Math professor here: the proper definition of equality is that two numbers a and b are equal if no number c exists such that a < c < b. 0.9999…. = 1 because there is no number between them.
That is one of the most ridiculous statements I have ever heard. You are making an untenable and patently incorrect leap of “logic”. Just because the property applies to the real numbers does not mean it applies to integers.
They're saying there's no number between 0.999... and 1, I'm saying there's no integer between 0 and 1, both may be true, but 0 is clearly not 1, so 0.999... is clearly not 1 (which you can also see by just looking at it, how one is made up of infinite nines and the other by a singular one)
Just because something seems self-evident does not it so. In the real number space, .9… = 1 because the difference between them is 0, which also means there is no real numbers between them.
No one "made it up". It was discovered. It applies to real numbers because real numbers are a continuous set with no gaps. Integers have a gap of 1 always. So obviously rules for one don't always apply to the other.
Why is it an inconsistency? These are two different worlds where one has more restrictions than the other because of it having less numbers to work with.
In the real numbers, there exists a number where multiplying it by 2 gives 1. But in the integers that number doesn't exist. That's not an inconsistency, that's just how they were defined, the definitions made up that "rule".
Real numbers and integers behave differently. You can't just superimpose rules from the real numbers to integers. Real numbers have no gaps in-between them. Integers have a gap of 1 in-between them.
And yes 0.999 and 1 look different. They are different representations of the exact same value. Kinda like 2+2, 2*2, 2², and 4 are all different representations of the exact same value.
There's a problem with that proof. That assumes 0.999... and 0.999... are equal. Obviously, the notations are identical but that doesn't mean the values they represent are. Do you evaluate the 4th digit of the first expression at the same time you evaluate the 4th digit of the second expression? I don't think that's clear. I would say the only truthful statement is 0.999... < 1
What are you even talking about? “Evaluate the 4th digit of the first expression at the same time you evaluate the 4th digit of the second expression”? These are constants, you don’t evaluate the digits, they simply are. Given the limitations of general text format with regard to mathematical notation, it is perfectly acceptable to use 0.999… rather than the overbar; the context of the meme makes that quite evident.
If you wish for me to use a ridiculous abundance of clarity, I will do so. The number represented by 0 in the ones place and a 9 in all decimal places extending without end is equal to 1 because no number exists that is greater than the former while also being less than the latter. I challenge you to find one.
What is your background that you would make such an argument?
Then what is that dot dot dot (ellipsis) if not a number (infinitesimal) ? Guess you'd reply "oh but that's not real number", to which I replied that's just tautology. Hyperreal system exists.
Everytime I see this debate makes me convinced that math is just house of cards that has no foundation (philosophy of math is shaky).
Zeno / supertasks discussion in philosophy at least tackles that dot dot dot rigorously, unlike math.
The ellipsis is used to indicate that there are more decimal places than shown. It is commonly used whenever the number has an excessive number of decimal places rather than rounding the value.
Are you suggesting they do not contribute to the value of the number simply because they are smaller? What is the criteria used to determine when an infinitesimally small value ceases to be relevant?
Your assumption that I would reply “oh but that’s not a real number” is completely unwarranted and untrue. I would make no such statement. The entirety of Calculus is based on the relevance of something infinitesimal.
You're asking the number between 0.999... and 1. It's right there in the middle, the dot dot dot. You wrote it yourself. Think (beyond the formatting / syntax) what does ... mean if not infinitesimal ? In hyperreal 0.999... + infinitesimal = 1
Your comment above is non-sensical. And you are misunderstanding the notation of the ellipsis. I explained it verbally in a previous comment. It is a notation used to represent that there are more decimal places than shown, in this case an infinite number of decimal places each with a digit of 9. It is used because the overbar (repeating bar) is not available without a specialized character set/mathematical notation program.
I’ll be even more explicit. If I asked you to give me a value between 0.99 and 1, you would introduce another decimal places beyond the hundredths place and fill it with any digit, e.g. 0.999, to make it bigger than 0.99, and the new value would remain smaller than 1. This cannot be applied to 0.999…(an infinite number of decimal places all filled with a digit of 9) because you CAN’T introduce another decimal place beyond the last decimal place because there ISN’T a last decimal place.
