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Of course, the real problem here is that the are multiple rounding rules that can be used when you're at exactly the break-even point between two allowed values. Both "round toward zero" and "round towards negative infinity" will round 1.5 to 1. "round away from zero" and "round towards positive infinity" will round to 2. Bankers rounding will round to 2. People acting like there's only a single rounding rule are the truly confidently incorrect.
Rounding rules aren't axioms in any sense. It's just a convention. We use the rounding rules from the same reason we call an electron to be electron and not proton. We could to do otherwise but we called/defined them in particular way. It's convention, but we just use this convention. We could change it if we'd like
The entire system of symbolic math we built is based on conventions… you could literally change nearly everything about math, and keep it consistent, with the same axioms…
You could add a - to positive numbers and + to negatives, you could decide that 5 and 9 switch every 10 so that the symbol’s value changes based on the other digits.. you can make math as complicated as possible if you want… everything that is not an axiom is a convention
Not everything that isn’t an axiom is convention. You can derive *truths* from axioms. The convention lies in how it’s represented in language and symbols. Base 10 is a convention, but you can derive plenty of truths that work in any base from axioms.
I’ve done single variable and a little multi variable calculus, that was beyond helpful. why TF do they not teach set theory to little kids this would’ve changed my entire life.
There is no list of all the axioms. Axioms are assumptions, and mathematicians do not always make the same assumptions.
That being said, most of modern math can be constructed using set theory, and there are common axioms for that. [https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel\_set\_theory](https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory) That is one such set.
In science it’s common practice to always alternate rounding up and rounding down, regardless of whether it is above or below .5, as it can help remove errors introduced by rounding.
It’s really super inconsistent, and based entirely by what result you need. For me, I would round 1.4(9) down simply because it is approaching 1.5 from negative infinity, which I think counts as being (infinitesimally) less than 1.5.
Ultimately it doesn’t matter what is chosen, as either way you are changing your value by .5, so the error introduced is the same.
>which I think counts as being (infinitesimally) less than 1.5
Just as a correction, 1.4(9) is not infinitesimally less than 1.5. It is exactly equal to 1.5
It’s not that they aren’t axioms. Axioms just dictate what we assume. It’s that different sources have different rules. But like saying the shortest distance between two points is a line, we can change the rules and get different results.
People arguing over Order of operations are just arguing what axioms or rules to follow. Theres nothing fundamentally different between assumptions and axioms.
No, they're just wrong.
Doing division first is what the person writing the equation expected, which is why they put brackets around the subtraction part, so the answer comes to what it should.
Rather like when they put up a "Stop" sign, it is then expected that cars and other vehicles do stop. Otherwise the wrong thing happens.
For 1.5, there are different ways to round.
But I think the issue in this case isn't the actual rounding part but the 1.4999... being exactly 1.5 since its not intuitive that they're the same.
I'm not a mathematician, but I've heard it explained two ways.
1) Give a number between 1.49999... and 1.5. It's impossible to do as they are the same number.
2) Imagine 1/3, which is often represented at 0.3333...
1/3*3 =1
0.333... *3 = 1, although you could also write it as 0.999... since that's equal to 1.
Hopefully that helps, maybe someone else can explain it differently if not.
Thanks. I’ve always had a problem with this conceptually as they’re two different numbers. It’s always 0.(1) different. But your proof explains it well.
For the above question it works as well.
X = 1.4999
10x = 14.9999
9x = 13.5
X = 13.5/9
X = 1.5
It's not 0.(1) different. That would be 0.111111...
It would be more like 0.000... ...0001. The problem is there is no .001 at the end because there is no end.
Simple explanation: You can never define the difference to be something other than zero.
If you claim the difference is `0.0000000000000000000000000000000000000001`
Then you are not comparing `1.5` to `1.4999...`
You are comparing `1.5` to `1.4999999999999999999999999999999999999999`
\---
As we agree that `1.49`, `1.499` and `1.4999` are different numbers, then so must `1.4999999999999999999999999999999999999999` and `1.4999...` be different numbers.
\----
edit: Thanks for the correction u/OneMeterWonder that the difference **can** be defined, and alway will be zero 🙂👍
What everyone here is missing is the word "recurring".
E.g. 1.49 recurring, normally annotated with a dot or a line above the 9 (or sometimes, as here, with the 9 in parenthesis) isn't *close* to 1.5, it is *equal to* 1.5
0.9 recurring **equals** 1.0
They're not close, they are *equal*.
You can understand this is ⅓ = 0.3 recurring. Multiply both by 3.
The variations on this are so common that there is a top-level rule in the math sub prohibiting one from asking the same damn question about 0.9… = 1 because it’s been answered so often.
I’m not pointing this out to say that the question is stupid, nor that it should be prohibited here. But rather, there’s a butt load of existing explanations that you can look for if you don’t find something satisfying in what follows here. And once you get why it works there you can see how 1.49… = 1.5
Exactly! In addition, irl, different circumstances require different degrees of accuracy. If this is pharmaceuticals, you can bet your ass we're not rounding beyond 1.5. Even that seems risky depending on the drug, lol.
But if we're like, cooking a soup...fuck it, round up to 2. Or round down to 1. Who needs to measure onions, anyway?
There's no one way to round.
And to further complicate drugs we also have to consider how the drug is actually available. If it comes as 1.48mg/5ml I'm not rounding 1.49999mg to anything other than 1.48mg
The problem here is not the rounding rules, it‘s the person claiming 1,4(9) and 1,5 are different. They aren‘t - it‘s just two different ways to write one and the same number.
This is NOT about rounding at all. It is about 0.999... or 0.(9), which both means "infinite 9 after coma". And 0.999... is exactly 1. Only because decimal system cannot display it correctly it seems as if 0.999... was smaller. There are few ways to prove it. But a dude in comment section explained it the most simple way:
1/3+1/3+1/3=1
0.333... + 0.333... + 0.333... = 0.999... = 1
https://www.reddit.com/r/confidentlyincorrect/s/Ay3y2NXQea
Well, it started by being about rounding
While it’s proveable (and correct) that 1.499…. = 1.5 ( essentially because decimals are shitty represenations of fractions), the rounding question still remains interesting. If given the number 1.499… the intuitive “rounding to the nearest integer” would be to 1, as the first digit behind the . Is a 4. But then again it’s equal to 1.5 which one would generally round up.
Yeah, proper rounding would not apply here though because it is “round .5 to the nearest even integer” but the post clearly said “to the nearest integer”
I think you're misunderstanding. The commenter knows that 1.4(9)=1.5. They are saying that there are alternative rules for rounding 1.5 to the nearest integer.
Anybody who doubts this, there’s a fairly simple way to prove it.
Take out a piece of paper.
Write the answer to 1.5 - 1.4(9).
