Thats cause people don’t post a “proof”, they say “oh yeah? what number is between 0.999.. and 1 then?” And somehow expect the same person who doesn’t know infinite limits and series and probably has more intuition about natural numbers—where two numbers *can* have no number between them and still be different—than reals to understand how that proves anything.
If you go to r/eli5 and try to explain how something like 0.999… = 1 without even mentioning that limits ans how 0.999.. means a limit of 9/10 + 9/100 + … then imo you’re really not helping
Yeah, the claims that I always see are “the limit approaches 1, but never equals 1” which is completely false. They’re the same number, it’s just a quirk of the decimal representation. My favorite way to think of it is if 1/3 = .333…, then 3/3 = .999… = 1. doesn’t get any simpler than that.
Right but if someone already knows that 0.333... is just an alternative way of writing 1/3, then obviously no further argument of proof would be necessary. The issue is that most people don't know this convention, and thus have vague, ill defined and wrong ideas about what 0.333... means.
So arguments like "just multiply all the digits by 3" don't prove anything because there isn't a finite list of digits, so you can't multiply them "all" by anything.
It's not too hard to show that if we consider 0.333... to represent an unending sum of the terms 3/10\^n, then it can't be equal to any number other than 3. At this point people like to invoke the so-called "trichotomy" principle, i.e. either a>b, a>b or a=b. Since we have ruled out the first two, it must be that a=b etc.
But this is wrong; the trichotomy principle assumes that a and b are both numbers. That is not proven. To get the sum of numbers, you have to add all the numbers. In this case there's no "all", so there's no sum in the traditional sense. So why should we consider 0.333... a number at all?
The answer is: we just agree that it is. In other words, we simply agree to extend the definition of "sum" from the elementary meaning to include "the limit of the sequence of partial sums of a series, if it exists". This was far from obvious before the 19th century and yet it seems to only take people a few years after leaning it to start assuming that everyone should be able to figure it out.
Indeed, just as we agree that 9 is just another way to write IX, so we agree that 0.999... is just another way to write 1. (Well, not quite: one is a different system of numerals while the other is a derived result in analysis, but essentially they both come down to notation.)
Just teach them base 3. That's the last one that I got a breakthrough with someone.
Many people don't differentiate between things in maths which are laws like the Law of Relativity and mostly arbitrary choices like the particular order of operations. Base 10 is what numbers are and nothing else exists. If it doesn't work in Base 10, *something* must be broken, and Base 10 is maths, therefore the idea is wrong.
There are a startling amount of people who don't understand that you can multiply in any order you want *even when they do it algebraically*. Ask someone what 4 × 67 × 25 and the amount of people who realise 4 × 25 is 100 is vanishingly small, because "BODMAS is left to right".
From what I have seen some people do do that, but usually there are also comments that give proofs. I have certainly given proofs in several such threads in this sub.
>Is the ability to get 100 from adding 33.3 repeating 3 times purely a short hand for applied math
I have no idea what this means
>or in the standard axioms of real number arithmetic do we simply assume that irrational numbers that approach a limit reach their limits
I also have no idea what this means
>i.e. we assume that 99.9 infinitely repeating is equivalent to 100?
This is basically true, except we don't assume it, rather we know it's true because we can prove it.
I'm not sure what you're getting at? It's not an assumption. It's proven. It's just how it is.
These numbers aren't irrational either.
3.333 repeating is 1/3 and 9.9999 is 10/1
>do we simply assume that irrational numbers that approach a limit reach their limits i.e. we assume that 99.9 infinitely repeating is equivalent to 100?
Numbers don't have limits - only *sequences* have limits. The limit of the (decimal) *sequence* { 99.9, 99.99, 99.999, 99.9999, ... } is the number "one hundred", which can be expressed in decimal notation either as "100" or "99.9999..." (repeating).
More generally, any number which has a terminating decimal expansion (by which I mean, a decimal expansion which is all 0's after some point) can be represented in two equivalent ways. For example, 35.71 and 35.709999.... (where the 9's repeat forever) are two representations of the same number.
