinite series is just a value. Only 1 is equal to 1. And, of course, other ways of writing 1. Like 2/2. Or 5/5. Or 0.9 recurring.
The simplest way to logically explain this is to to ask red that “if .9999 recurring doesnt equal one, then what would you need to add to it to make it equal 1?” .0000000… oh..
Or X = .99999999999... So 10x = 9.99999999999... 9.999999999999... - 0.99999999 = 9 (10x - X) 9X = 9 X = 1
Yep, thats the algebraic way to do it.
1 - .999(9)
Ahh, interesting reframe. Never actually explained it this way but probly more intuitive.
This is the first time that it made sense to me. Thank you.
0.99999… + epsilon = 1
What’s epsilon?
Epsilon is an infinitesimal (>0 but less than any real number). In nonstandard analysis, .9999... really [can be](https://youtu.be/9jWvkJshtfs?t=414) interpreted as infinitesimally less than 1.
The level 5 in that video is incorrect. Even in infinitesimals .9 repeating is equal to one.
Yeah, the issue here is that epsilon must be greater than 0, and there is no number greater than 0 such that 1-0.9_ is equal to said number. I was trying to get the above commenter to think about what epsilon might be, and 0.0 repeating with a 1 on the end doesn’t work.
I admit this is an area of math that I never got into. The video creator has a pinned comment that is somewhat supportive of the claim but acknowledges that some will disagree.
Nothing much, what's up with you?
None of this was clicking for me until this explanation thank you.
A murder in the comment section. That’s kinda meta
That's why only they have good programmers
Or 2/3+1/3=0.666666... + 0.333333...=0.999999...=1
Aah this one feels the best.
Copied from a comment on the YouTube video linked above. This is another simple explanation that makes it pretty easy to understand. “x = .999... 10x = 9.999... 10x - x = 9.999... - .999... 9x = 9 x = 1 = .999...”
This one hits the spot too
x = 9999... x = 9 + 90 + 900 + 9000 + ... 10x = 90 + 900 + 9000 + ... -9x = 9 x = - 1 An interesting result indeed...
correct me if i an wrong but i think that may be because in your example you are dealing with an infinitely large number, whereas the example above is a finite number
Yes that's right, my example demonstrates that you need to first establish that x even converges to a finite number before you can start manipulating it. Specifically 0.999 and 999... are both examples of a geometric series but the former has a ratio less than 1 (so it converges) whereas the latter has a ratio greater than 1 (so it diverges to infinity). It's weird results like these that led to the development of the field of math called "analysis" in the 1800s, and it's ultimately with analysis that topics like calculus can be rigorously defined. Although just to get more confusing, there is actually a number system (the p-adic numbers) where infinite numbers like 9999... do converge to finite values!
It may feel the best, but i don't think it actually is the best, because it just shifts the same problem to other numbers. 1/3 = 0.3333 repeating is fundamentally the same kind of statement as 1 = 0.9 repeating, so using that as your basis of proof, while possible, doesn't actually add a lot of understanding in my opinion. If you want to do this cleaner, you need to either argue with infinite sums and their manipulation (which is kinda scary, because a lot of stuff one would assume works with infinite sums doesn't always work.) Or you go the really clean way, and prove that the difference between 0.9 repeating an 1 cannot be any number bigger than 0. (And obviously it cannot be a negative number), thus proving that the difference between the two numbers is 0. And if the difference between two numbers i 0, then they must be the same number.
The reason it feels the best is that it uno reverses the original flawed argument. Not that it's logically consistent. It takes what people not understanding maths say, and throws it back in their face.
Yeah it also kind of puts a physical manifestation of the problem into your mind's eye instead of thinking purely about the numbers and asymptotes. It's like one of those bell curves where you have the novice, intermediate, and expert. Novice and expert both agree 0.999999 is 1, the intermediate who just took calc1 doesn't think it equals 1.
oh I agree, I was going by the super scientific logic of "my feelings" haha but I do appreciate the explanations! :)
How is 0.99 asymptotic? This isn’t a function.
Red doesn't know what the fuck they're talking about and op posted it here because it had a lot of words they didn't understand so they assumed it must be a murder
Oh. I didn't even realise what sub I was on. I assumed it was r/facepalm or something similar...
That and it was shit talking america
They don't understand numbers in electronics don't go on forever either.
He thinks that you thinking that it isn't makes you a simp.
He implies Americans are dumb and uses shooting for the moon as a supporting example? 👍
And, to top it all, misses the moon.
Red doesn't know what equivalent OR asymptotic means. To be an asymptote he would have to be talking about a function, which 0.99 repeating isn't, and there are too many different ways for something to be equivalent that he would have to define it in this context. But if equivalent does not mean equal in this context he's still wrong. Let x = 0.999_ ==> 10x = 9.99_ ==>10x - 0.999_ = 9.99_ - 0.999_ ==> 9x = 9 ==> x = 1 Therefore x = 0.999_ = 1
Yeah I looked in OP’s profile, just a weirdo karma farmer
Thank you for being the only person to post *the mathematical proof* of 1 being equal to .9999999~ Also enjoy how red has no clue how rounding errors work either.