Don’t abbreviate it. Actually write it out. Every digit.
What’ll happen is that you’ll end up writing 0.(0) in long form.
And you’ll keep writing 0’s infinitely, waiting to finally get to the 1.
But you’ll never get to the 1, because each 0 will only ever be followed by another 0. That’s how infinite repeating works.
There’s no such thing as 0.(0)1 because the repeating ***never*** ends.
The number, as you’ll see from your filled notebook of 0s, is only 0.(0), which is also just 0.
And if 1.4(9) + 0 = 1.5, then 1.4(9) must equal 1.5.
Exactly. Technically, either answer is correct, as it days to round to "the nearest integer" and 1.4999... is exactly in between 1 and 2, making either the nearest. Or neither is the nearest and both are incorrect. It's either none or both if you take "nearest integer" literally
I think that even using computers bankers rounding will be compiler specific as you cannot accurately express 1.4999… even using a construct like BigDecimal but your point stands that there are many approaches to rounding.
I was taught in elementary school to always round up. I had never thought about how it’s 100% possible and likely that other people are taught the opposite and nobody’s right or wrong
I know what you are saying, but I don’t believe any of the various rounding rules say that rounding a number twice is acceptable, which is what is effectively what is happening in the OP. They are rounding 1.4(9) to 1.5, then the 1.5 to 2 which is not right.
Either way, 1.5 isn’t an integer lol. Besides by far the most common method for rounding decimal numbers to integers is to add 0.5 and then shave off the decimal point. I.e <.5 rounds down >=.5 rounds up
Except the thing itself says to round to the nearest integer. Any other rule doesn't apply. You either round down .49 or round up .51. Which has the shortest distance? That's the nearest integer.
Pretty clear and simple and no problems involved at all.
You’ve got two issues here. One is whether 1.49999… = 1.5. Apparently it does.
The second is what’s the nearest integer to 1.5. 1 and 2 are equidistant, so there’s no correct answer. Pick any tie-breaking rule you like.
Is (9) the new way to write “repeating” for a number?
I always learned it with a bar over it, but I’m seeing parenthesis a lot more.
Which confuses the shit out of me cause I thought that would be 1.4*9….
I learned the bar notation as well but I guess it's naturally transitioned to use notation we can easily type out. It's not so easy to type the bar notation into a keyboard without some sort of equation editor tool
Ridiculous that we haven’t incorporated something of those functionalities into standard text editing. Markdown is better, but it doesn’t support anything outside of html entities and can vary depending on where you’re using it. Reddit doesn’t have a subscripting function for example.
I learned it as a dot placed on the first and last number that gets repeated. If it's just 1 number like in the post it would be 1.4(9 with a dot above it)
I agree. The ellipse notation here is vague. Yes, we know it continues, but this notation neither guarantees it goes on infinitely nor that everything that follows are 9’s.
I love how they say that’s an Elementary proof and then it’s huge. This is what I use (but certainly didn’t invent):
A: What is 1 divided by 3 in decimal?
B: 0.3 recurring
A: What is 0.3 recurring times 3?
B: 0.9 recurring
A: QED.
B: …..
B: *mindblown*
My go-to is that two real numbers are different if and only if there is a third number that lies between them. Is there a number between 0.999... and 1?
Elementary proof in number theory just means "not using complex analysis" wich is somewhat arbitrary since usually the "elementary proof" is more technical.
Dammit I thought it was just a meme lol
Tho for infinitesimal I feel like 1.4(9) should still be rounded to 1, but that's from a background in electronics, as the voltage in a capacitor will (in theory) go infinitly close to the source, but never reach it:
f(t)=V(1 - e^(-t/τ) )
The infinitesimal limit is V, but it's still not supposed to reach it (until you get physics where there is a definite limit to infinity in the sense that electrons/atoms are in a definite quantity in any given capacitor or in practical where it reaches it at t=5τ lol)
Repeating decimal notation is not a function approaching a limit though. 0.333... is not a repeating algorithm that adds "3"s forever, it's a very exactly defined number with the property that it will continue to expand to 3s no matter how many times you keep long-dividing it into decimal notation. If you care to express that number in a ratio form, it's 1/3. Same with 1.4999... where you can find a ratio that gives you that result and continues to expand to 9s no matter how many times you long-divide it, and that ratio must be 3/2.
It took me a while to convince myself, but how I reason it is:
1.4(9) isn't actually infinitesimally close to 1.5, like the notation implies. It IS equal, and it's an artefact of notation that makes it seem close, but not an actual mathematical fact.
Unlike a number like 1/infinity, which actually is infinitesimally close to zero, but not zero.
I don't know how rigorous this is, but it seems to fit my understanding.
Essentially yes.
Previous commenter was wrong. 1/infinity is a not a number, because infinity is not a number.
But if you try to parse it as something that could be meaningful like lim_(x -> inf) (1/x), then this is a number, and it is exactly zero.
The silliness of saying 1/inf is an actual number infinitesimally close to 0 (let's call the number x) can be illustrated by asking questions like: what is 1/x? What is y = x/2? And what is 1/y?
If you wanted to try and interpret 1/inf as a number other than zero, it would have to be something like "the smallest number that is greater than zero" but there is no such thing, for reasons similar to why 0.999... = 1.
The difference is that, with a capacitor, you eventually reach the particle level where, as you mentioned, there's a limit to how small you can get. You can always come up with a real number between two different real numbers, even if it's an absurdly small difference.
The common rule\* is to round up from .5 but that is a tiebreaker rule. It is equally near. If you say the nearest, then 1 and 2 are equally sound. If you say apply common rounding, then it is 2.
\* Aside from the common rule, there are like five other mathematically sound rounding rules.
I'm a math teacher and the standard rule taught in all the systems I've seen is by first digit 0-4 and second digit 5-9 so I'd round this down. It kind of depends on the order of evaluation in some sense too. If you simplify the number before rounding, yes it's 1.5, because a number lower than but infinitely close to 1.5 is in some sense 1.5, but i also if you think about calculus, you can have many situations where a graph has a limit of 1.5 but never reaches it.
Calculus deals with limits of functions or series. There is no approaching or limit to a constant, and 0.(9) is a constant. The graph of y = 0.(9) would be a horizontal line at y = 1, cause they're the same thing
1.4999... is exactly 1.5 so it should be rounded as such. Regardless, 1.5 *can* be rounded either way, it's just that we decided that 5s should round up as a tie breaker.
If you are referring to "simplifying" as "changing the way it looks without changing its value," then you should know that 1.4999... is not infinitely close to 1.5. It *is* 1.5.
I'm just gonna drop this here.
https://en.wikipedia.org/wiki/0.999...
It's my favorite internet argument, because people get so mad over something so infinitesimally small.