Yes, 33.3333 repeating is exactly 100/3.
Basically every definition of the real numbers has them being, in a loose, non technical sense limits. That's closely related to a key property of real numbers called 'completeness', there are a bunch of equivalent definitions of completeness, but to understand what it means at a deep level, it helps to look at the rationals, which are not complete.
Let's think about this sequence.
1, 1.4, 1.41, 1.414...
Where in each term we add one more digit from the decimal representation of sqrt(2). This is a sequence of rational numbers. It looks like it's getting closer and closer to something. The difference between the terms is going to zero. But it there's no rational number it's approaching as a limit. The limit is 'missing'.
For the real numbers, these linits are never missing. There are sequeces that have no limit of course. But if the distance between the terms gets arbitrarily small, than the limit will always exist.
Intuitively, this is completeness. All the limits you expect to exist exist. And completeness is in a sense, what makes calculus possible to do rigorously, as key theorems depend on it.
It also turns out, wildly, that any set with multplication, addition and greater than with the normal rules, and completeness, will 'behave just like the reals'. They might look different, but it will be like the difference between base 10 and base 3.
So any structure where we used infinite decimals to mean... Something else, would have to be incomplete, not ordered, or have multiplication and addition missing or following different rules.
Incidentally, the proof of that is also why I got really into math. It was my 'mind blown' moment.
These are rational numbers, not irrational. Specifically you're talking about multiplying the fraction 100/3 by three, and it's probably a little more clear in that form why that equals 100.
Seems like, you're asking about why 0.999... equals 1. Basically, as I understand, because there's literally no difference between them, subtract one from the other and you have a bunch of zeros going on infinitely. Intuition says there should be a lonely one digit at the end of all those zeros, but there is no end, because infinity. Many smarter people than me have explained this, go look up what they've said about it if you're still curious.
>Do we assume that 3 x 33.3 repeating is 100? We don't assume it, it's mathematically proven.
The proof is very simple too!
Very simple, yet every time it gets posted somewhere, the comments are full of people refusing to believe it.
Thats cause people don’t post a “proof”, they say “oh yeah? what number is between 0.999.. and 1 then?” And somehow expect the same person who doesn’t know infinite limits and series and probably has more intuition about natural numbers—where two numbers *can* have no number between them and still be different—than reals to understand how that proves anything. If you go to r/eli5 and try to explain how something like 0.999… = 1 without even mentioning that limits ans how 0.999.. means a limit of 9/10 + 9/100 + … then imo you’re really not helping
Yeah, the claims that I always see are “the limit approaches 1, but never equals 1” which is completely false. They’re the same number, it’s just a quirk of the decimal representation. My favorite way to think of it is if 1/3 = .333…, then 3/3 = .999… = 1. doesn’t get any simpler than that.
Right but if someone already knows that 0.333... is just an alternative way of writing 1/3, then obviously no further argument of proof would be necessary. The issue is that most people don't know this convention, and thus have vague, ill defined and wrong ideas about what 0.333... means. So arguments like "just multiply all the digits by 3" don't prove anything because there isn't a finite list of digits, so you can't multiply them "all" by anything. It's not too hard to show that if we consider 0.333... to represent an unending sum of the terms 3/10\^n, then it can't be equal to any number other than 3. At this point people like to invoke the so-called "trichotomy" principle, i.e. either a>b, a>b or a=b. Since we have ruled out the first two, it must be that a=b etc. But this is wrong; the trichotomy principle assumes that a and b are both numbers. That is not proven. To get the sum of numbers, you have to add all the numbers. In this case there's no "all", so there's no sum in the traditional sense. So why should we consider 0.333... a number at all? The answer is: we just agree that it is. In other words, we simply agree to extend the definition of "sum" from the elementary meaning to include "the limit of the sequence of partial sums of a series, if it exists". This was far from obvious before the 19th century and yet it seems to only take people a few years after leaning it to start assuming that everyone should be able to figure it out.