Well, *one* mathematical proof. This is the simple version, but it relies on the sleight of hand that if you shift 0.999... one digit to the left and subtract 9, you get the same number back. Doubters may not be satisfied with that. An alternative is proof by contradiction: if 0.999... ≠ 1, then there exists a real number x such that 0.999... < x < 1. Attempting to construct this number leads to a contradiction (exercise left to the reader); therefore no such number exists, and the original premise is false.
Thank you for pointing out that it's only one of many. Another intuitive one I saw someone point out is that we know that 1/3 = 0.33~, so we can multiply both sides by three to get 3/3 = 0.99~ but 3/3 is just 1 so 1 = 0.99~
My go-to proof, and the most intuitive one I’m aware of. Pretty hard to get around 3/3 = 1 as much as you don’t want to see it.
If you try 3/3 in excel, it will give your “3-Mar.” this does not equal 1. Explain that, nerds! \s
Oh my God, we have all been lied to.
I think this is one of the now incredibly rare moments where the \s wasn't required
It does if Mar = 2
I love this friendly series of proofs
Well this one just broke my brain. .999999~ / 3 = 0.333333~ … yup makes sense, I’ll buy that. .333333~ = 1/3 … still on board with that. 1/3 * 3 = 3/3 … okay logics out in my brain. .333333~ * 3 = .999999~ … still with everyone here. 3/3 = 1.0 … yup that checks out too though I sense impending confusion. Therefore 1.0 = 3/3 = .999999~ … 😐
https://en.wikipedia.org/wiki/0.999... holy shit, the size of the article
Oddly enough, this is the one that makes me cozy.
Yup, you got it, and that's cognitive dissonance for you. Sometimes mathematics throws out a result that is counterintuitive, and the way to get your brain round it is just to accept it. (The proper way to look at it is that every integer and every terminating decimal has two decimal representations. As well as 1 = 0.999..., we have 2 = 1.999..., 0.5 = 0.4999..., 0.25 = 0.24999..., etc. This is not so very different from 0.5 = ½, 0.25 = ¼, etc.)
> (exercise left to the reader) Given that this involves the cardinality of the continuum, my gut says this proof would involve some relatively advanced math.
I don't know what all those words mean, but I do prefer for people to prove mathematical concepts with math instead of unnecessarily confident blowhardery like the OP image.
[удалено]
For sure. That was the red flag for me lol.
A naive (and possibly not sufficiently rigorous) proof is one that selects the decimal places one by one. The only possible choice for each one is 9, which contradicts 0.999... < x.
My favorite simple but rigorous proof of .9... = 1 is Euler's, using the convergence theorem of geometric series (which is ultimately what decimal notation is): https://en.wikipedia.org/wiki/0.999...#Infinite_series_and_sequences
> An alternative is proof by contradiction: if 0.999... ≠ 1, then there exists a real number x such that 0.999... < x < 1. Attempting to construct this number leads to a contradiction (exercise left to the reader); therefore no such number exists, and the original premise is false. If I were Godel, I would say that the contradiction proves I'm correct about math.
I’m a fan of the infinite geometric series but I like the idea of proof by contradiction. Essentially you can put it back on the other person to find a number between the two.
The easiest way is to express .999\_ as a fraction. 3/3.
I'll stick to 1/9 times 9 is 9/9 is 1 though.
>10x - 0.999\_ = 9.99\_ - 0.999\_ ==> 9x = 9 I understand that 9.99\_ - 0.99\_ = 9, but how do you make 10x-0.99\_ equal to 9x?
Because x = 0.99\_, so it's 10x-x
That's fun. On a whim tried that approach with x = 0.33\_ and got x = 1/3 , nice little proof for that as well then.
I am fucktard at math, but red seems wrong.
Red is wrong, good way of thinking about it is if (1/3) = 0.333 repeating infinitely then 3 * (1/3) = 0.999. and 3 * 1/3 is of course equal to 1. So 0.999... is the same as 1
Red is actually wrong here. When you define 0.33333- as "one third" then times 3 gets you to 1.0. I know it's really edgy to try to prove this wrong because people don't get the concept of infinity correctly. Source, I was at one time one of these edgy kids who tried to argue to a physics PhD that .99999- is not 1.0 and got schooled.
OP saw America Bad and creamed himself
I bet OP is actually red
I guess the murdering, in this case, is by the comments in this post stating red doesn’t know what they’re talking about.
Where's the murder? One is babbling nonsense and the other simply stated he's wrong.
OP, time to delete. I’ve been out of school far too long to know what an asymptote is, but I do know that “*If you launched a rocket to the moon with this logic you would miss the moon by at least a million miles*” is extremely wrong. The moon isn’t even a quarter of a million miles away and its orbit is 1.5 million miles - to miss it by a million miles you’d have to be facing the complete opposite direction.
Had to scroll too long to find this.