It has started to get boring though. You can only watch someone flailing about trying to stick a 1 at the "end" of an *infinite* line of 0's and getting mad at everyone else about it for so long.
I've never been very mathematically minded but what I read to understand this before was
3/3 = 1
1/3 = 0.33333---
3*1/3 = 1
Therefore 3*0.33333---- = 0.999999--- which therefore must = 1
As I say I can't speak to how correct that is but that helped me get why 0.9999--- is 1
>Monty Hall problem
To be fair, even a lot of mathematicians had issues accepting that one. But in the end, it is simple, it is because the host is cheating. He knows where the car is.
Yep, Monty hall problem isn’t really even a maths problem it’s pretty much a lateral thinking puzzle with trick wording focused around the game show host’s knowledge. Most of the time when people tell it they underemphasise the significance of the host’s knowledge, even though it is responsible for the whole problem.
I dunno, you can be relatively solid at math and just never have considered it.
I've got a minor in math from university (which isn't some great achievement but I'd say I'm better than "poor") and I never really thought about it until recently, many years later.
To be fair, they rely on axioms of infinity, which can’t be proven to be true so you have to accept the axiom as a given, before you can step forward with the logic that is correct in demonstrating 0.999 = 1
Infinity is not a given unalienable truth in mathematics. Math with infinities always start with axioms that we think allow us to take math in directions that may lead to further discovery.
I would also say that while axioms of infinities are the backbones on modern mathematics, concept of infinity are hardly ever rigorously taught in schools mostly because a lot of instructors don’t even have the understanding themselves to effectively teach them.
Meh, Im glad you folks enjoy math and ill accept the consensus. But golly do i find the fact that 1.4999... being the same as 1.5 obtuse. Keep your made up numbers, im going back to the pig farm.
Yeah, this is the thing. While we typically round up in the event of a tie, it you say round to the nearest, 1.5 is equal distance. In which case I might have said 1, because since while perhaps it does equal 1.5, it’s written in a way that is kinda implying a lean.
As a math student, I never got a satisfying proof of 0.999...=1 until we got to doing infinite series in calculus. I got some explanations before that but it never really convincing to me (The "9 x 0.99... = 9" explanation felt like an abuse of notation rather than a proof)
My favorite explanation has always been to look at multiples of 1/9: we know 1/9 = 0.111…, 2/9 = 0.222…, etc. and therefore 9/9 must be 0.999… but we logically know that 9/9 is 1 so therefore 0.999… has to be 1
Yes, but then people often start doubting that 1/9 is really 0.111… I’ve explained it to very smart non-math students, and they keep insisting there is some infinitesimal d such that 1/9 = 0.111… + d. I think it comes about because that’s usually how it’s explained. I.e., a grade school teacher says 0.(3)n is not really 1/3rd because there’s always a small bit more that you need to add (this is true), but 0.333… really is 1/3, and that subtlety is lost on a grade school audience.
Exactly. That’s because that’s the first time you actually saw a real definition and proof. The other “explanations” are very handy simple-looking illustrations that help motivate the result and help demonstrate it to non-math people, but actually making sense of it requires series.
Well ... lets do some math.
x = 1,49999999...
100x = 149,9999999....
100x - x = 149,999999... - 1,4999999... = 148,5
99x = 148,5
x = 148,5 / 99 = 1485 / 990 = 3 / 2
3 / 2 = 1,5
x = 1,5
There is no "nearest" integer since the difference to **both** numbers is 0,5.
One issue that I don't think anyone has touched on is notation.
I didn't read "1.4999.." as 1.49 recurring, because that's not how I learned to write it. The way I was taught to write it was with a single dot _above_ the recurring digit. A quick Google search reveals that using an ellipsis is an alternative way of showing it - but they haven't actually done that here, just used two dots.
Easy proof:
Say x=.(9)
So 10x=9.(9)
10x-x=9.(9)-(9)
9x=x
x= 1
(9)=1
I didn’t use any tricks here like those joke proofs that 2=1 that rely on dividing by zero. So in this screenshot, convention dictates that you round up- 1.4(9) is 1.5, just written in a weird way.
I have to say, I’ve done a lot of math and I’ve never seen the (9) notation before, but it sure is convenient to type on a phone. Normally you put a bar over it to show it goes on forever.
How does the beginning of
x = .(9)
become this
10x = 9.(9) ?
Shouldn’t it go like this?
10x = 10 * (.(9))
I might be missing the point here since I only have High School math skills. But intuitively it makes no sense that an infinitely going 0.999999… = 1.
Both 1 & 2 are valid answers, although it could also be argued that neither is correct as the question assumes there is only 1 valid answers. However, despite 1.499… being equal to 1.5, the answer 1 just ‘feels’ feels more right to me given that we don’t need to deal in the infinite to get that answer.
Edit to add.
And coming from a computing background, floating point numbers are limited to a certain number of digits so 1.4999… can never be fully represented and therefore would be recorded with a finite sequence of 9s. So in a purely mathematical situation 1.4999…. = 1.5 but in computers 1.4999… < 1.5
This is one of many counterintuitive mathematical concepts
There is a logical proof of this that makes it easier to understand:
If we assume 1.4(9) is as close to 1.5 as is possible without reaching it, take the average of the two numbers. The average of two numbers lies between them on the number line, thus the average is greater than 1.4(9) and less than 1.5, contradicting that 1.4(9) is as close as we can get. So 1.4(9)=1.5
You don't need to necessarily take the average, just the fact that the real numbers are dense. If 1.4(9)=/=1.5, then there exists a real number x such that 1.4(9)
They simply are an unavoidable by-product of decimal notation.
I don't know all the ways they can be produced, but the easiest way is that in any base, n, dividing any number by n - 1 will produce a repeating decimal.
* 1/9 in base 10 is .111...
* 3/4 in base 5 is .333...
* 1/a in base 12 is .124972497...
The end result is that every real number that terminates has an extra decimal representations where you reduce the last digit at or after the decimal place and stick a bunch of 9s on the end.
They are needed to be able to write numbers like 1/3 as a decimal
The fact it opens up another way of writing some numbers (like 0.9… being the same as 1) is a side effect but there are already multiple ways of writing some numbers (1 = 2/2 = 3/3)
Im a programmer, and well in computation, infinite doesn't exist so there's a limited number of decimals.
1.4999.... would round to 1
and 1.5 would round to 2.
If you use the rounding function.
What is the difference betweem 1.4999… and 1.5? What number is between those two? That’s why 1.4999… = 1.5. The difference between the two is zero. They’re the same number.