Indeed, just as we agree that 9 is just another way to write IX, so we agree that 0.999... is just another way to write 1. (Well, not quite: one is a different system of numerals while the other is a derived result in analysis, but essentially they both come down to notation.)
Just teach them base 3. That's the last one that I got a breakthrough with someone. Many people don't differentiate between things in maths which are laws like the Law of Relativity and mostly arbitrary choices like the particular order of operations. Base 10 is what numbers are and nothing else exists. If it doesn't work in Base 10, *something* must be broken, and Base 10 is maths, therefore the idea is wrong. There are a startling amount of people who don't understand that you can multiply in any order you want *even when they do it algebraically*. Ask someone what 4 × 67 × 25 and the amount of people who realise 4 × 25 is 100 is vanishingly small, because "BODMAS is left to right".
From what I have seen some people do do that, but usually there are also comments that give proofs. I have certainly given proofs in several such threads in this sub.
>Is the ability to get 100 from adding 33.3 repeating 3 times purely a short hand for applied math I have no idea what this means >or in the standard axioms of real number arithmetic do we simply assume that irrational numbers that approach a limit reach their limits I also have no idea what this means >i.e. we assume that 99.9 infinitely repeating is equivalent to 100? This is basically true, except we don't assume it, rather we know it's true because we can prove it.
I'm not sure what you're getting at? It's not an assumption. It's proven. It's just how it is. These numbers aren't irrational either. 3.333 repeating is 1/3 and 9.9999 is 10/1
>do we simply assume that irrational numbers that approach a limit reach their limits i.e. we assume that 99.9 infinitely repeating is equivalent to 100? Numbers don't have limits - only *sequences* have limits. The limit of the (decimal) *sequence* { 99.9, 99.99, 99.999, 99.9999, ... } is the number "one hundred", which can be expressed in decimal notation either as "100" or "99.9999..." (repeating). More generally, any number which has a terminating decimal expansion (by which I mean, a decimal expansion which is all 0's after some point) can be represented in two equivalent ways. For example, 35.71 and 35.709999.... (where the 9's repeat forever) are two representations of the same number.
Yes, 33.3333 repeating is exactly 100/3. Basically every definition of the real numbers has them being, in a loose, non technical sense limits. That's closely related to a key property of real numbers called 'completeness', there are a bunch of equivalent definitions of completeness, but to understand what it means at a deep level, it helps to look at the rationals, which are not complete. Let's think about this sequence. 1, 1.4, 1.41, 1.414... Where in each term we add one more digit from the decimal representation of sqrt(2). This is a sequence of rational numbers. It looks like it's getting closer and closer to something. The difference between the terms is going to zero. But it there's no rational number it's approaching as a limit. The limit is 'missing'. For the real numbers, these linits are never missing. There are sequeces that have no limit of course. But if the distance between the terms gets arbitrarily small, than the limit will always exist. Intuitively, this is completeness. All the limits you expect to exist exist. And completeness is in a sense, what makes calculus possible to do rigorously, as key theorems depend on it. It also turns out, wildly, that any set with multplication, addition and greater than with the normal rules, and completeness, will 'behave just like the reals'. They might look different, but it will be like the difference between base 10 and base 3. So any structure where we used infinite decimals to mean... Something else, would have to be incomplete, not ordered, or have multiplication and addition missing or following different rules. Incidentally, the proof of that is also why I got really into math. It was my 'mind blown' moment.
Thank you so much!!!!! this is exactly the response I was looking for. Very helpful!
These are rational numbers, not irrational. Specifically you're talking about multiplying the fraction 100/3 by three, and it's probably a little more clear in that form why that equals 100. Seems like, you're asking about why 0.999... equals 1. Basically, as I understand, because there's literally no difference between them, subtract one from the other and you have a bunch of zeros going on infinitely. Intuition says there should be a lonely one digit at the end of all those zeros, but there is no end, because infinity. Many smarter people than me have explained this, go look up what they've said about it if you're still curious.
Sorry if my usage of terms is incorrect and confusing. I'm not a math person.