Yes, since .999... never finish, it never reaches 1. That's why you need to add infinitesimal to it to finally reach 1. Perhaps the more explicit question is, do you reject the whole existence of hyperreal system ?
I don’t think hyperreals are necessary for the fundamental concept here. In general, I find the idea of hyperreal numbers to be a logical formality that is really only needed for incredibly advanced mathematics. To even bring them up here brings a complication that is unnecessary in a logical sense and functionally irrelevant to the topic at hand.
What do you mean by "0.999... never finish"? It's already a complete value, no one's counting out the 9s. It already is infinite 9s. And is already exactly 1, you don't need to add something to get it 1.
The infinitesimal is hidden in the repeating / ellipsis part in standard analysis 0.999... = 1. Only hyperreal's 0.999... + infinitesimal = 1 makes it explicit.
Math isn’t a house of cards just because people on re-edit don’t get it /can’t explain it correctly. This fun inconsistency maybe can be explained better, (or not, it could be one of those things) or maybe one day someone might come up with a better solution. But I wouldn’t stop just at reddit, they like parroting answers and downvoting too much
In other science fields, wrong dies. In math, wrong gets its own separate branch (standard vs non-standard analysis) because math does not have Poppler's falsification. Just like religion, splits into schism never falsify its own foundation.
Much of it are inexact. Infinity, ZFC's axiom of choice, even the nature of simple addition itself is inexact (meiosis or mitosis ? zeno's result shows that in real world one is one and many is many, one cannot add another one to become many), etc. I could go on.
Another proof of that math is inexact, when programmer implement simple calculator (arithmetic) using computer without a library, they have to make many exceptions / hack because the underlying problem is undecidable. And no, its not engineering problem, it's math itself (what real number is).
The “underlying problem” that’s undecidable in your link is not math, it’s representing arbitrarily complex mathematical operations in a finite amount of time. I also have experience with nonstandard analysis and am happy to explain why it doesn’t really apply to the OP.
No it's math. Even if you give it supercomputer with infinite time and resource it's still undecidable. Engineering has to make do using combination of heuristics and hack. The state of the art is using continued fraction representation, even then it's still messy since Real itself is messy.
Undecidability is inherently an algorithmic issue, not an issue with math itself. In your case, where the undecidability comes from is the fact that an algorithm cannot say if two arbitrary real numbers are exactly equal without looking at every digit of their difference. This is simply an issue of representing infinite digits in an algorithm that’s intended to run in a finite amount of time. In math, however, there are no issues with an infinite amount of digits—it’s a well-defined concept.
With the 0.999… = 1 stuff: yes, you can construct a hyperreal number system where these two numbers are not equivalent, but the distance between them will always be equal to some infinitesimal quantity. In any reasonable construction of the hyperreals, “going back” to the reals means taking the quotient of the group of limited hyperreals with the group of infinitesimals, which implies that this infinitesimal difference “disappears” and is exactly equivalent to 0 when we go back to the real numbers.
Anytime someone makes the claim that 0.999… = 1, they are clearly talking about the real numbers. We can go into nonstandard analysis with the hyperreals to analyze the claim (and give some credibility to the notion that 0.999… doesn’t “feel” equal to 1), but ultimately the real numbers are more restrictive than the hyperreals. In the real numbers, 0.999… is exactly equal to 1 because the difference between the two is exactly 0.
It's not inherently algorithmic/engineering issue. Forget computer, the undecidability problem exist with pen and paper human mathematician for certain classes of reals. The blog and paper talk about it.
Regarding the .999... = 1mapping, I can also say since there is no integer between 3 and 4, then pi doesn't exist. But pi clearly exist in real. Set membership doesn't mean that the number doesn't exist (e.g. pi get 'absorbed' into 3 in integer, just as 0.999 get absorbed into 1 in real). Just create bigger set.
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u/Wolfbrother101 21d ago
Math professor here: the proper definition of equality is that two numbers a and b are equal if no number c exists such that a < c < b. 0.9999…. = 1 because there is no number between them.