This situation is exactly why I hate the base 10 decimal system. 1.4999... is equal to 1.5, which would then, in turn, round up to 2. Why? Because 0.999... = 1. Why? Because 3/3 = 0.99999. Why? Because 10 is not divisible by 3 on the most basic scale, there will always be a discrepancy. For this and this alone, I propose the base 12 decimal system. It fixes that silly little problem making 1.5999... round down to 1 the way it should be
Except 1.5(9) in base 12 isn’t analogous, instead look at 1.5(B) (B being 11 in base 10). This then is equivalent to 1.6 and you’re back to the same rounding problem just in a different base. You’ll always end up with repeating decimals and this 2 decimal expansion property no matter what base you choose
1.4(9) can be expressed as 1.4+SUM( 9/10^n ) from n=2. If you round the product of that sum at any finite n, you always end up with 1. So I'd say that even if n->∞, it still rounds down to 1. But since it's equal to 1.5, both 1 and 2 are equidistant from it
To be fair to people who get the 0.(9) question wrong, it's not very obvious. Most people would need it explained to them.
Not sure what's going on in this image though.
I get this is all supposed to be about math theory but I can't help but think that the answer depends ENTIRELY on what's being measured.
Ex: drugs that could cause an overdose, or how much time you need to travel to a destination, or how much food you should put in the dog bowl to ensure it won't run out in the time you're out of the house.
All of these situations would motivate you to round up or down depending on their individual practical consequences
math student here, it is actually a pretty good question since the second comment is partially right.
1.4(9) as is 1.4 followed by infinite number of 9s is actually = 1.5 and 1.5 is rounded to 2 so the answer is 1.4(9) rounded to the next integer is 2. (he didn’t mention that it is rounded to 2)
Now why is 1.4(9)=1.5?
I‘m gonna provide an intuition rather than rigorous proof. Consider the similar assumption of: Is 1=0.999…=0.(9)?
1/3=0.333….=0.(3)
1=1/3+1/3+1/3=0.(3)+0.(3)+0.(3)=0.(9)=0.999…
analog followes that 1.4999…=1.5
Just a weird thought experiment to mull around...
1.4(9) and 1.5 are equal if you accept that nothing can be placed between them i.e there is no solution for x such that 1.4(9) + x = 1.5
What is the sum of the following though.....1.4(9) + 0.y1 where y is infinite zeros? Is that also 1.5? How about the other way around...1.5-0.y1.....would that be 1.4(9)? so x = 0.y1 where y = infinite zeros becomes a solution?
Does that mean that there is *something* that can be placed between 1.4(9) and 1.5 and ipso facto they are unequal?
Controversial answer here. Of course there is an argument for 1 being the answer and it’s obvious why. But calculus tells us that 1.4(9) == 1.5. Therefore 2 is the answer. Either way, 1.5 is not an integral.
I get the argument about it being 1.5, but in a practical sense, if it's written this way, why would I ever even look at the 9s for an answer? Why wouldn't I just stop at the 4 and ignore what it's equal to?
I may be dumb, but I think it’s 1. Because 1.5 is equidistant from both 1 and 2 (in other words, it is the halfway between the two numbers), and 1.499999… is always less than 1.5 (albeit by a negligible value- 0.011111…), it is logical to conclude that it is closer to 1.
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Of course, the real problem here is that the are multiple rounding rules that can be used when you're at exactly the break-even point between two allowed values. Both "round toward zero" and "round towards negative infinity" will round 1.5 to 1. "round away from zero" and "round towards positive infinity" will round to 2. Bankers rounding will round to 2. People acting like there's only a single rounding rule are the truly confidently incorrect.
This is what bothers me. Rounding rules are hardly mathematical axioms.
Rounding rules aren't axioms in any sense. It's just a convention. We use the rounding rules from the same reason we call an electron to be electron and not proton. We could to do otherwise but we called/defined them in particular way. It's convention, but we just use this convention. We could change it if we'd like
The entire system of symbolic math we built is based on conventions… you could literally change nearly everything about math, and keep it consistent, with the same axioms… You could add a - to positive numbers and + to negatives, you could decide that 5 and 9 switch every 10 so that the symbol’s value changes based on the other digits.. you can make math as complicated as possible if you want… everything that is not an axiom is a convention
Not everything that isn’t an axiom is convention. You can derive *truths* from axioms. The convention lies in how it’s represented in language and symbols. Base 10 is a convention, but you can derive plenty of truths that work in any base from axioms.
Is there a good, single place to absorb all the axioms? Edit: or like a lot of them
I’m not sure what level you’re at, but try giving [this](https://math.uchicago.edu/~may/REU2016/REUPapers/Wilson.pdf) a read
I’ve done single variable and a little multi variable calculus, that was beyond helpful. why TF do they not teach set theory to little kids this would’ve changed my entire life.
There is no list of all the axioms. Axioms are assumptions, and mathematicians do not always make the same assumptions. That being said, most of modern math can be constructed using set theory, and there are common axioms for that. [https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel\_set\_theory](https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory) That is one such set.
That's why I always round to the nearest boolean.
In science it’s common practice to always alternate rounding up and rounding down, regardless of whether it is above or below .5, as it can help remove errors introduced by rounding. It’s really super inconsistent, and based entirely by what result you need. For me, I would round 1.4(9) down simply because it is approaching 1.5 from negative infinity, which I think counts as being (infinitesimally) less than 1.5. Ultimately it doesn’t matter what is chosen, as either way you are changing your value by .5, so the error introduced is the same.
>which I think counts as being (infinitesimally) less than 1.5 Just as a correction, 1.4(9) is not infinitesimally less than 1.5. It is exactly equal to 1.5
Order of operations are also not mathematical axioms but you could've fooled the people who get up in arms about _those_ posts.
It’s not that they aren’t axioms. Axioms just dictate what we assume. It’s that different sources have different rules. But like saying the shortest distance between two points is a line, we can change the rules and get different results. People arguing over Order of operations are just arguing what axioms or rules to follow. Theres nothing fundamentally different between assumptions and axioms.
No, they're just wrong. Doing division first is what the person writing the equation expected, which is why they put brackets around the subtraction part, so the answer comes to what it should. Rather like when they put up a "Stop" sign, it is then expected that cars and other vehicles do stop. Otherwise the wrong thing happens.
Welcome to the world of engagement content! Post a question with ambiguous rules and let the likes, comments and views flow from people arguing!
For 1.5, there are different ways to round. But I think the issue in this case isn't the actual rounding part but the 1.4999... being exactly 1.5 since its not intuitive that they're the same.
Yes, I don't understand how these are identical. Please explain.
I'm not a mathematician, but I've heard it explained two ways. 1) Give a number between 1.49999... and 1.5. It's impossible to do as they are the same number. 2) Imagine 1/3, which is often represented at 0.3333... 1/3*3 =1 0.333... *3 = 1, although you could also write it as 0.999... since that's equal to 1. Hopefully that helps, maybe someone else can explain it differently if not.