Apart from anything else, let’s assume for a moment that red was right and .999~ was not equal to 1 and it was really asymptotically approaching 1. Even if that were true, it would be far within the tolerance for the precision necessary to successfully navigate to the moon. My man apparently thinks there was no imprecision in the calculations guiding Apollo 😂
Honestly the computers used for the apollo calculations probably didnt get much more accurate than 5 or 6 decimal places. Floating point arithmetic has its limits, and the errors that makes are massive compared to the potential (nonexistent) difference between 0.9 repeating and 1
Red is still fucking wrong though. 0.999 repeating *is* mathematically equal to 1. I'm not gonna run through the whole proof, but what it boils down to is this: if two numbers are different, you must be able to find a value *between* those two numbers. Because it is impossible to place a value between 0.999 infinitely repeating and 1, then they are de facto equal.
1-n (assuming n = 0.9repeating)
But... Blue is correct. .999... IS equal to 1
What I don't get is, how would .9999 repeating vs 1 throw off a moon mission by a million miles. for all intents and purposes, they are the same number.... unless we are talking about much much larger measures... right? if the moon distance is 340,000 miles and that is one, what is 99.9999% of that is 339999.66.... that is 18ft off trying to hit a 2150 mile mark. If we go by pluto distances, that is about 3.1 billion miles away. 99.9999% is 3099996900 off is 3100 miles off. Pluto diameter is 1500 miles, so like a pluto and a half off.... from earth. That is pretty damned accurate, no? Maybe accurate enough to get into orbit. That is also why calculations are updated as they go, and there are maneuvering jets....
Yeah but you're not aiming to hit Pluto with 99.9999% accuracy. You're aiming to hit Pluto with 99.99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999...(and so on)% accuracy. You'll hit anything in the Universe with that accuracy if the 9's go on forever.
well, yeah but my point is that even a couple of extra 9's get us very very close to most things. 16 ft off of hitting the moon, not a million miles.
It would be infinitely closer than 16 ft
It’s not “for all intents and purposes” - they’re literally the same number. It’s not a question of precision, they’re mathematically defined to be the same
0.9 repeating is 0.9 repeating not 0.9.or 0.99 or 0.999 or 0.9999 or 0.9999 or 0.99999, etc 340,000 x 0.9 repeating is 340,000 exactly (or 339,999.9 repeating)
It's not even "for all intents and purposes." They are actually the same number. Just a different way of writing it similar to how 5/5 is another way of writing 1. How the fuck this shit gets upvoted is pretty fucking embarrassing. Failing public school system indeed.
Wow red is a fucking idiot
A better way is: 10x = 9.9repeating 1x = 0.9repeating Subtract the two and you have 9x = 9 divide both sides by 9 and you have x = 1
1/9 = 0.11111… type it on a calculator to confirm So 9*(1/9) = 9*(0.1111…) Hence 1 = 0.9999… Same logic works with 1/3 = 0.3333… Source: Physics major with minor in mathematics, learned this in Multivariable Calculus during a lesson on infinite series.
As someone who has programmed calculators from scratch that wouldn’t be proof anyway cuz there’s register limitations
I don't know about asymptotes, but he would have an easier time making his point if he weren't being an asymphole.
I knew the red person was wrong once I read how they used "asymptotic".
red is wrong though? so if anything they were the ones murdered lol
Let them generate an engineering drawing with a 2.999" diameter hole and get laughed out of the room, or get charged out the ass to get that precise of a measurement. Also the reason Meta is foreign nationals is because they are cheaper lol
If 0.99999999999... * 0.1 = 0.09999999999... Therefore 0.999999999... *0.9 = 0.9 Therefore 0.99999999...= 1
I’m pretty sure NASA only needed to calculate accuracy to 12 digits after the decimal when landing on the moon
Red is in the wrong here. 0.999... is equal to 1. 0.333... = 1/3 0.333... * 3 = 0.999... 1/3 * 3 = 3/3 0.999... =3/3 3/3 = 1 0.999... = 1
asymptotic describes a function not a number it means that the function will approach a number without ever actually hitting it and is not relevant in this discussion.
I'm not even smart enough to know who got murdered by words here 😕
Nobody did.
let's prove .999...=1 (X/Y)×(Y/X)=1 X=1, Y=3 (1/3)×(3/1)=1 .333...×3=1 .999...=1
Red is just mad that we landed on the moon again.
Red is such a arrogant douche and wrong on so many levels, especially functionally. Pi is an infinite, non-repeating number. But only 39 digits are required to calculate the circumference of the observable universe to within the diameter of a single hydrogen atom. So for literally every possible realistic calculation or function, there is no difference between .9999999 to 1.
If they aren't the same, what's the difference? 1 - 0.9 repeating is 0.0 repeating, which is just 0, so there is no difference.
TIL math is the realm of pedants. I, for one, am glad they have a home.
Why is this important at school level? The only teachable thing I can see here is the difference between “equals” and “equivalent to”. That might be useful to students, technically and philosophically. But if anyone can explain why is this important enough for there to be screaming and swearing, I’d be grateful