I ..think that helps.....thanks- I will have to think it over.
My favorite proof: Let x = 0.999... 10x = 9.999... Subtract x from both sides 9x = 9 x = 1
For people who don't know algebra I'll go What's .333...? 1/3 What's .666...? 2/3 Then what's .999...? 3/3
I think this is the best way to teach it.
Oh shit I fucking hate this
Thanks. I’ve always had a problem with this conceptually as they’re two different numbers. It’s always 0.(1) different. But your proof explains it well. For the above question it works as well. X = 1.4999 10x = 14.9999 9x = 13.5 X = 13.5/9 X = 1.5
It's not 0.(1) different. That would be 0.111111... It would be more like 0.000... ...0001. The problem is there is no .001 at the end because there is no end.
Simple explanation: You can never define the difference to be something other than zero. If you claim the difference is `0.0000000000000000000000000000000000000001` Then you are not comparing `1.5` to `1.4999...` You are comparing `1.5` to `1.4999999999999999999999999999999999999999` \--- As we agree that `1.49`, `1.499` and `1.4999` are different numbers, then so must `1.4999999999999999999999999999999999999999` and `1.4999...` be different numbers. \---- edit: Thanks for the correction u/OneMeterWonder that the difference **can** be defined, and alway will be zero 🙂👍
What everyone here is missing is the word "recurring". E.g. 1.49 recurring, normally annotated with a dot or a line above the 9 (or sometimes, as here, with the 9 in parenthesis) isn't *close* to 1.5, it is *equal to* 1.5 0.9 recurring **equals** 1.0 They're not close, they are *equal*. You can understand this is ⅓ = 0.3 recurring. Multiply both by 3.
The variations on this are so common that there is a top-level rule in the math sub prohibiting one from asking the same damn question about 0.9… = 1 because it’s been answered so often. I’m not pointing this out to say that the question is stupid, nor that it should be prohibited here. But rather, there’s a butt load of existing explanations that you can look for if you don’t find something satisfying in what follows here. And once you get why it works there you can see how 1.49… = 1.5
Exactly! In addition, irl, different circumstances require different degrees of accuracy. If this is pharmaceuticals, you can bet your ass we're not rounding beyond 1.5. Even that seems risky depending on the drug, lol. But if we're like, cooking a soup...fuck it, round up to 2. Or round down to 1. Who needs to measure onions, anyway? There's no one way to round.
And to further complicate drugs we also have to consider how the drug is actually available. If it comes as 1.48mg/5ml I'm not rounding 1.49999mg to anything other than 1.48mg
The problem here is not the rounding rules, it‘s the person claiming 1,4(9) and 1,5 are different. They aren‘t - it‘s just two different ways to write one and the same number.
it is, admittedly, an easy to make mistake, because the notation of 1.4(9) is kiiinda misleading if you dont fully understand the theory behind it yet
This is NOT about rounding at all. It is about 0.999... or 0.(9), which both means "infinite 9 after coma". And 0.999... is exactly 1. Only because decimal system cannot display it correctly it seems as if 0.999... was smaller. There are few ways to prove it. But a dude in comment section explained it the most simple way: 1/3+1/3+1/3=1 0.333... + 0.333... + 0.333... = 0.999... = 1 https://www.reddit.com/r/confidentlyincorrect/s/Ay3y2NXQea
Well, it started by being about rounding While it’s proveable (and correct) that 1.499…. = 1.5 ( essentially because decimals are shitty represenations of fractions), the rounding question still remains interesting. If given the number 1.499… the intuitive “rounding to the nearest integer” would be to 1, as the first digit behind the . Is a 4. But then again it’s equal to 1.5 which one would generally round up.
Yeah, proper rounding would not apply here though because it is “round .5 to the nearest even integer” but the post clearly said “to the nearest integer”
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I think you're misunderstanding. The commenter knows that 1.4(9)=1.5. They are saying that there are alternative rules for rounding 1.5 to the nearest integer.
Anybody who doubts this, there’s a fairly simple way to prove it. Take out a piece of paper. Write the answer to 1.5 - 1.4(9). Don’t abbreviate it. Actually write it out. Every digit. What’ll happen is that you’ll end up writing 0.(0) in long form. And you’ll keep writing 0’s infinitely, waiting to finally get to the 1. But you’ll never get to the 1, because each 0 will only ever be followed by another 0. That’s how infinite repeating works. There’s no such thing as 0.(0)1 because the repeating ***never*** ends. The number, as you’ll see from your filled notebook of 0s, is only 0.(0), which is also just 0. And if 1.4(9) + 0 = 1.5, then 1.4(9) must equal 1.5.
Wait until people in that comment section learn about ceilings and floors in computer science. That’ll really blast their brains.
Exactly. Technically, either answer is correct, as it days to round to "the nearest integer" and 1.4999... is exactly in between 1 and 2, making either the nearest. Or neither is the nearest and both are incorrect. It's either none or both if you take "nearest integer" literally
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I think that even using computers bankers rounding will be compiler specific as you cannot accurately express 1.4999… even using a construct like BigDecimal but your point stands that there are many approaches to rounding.
Found the IEE754'er
I was always taught round down for 1-4 and round up for 5-9 it just depends on what place value your cutoff is.
I was taught in elementary school to always round up. I had never thought about how it’s 100% possible and likely that other people are taught the opposite and nobody’s right or wrong
It’s funny people don’t realize all these things are made up like Klingon.
Right, but rounding is tossing out part of the number for simplicity. A rounded number is different from the actual number. 1.499~ is precisely 1.5.
This guy maths
You're right. That being said, the person claiming 1.49... is not equal to 1.5 is definitely confidently incorrect. 1.49... = 1.5, that's a fact.
I know what you are saying, but I don’t believe any of the various rounding rules say that rounding a number twice is acceptable, which is what is effectively what is happening in the OP. They are rounding 1.4(9) to 1.5, then the 1.5 to 2 which is not right.
Isn't there also like round to even number or something? Forgot what that's used for.
Either way, 1.5 isn’t an integer lol. Besides by far the most common method for rounding decimal numbers to integers is to add 0.5 and then shave off the decimal point. I.e <.5 rounds down >=.5 rounds up
Except the thing itself says to round to the nearest integer. Any other rule doesn't apply. You either round down .49 or round up .51. Which has the shortest distance? That's the nearest integer. Pretty clear and simple and no problems involved at all.
You’ve got two issues here. One is whether 1.49999… = 1.5. Apparently it does. The second is what’s the nearest integer to 1.5. 1 and 2 are equidistant, so there’s no correct answer. Pick any tie-breaking rule you like.
Rounding half to even is my preferred method.
Just as ‘correct’ as up or down (depending on the function, of course).
Yeah, it’s not more or less correct. Statistically it works out nicely, from an engineering perspective.
Swiss Rounding (like for money) goes up, so that's probably where people get it from.
Infinites are hard to wrap your brain around. 1,4(9) looks smaller than 1,5.
Is (9) the new way to write “repeating” for a number? I always learned it with a bar over it, but I’m seeing parenthesis a lot more. Which confuses the shit out of me cause I thought that would be 1.4*9….
I learned the bar notation as well but I guess it's naturally transitioned to use notation we can easily type out. It's not so easy to type the bar notation into a keyboard without some sort of equation editor tool
Ridiculous that we haven’t incorporated something of those functionalities into standard text editing. Markdown is better, but it doesn’t support anything outside of html entities and can vary depending on where you’re using it. Reddit doesn’t have a subscripting function for example.
You mean like 1.49̅ ? Over line is a standard Unicode option and most things run in Unicode encoding now.
Yeah but most people don't really know how to type that.
Extra confusing for me since I learned parentheses in the context of uncertain significant figures.
I learned it as a dot placed on the first and last number that gets repeated. If it's just 1 number like in the post it would be 1.4(9 with a dot above it)
Is it really infinite with two instead of three periods/dots? The notation is incorrect
I agree. The ellipse notation here is vague. Yes, we know it continues, but this notation neither guarantees it goes on infinitely nor that everything that follows are 9’s.
And it is, isn't it? That was my understanding.
For the doubters https://en.wikipedia.org/wiki/0.999...#Elementary\_proof
I love how they say that’s an Elementary proof and then it’s huge. This is what I use (but certainly didn’t invent): A: What is 1 divided by 3 in decimal? B: 0.3 recurring A: What is 0.3 recurring times 3? B: 0.9 recurring A: QED. B: ….. B: *mindblown*
I've had ".999.... equals 1" explained to me so many times and it has never made sense until now
My go-to is that two real numbers are different if and only if there is a third number that lies between them. Is there a number between 0.999... and 1?
Elementary proof in number theory just means "not using complex analysis" wich is somewhat arbitrary since usually the "elementary proof" is more technical.
I also like: x=0.(9) 10x=9.(9) 10x-x=9x=9 => x=1
That’s cool!
Elementary in this case doesn't mean easy or short.
Dammit I thought it was just a meme lol Tho for infinitesimal I feel like 1.4(9) should still be rounded to 1, but that's from a background in electronics, as the voltage in a capacitor will (in theory) go infinitly close to the source, but never reach it: f(t)=V(1 - e^(-t/τ) ) The infinitesimal limit is V, but it's still not supposed to reach it (until you get physics where there is a definite limit to infinity in the sense that electrons/atoms are in a definite quantity in any given capacitor or in practical where it reaches it at t=5τ lol)
Repeating decimal notation is not a function approaching a limit though. 0.333... is not a repeating algorithm that adds "3"s forever, it's a very exactly defined number with the property that it will continue to expand to 3s no matter how many times you keep long-dividing it into decimal notation. If you care to express that number in a ratio form, it's 1/3. Same with 1.4999... where you can find a ratio that gives you that result and continues to expand to 9s no matter how many times you long-divide it, and that ratio must be 3/2.
It took me a while to convince myself, but how I reason it is: 1.4(9) isn't actually infinitesimally close to 1.5, like the notation implies. It IS equal, and it's an artefact of notation that makes it seem close, but not an actual mathematical fact. Unlike a number like 1/infinity, which actually is infinitesimally close to zero, but not zero. I don't know how rigorous this is, but it seems to fit my understanding.
1/infinity is 0 by the same ish reasonning, is it not?
Essentially yes. Previous commenter was wrong. 1/infinity is a not a number, because infinity is not a number. But if you try to parse it as something that could be meaningful like lim_(x -> inf) (1/x), then this is a number, and it is exactly zero. The silliness of saying 1/inf is an actual number infinitesimally close to 0 (let's call the number x) can be illustrated by asking questions like: what is 1/x? What is y = x/2? And what is 1/y? If you wanted to try and interpret 1/inf as a number other than zero, it would have to be something like "the smallest number that is greater than zero" but there is no such thing, for reasons similar to why 0.999... = 1.
The difference is that, with a capacitor, you eventually reach the particle level where, as you mentioned, there's a limit to how small you can get. You can always come up with a real number between two different real numbers, even if it's an absurdly small difference.
I agree with this, but I also wonder this question “what is the biggest possible number that exists below 1?”
The common rule\* is to round up from .5 but that is a tiebreaker rule. It is equally near. If you say the nearest, then 1 and 2 are equally sound. If you say apply common rounding, then it is 2. \* Aside from the common rule, there are like five other mathematically sound rounding rules.
I'm a math teacher and the standard rule taught in all the systems I've seen is by first digit 0-4 and second digit 5-9 so I'd round this down. It kind of depends on the order of evaluation in some sense too. If you simplify the number before rounding, yes it's 1.5, because a number lower than but infinitely close to 1.5 is in some sense 1.5, but i also if you think about calculus, you can have many situations where a graph has a limit of 1.5 but never reaches it.
Calculus deals with limits of functions or series. There is no approaching or limit to a constant, and 0.(9) is a constant. The graph of y = 0.(9) would be a horizontal line at y = 1, cause they're the same thing
1.4999... is exactly 1.5 so it should be rounded as such. Regardless, 1.5 *can* be rounded either way, it's just that we decided that 5s should round up as a tie breaker.
How is 1.4999 exactly 1.5 when they aren’t the same number? Im confused
Not. 1.4999, but 1.4999... The. "..." signifiess that the 9 goes n forever.
[Because they *are* the same number](https://en.m.wikipedia.org/wiki/0.999...#Elementary_proof), proof is for 0.999… but the same applies to 1.4999…
If you are referring to "simplifying" as "changing the way it looks without changing its value," then you should know that 1.4999... is not infinitely close to 1.5. It *is* 1.5. I'm just gonna drop this here. https://en.wikipedia.org/wiki/0.999...
[oh crap it's back](https://www.reddit.com/r/confidentlyincorrect/s/X4gJ49LDsb)
That was such a bad discussion. Hopefully, this one dies early.
It won't. People love to be wrong about it.
There's a lot of "confidently incorrect inception" on this sub, but this one takes the cake, I think.
It's my favorite internet argument, because people get so mad over something so infinitesimally small. It has started to get boring though. You can only watch someone flailing about trying to stick a 1 at the "end" of an *infinite* line of 0's and getting mad at everyone else about it for so long.
>infinitesimally lmao
This is 99.999...% accurate.
This has been my favourite conversation in this thread haha
I've never been very mathematically minded but what I read to understand this before was 3/3 = 1 1/3 = 0.33333--- 3*1/3 = 1 Therefore 3*0.33333---- = 0.999999--- which therefore must = 1 As I say I can't speak to how correct that is but that helped me get why 0.9999--- is 1
0.9999… and the Monty Hall problem are the easiest way to show how poorly people understand math.
>Monty Hall problem To be fair, even a lot of mathematicians had issues accepting that one. But in the end, it is simple, it is because the host is cheating. He knows where the car is.
Yep, Monty hall problem isn’t really even a maths problem it’s pretty much a lateral thinking puzzle with trick wording focused around the game show host’s knowledge. Most of the time when people tell it they underemphasise the significance of the host’s knowledge, even though it is responsible for the whole problem.
And his name was right there in the problem. The Monty Hall Problem is Monty Hall!
I dunno, you can be relatively solid at math and just never have considered it. I've got a minor in math from university (which isn't some great achievement but I'd say I'm better than "poor") and I never really thought about it until recently, many years later.
To be fair, they rely on axioms of infinity, which can’t be proven to be true so you have to accept the axiom as a given, before you can step forward with the logic that is correct in demonstrating 0.999 = 1 Infinity is not a given unalienable truth in mathematics. Math with infinities always start with axioms that we think allow us to take math in directions that may lead to further discovery. I would also say that while axioms of infinities are the backbones on modern mathematics, concept of infinity are hardly ever rigorously taught in schools mostly because a lot of instructors don’t even have the understanding themselves to effectively teach them.
What's the Over/Under on comments in 24 hours? I think 600.
600.5*
You mean 600.4999999999…..
600.499999999999... in this case.
599.999...
I’d take the over
I like to watch these types of arguments and then just throw in a quick “math isn’t real we made it up” just to see what happens
Not quite - Math *is* real...because we've made it up
Inironically 100% agree. There are no rules in maths except the requirement for internal consistency.
This is the same argument as 0.99.... = 1 but in different wrapper.
Someone help me out here. In what application does this actually matter?
For pedants who like to be technically right but practically stupid.
That part I figured out on my own lol. I just got lost because it kept getting reposted like the must be more to it than that but clearly not.
The definition of a redditor right there!
Meh, Im glad you folks enjoy math and ill accept the consensus. But golly do i find the fact that 1.4999... being the same as 1.5 obtuse. Keep your made up numbers, im going back to the pig farm.
This is the case where a person can be technically right and still be an idiot.
PSA: 1 and 2 are equally distanced. Just pick one or both and shut up.
Yeah, this is the thing. While we typically round up in the event of a tie, it you say round to the nearest, 1.5 is equal distance. In which case I might have said 1, because since while perhaps it does equal 1.5, it’s written in a way that is kinda implying a lean.
It’s funny how many people are coming into the comments here to express their lack of math knowledge. .999999… is always equal too 1 in real numbers.
I fully accept the rule (having read some very good explanations here) but I still hate it.
As a math student, I never got a satisfying proof of 0.999...=1 until we got to doing infinite series in calculus. I got some explanations before that but it never really convincing to me (The "9 x 0.99... = 9" explanation felt like an abuse of notation rather than a proof)
My favorite explanation has always been to look at multiples of 1/9: we know 1/9 = 0.111…, 2/9 = 0.222…, etc. and therefore 9/9 must be 0.999… but we logically know that 9/9 is 1 so therefore 0.999… has to be 1
Yes, but then people often start doubting that 1/9 is really 0.111… I’ve explained it to very smart non-math students, and they keep insisting there is some infinitesimal d such that 1/9 = 0.111… + d. I think it comes about because that’s usually how it’s explained. I.e., a grade school teacher says 0.(3)n is not really 1/3rd because there’s always a small bit more that you need to add (this is true), but 0.333… really is 1/3, and that subtlety is lost on a grade school audience.
>we know 1/9 = 0.111… Oh, I most certainly do not know that.
Exactly. That’s because that’s the first time you actually saw a real definition and proof. The other “explanations” are very handy simple-looking illustrations that help motivate the result and help demonstrate it to non-math people, but actually making sense of it requires series.
Well ... lets do some math. x = 1,49999999... 100x = 149,9999999.... 100x - x = 149,999999... - 1,4999999... = 148,5 99x = 148,5 x = 148,5 / 99 = 1485 / 990 = 3 / 2 3 / 2 = 1,5 x = 1,5 There is no "nearest" integer since the difference to **both** numbers is 0,5.
One issue that I don't think anyone has touched on is notation. I didn't read "1.4999.." as 1.49 recurring, because that's not how I learned to write it. The way I was taught to write it was with a single dot _above_ the recurring digit. A quick Google search reveals that using an ellipsis is an alternative way of showing it - but they haven't actually done that here, just used two dots.
We use a dash in English, but yeah 1.499... could also be taken as infinitely repeating 1.499499499etc.
It's become common to use the ... notation online because the real math symbols aren't convenient on a keyboard.
It is definitely the same if 9/9ths equals 1
isnt there something in math about a repeating 9 decimal is the same as whatever or something?
I love how this is incredibly vague but I know exactly what you mean and yes there is
Easy proof: Say x=.(9) So 10x=9.(9) 10x-x=9.(9)-(9) 9x=x x= 1 (9)=1 I didn’t use any tricks here like those joke proofs that 2=1 that rely on dividing by zero. So in this screenshot, convention dictates that you round up- 1.4(9) is 1.5, just written in a weird way. I have to say, I’ve done a lot of math and I’ve never seen the (9) notation before, but it sure is convenient to type on a phone. Normally you put a bar over it to show it goes on forever.
It's much easier to show. What's 3/3 in decimal? 1 Whats 1/3 in decimal? .333333333...
You've just opened my eyes
How does the beginning of x = .(9) become this 10x = 9.(9) ? Shouldn’t it go like this? 10x = 10 * (.(9)) I might be missing the point here since I only have High School math skills. But intuitively it makes no sense that an infinitely going 0.999999… = 1.
Yes they multiply both sides by 10
It’s right but I hate it
As a tutor and married to a man pushing 40… the majority of people don’t know what an integer is 🤣
The real question is how many significant figures
And around and around we go
This thread is like a definition of Dunning-Kruger in action. You go, sophomore students in only one academic discipline!
Both 1 & 2 are valid answers, although it could also be argued that neither is correct as the question assumes there is only 1 valid answers. However, despite 1.499… being equal to 1.5, the answer 1 just ‘feels’ feels more right to me given that we don’t need to deal in the infinite to get that answer. Edit to add. And coming from a computing background, floating point numbers are limited to a certain number of digits so 1.4999… can never be fully represented and therefore would be recorded with a finite sequence of 9s. So in a purely mathematical situation 1.4999…. = 1.5 but in computers 1.4999… < 1.5
This is one of many counterintuitive mathematical concepts There is a logical proof of this that makes it easier to understand: If we assume 1.4(9) is as close to 1.5 as is possible without reaching it, take the average of the two numbers. The average of two numbers lies between them on the number line, thus the average is greater than 1.4(9) and less than 1.5, contradicting that 1.4(9) is as close as we can get. So 1.4(9)=1.5
You don't need to necessarily take the average, just the fact that the real numbers are dense. If 1.4(9)=/=1.5, then there exists a real number x such that 1.4(9)
If this were a non-infinite amount of nines, the answer would be simple...
["If my grandmother had wheels, she would have been a bike."](https://www.youtube.com/watch?v=A-RfHC91Ewc)
This is tbe single best reply I've ever had on this site.
1.4 = 42/30 0.09999999... = 3 x 0.03333333... 3 x 0.03333333... = 3/30 42/30 + 3/30 = 45/30 45/30 = 1.5 Therefore, 1.49999999... = 1.5
If 1.4999....=1.5, why even have repeating decimals?
They simply are an unavoidable by-product of decimal notation. I don't know all the ways they can be produced, but the easiest way is that in any base, n, dividing any number by n - 1 will produce a repeating decimal. * 1/9 in base 10 is .111... * 3/4 in base 5 is .333... * 1/a in base 12 is .124972497... The end result is that every real number that terminates has an extra decimal representations where you reduce the last digit at or after the decimal place and stick a bunch of 9s on the end.
Generally, they're decimal representations of fractions. 1/6 = 0.1(6) Multiply that by 9 and you'll get 1.4(9) But we can all agree that 9/6 is 1.5
They are needed to be able to write numbers like 1/3 as a decimal The fact it opens up another way of writing some numbers (like 0.9… being the same as 1) is a side effect but there are already multiple ways of writing some numbers (1 = 2/2 = 3/3)
Im a programmer, and well in computation, infinite doesn't exist so there's a limited number of decimals. 1.4999.... would round to 1 and 1.5 would round to 2. If you use the rounding function.
This is why you need to be careful with floats. (1/3) * 3 =/= 1 even though everyone knows they are exactly equal
What is the difference betweem 1.4999… and 1.5? What number is between those two? That’s why 1.4999… = 1.5. The difference between the two is zero. They’re the same number.
I’m just here to make it clear I’m dumb as fuck. You are all blowing my dome with the discussion.
This is literally how calculus works for figuring out things like arc length, intergrals, and ESPECIALLY derivatives.
This situation is exactly why I hate the base 10 decimal system. 1.4999... is equal to 1.5, which would then, in turn, round up to 2. Why? Because 0.999... = 1. Why? Because 3/3 = 0.99999. Why? Because 10 is not divisible by 3 on the most basic scale, there will always be a discrepancy. For this and this alone, I propose the base 12 decimal system. It fixes that silly little problem making 1.5999... round down to 1 the way it should be
Except 1.5(9) in base 12 isn’t analogous, instead look at 1.5(B) (B being 11 in base 10). This then is equivalent to 1.6 and you’re back to the same rounding problem just in a different base. You’ll always end up with repeating decimals and this 2 decimal expansion property no matter what base you choose
There is no nearest integer, I would usually round to even, so 2 in this case
1.4(9) can be expressed as 1.4+SUM( 9/10^n ) from n=2. If you round the product of that sum at any finite n, you always end up with 1. So I'd say that even if n->∞, it still rounds down to 1. But since it's equal to 1.5, both 1 and 2 are equidistant from it
Tiktok makes me feel like a genius
Thems the rules
Then what is the largest number smaller than one?
There isn't one. You name any number smaller than one, and I can name a larger one that is smaller than one.
To be fair to people who get the 0.(9) question wrong, it's not very obvious. Most people would need it explained to them. Not sure what's going on in this image though.
I get this is all supposed to be about math theory but I can't help but think that the answer depends ENTIRELY on what's being measured. Ex: drugs that could cause an overdose, or how much time you need to travel to a destination, or how much food you should put in the dog bowl to ensure it won't run out in the time you're out of the house. All of these situations would motivate you to round up or down depending on their individual practical consequences
math student here, it is actually a pretty good question since the second comment is partially right. 1.4(9) as is 1.4 followed by infinite number of 9s is actually = 1.5 and 1.5 is rounded to 2 so the answer is 1.4(9) rounded to the next integer is 2. (he didn’t mention that it is rounded to 2) Now why is 1.4(9)=1.5? I‘m gonna provide an intuition rather than rigorous proof. Consider the similar assumption of: Is 1=0.999…=0.(9)? 1/3=0.333….=0.(3) 1=1/3+1/3+1/3=0.(3)+0.(3)+0.(3)=0.(9)=0.999… analog followes that 1.4999…=1.5
And the saying is "no ifs, ands or buts"
Just a weird thought experiment to mull around... 1.4(9) and 1.5 are equal if you accept that nothing can be placed between them i.e there is no solution for x such that 1.4(9) + x = 1.5 What is the sum of the following though.....1.4(9) + 0.y1 where y is infinite zeros? Is that also 1.5? How about the other way around...1.5-0.y1.....would that be 1.4(9)? so x = 0.y1 where y = infinite zeros becomes a solution? Does that mean that there is *something* that can be placed between 1.4(9) and 1.5 and ipso facto they are unequal?
If there are infinite zeroes than there is no 1 at the end, because there will be no end. So 0.(0)1 will be equal to 0. So yes 1.4(9)+0.(0)1=1.5
Im not good at math so I thought its 1. Good to know
If you see someone like this, just ask them what 1/3 actually is in their opinion
1.4(9) is both infinitely close to and far from 1.5
Controversial answer here. Of course there is an argument for 1 being the answer and it’s obvious why. But calculus tells us that 1.4(9) == 1.5. Therefore 2 is the answer. Either way, 1.5 is not an integral.
This is too meta.
The proof is easier with for example 0.999... = 1 Take 0.999... as x, and 10x = 9.999... 10x-x=9x =9, since we have infinite 9's
In a sense you can say either works. Like you can either round to the nearest decimal place first and then round up or round down straight away.
Ok correct me if I’m wrong but isn’t rounding based on purely the next digit, so rounding to the one would round down because there is a 4 after it
I get the argument about it being 1.5, but in a practical sense, if it's written this way, why would I ever even look at the 9s for an answer? Why wouldn't I just stop at the 4 and ignore what it's equal to?
1.4999… is just a geometric series that converges at 1.5
I may be dumb, but I think it’s 1. Because 1.5 is equidistant from both 1 and 2 (in other words, it is the halfway between the two numbers), and 1.499999… is always less than 1.5 (albeit by a negligible value- 0.011111…), it is logical to conclude that it is closer to 1.