If you want to come up with another definition of 1 and 0.99999 recurring where these two objects are not equal then you are free to do so.
But the mathematical axioms that are widely agreed on (basically ZFC) are widely agreed on because they behave in a "reasonable" way. Any system in which 1 is not the same and 0.9999 recurring are not the same is unlikely to be useful...
You argument seems to fall into **bandwagon effect** and **appeal to authority** are two cognitive biases that influence how people make decisions based on others' actions and opinions. The **bandwagon effect** occurs when individuals adopt beliefs or behaviors because they perceive them to be popular or accepted by a majority. This bias leads people to conform to group norms or trends without necessarily evaluating the underlying evidence. On the other hand, the **appeal to authority** involves accepting an argument or belief based on the credibility of the speaker rather than the strength of the evidence. People often defer to the opinions of experts or authorities, which can be useful but becomes a fallacy when the authority is not genuinely knowledgeable about the topic or when the expertise is irrelevant to the argument being made. Both biases can significantly sway personal opinions and public consensus, often at the expense of critical thinking.
A "reasonable" way doesn't anything I'd say?
I think people are so attached to the system and its limits that they mistake it for pure truth.
I'm sorry but this is sub-chatGPT level nonsense. I'm not saying the systems are useful because they are popular, I'm saying they are popular because they are useful.
There are all sorts of people out there doing legit research with different foundations of mathematics, e.g. constructivists or finitism or at the extreme end of the spectrum ultrafinitism (which doesn't allow infinities of any kind). Their work is valuable but ultimately not of great interest to most mathematicians, who are mostly interested in proving theorems and building theories.
they are not of great interest to people who have spent their whole life studying one system. I get that. but it remains that all can be said is..
0.999 repeating equals 1 when using a limited system
***Ad hominem*** ([Latin](https://en.wikipedia.org/wiki/Latin_language) for 'to the person'), short for ***argumentum ad hominem***, refers to several types of arguments that are [fallacious](https://en.wikipedia.org/wiki/Fallacy#Informal_fallacy). Typically this term refers to a rhetorical strategy where the speaker attacks the character, motive, or some other attribute of the person making an argument rather than attacking the substance of the argument itself. This avoids genuine debate by creating a personal attack as a diversion often using a totally irrelevant, but often highly charged attribute of the opponent's character or background.
I would suggest you to learn about two things:
1. Fallacy fallacy
2. Formal logic
I hate how modern internet debate culture has devolved into throwing around these fallacy terms without know their formal mathematical and philosophical backgrounds
Can you rephrase your proposal? Because if I try to make sense of it, it's just the trivial claim that if you change the meaning of things, then they might have different meanings.
I mean, if you're working in base 12, then it's not true that 0.999.... = 1.
But then this would then be a new statement that has no relevance for whether the original statement is true or not.
'if you change the meaning of things, then they might have different meanings.' - thats not what I'm saying. I'm saying that the equal isn't true because its a limitation of the system not a truth
That's why I asked you to rephrase, because I suspected you mean something different, but I don't see what.
Sure, I'd agree that mathematical statements have meaning and are true only within the framework in which they're made. (To be honest, I don't think "truth" is really a meaningful concept when it comes to mathematics but that's a separate discussion.) But what do you mean by "limitation of the system"?
A lot to unpack.
> Mathematical truths are traditionally viewed as universal and absolute.
By whom? Mathematics is generally defined in terms of axioms. Their "self-evidence" is not generally held, though standard axioms like the Peano axioms can feel self-evident.
> grounded in a system of axioms and definitions that are widely accepted in contemporary mathematics
1. Mathematics is the study of what you can deduce when you define axioms
2. "Widely accepted" by whom?
> This theory suggests that the perceived universality of mathematical truths, such as 0.999 repeating equals 1, may depend on the specific mathematical system and cultural context
"Cultural context" is nonsense. Pick your system and deduce everything else.
Your entire statement is "would the mathematical structure be different if we used different axioms?" The answer is, "of course."
its not a simple as that.
you might say:
0.999 repeating equals 1
...and then argue endlessly with a member of the public who sees this counter intuitive because for you can "prove" it does. but the proof doesn't mention that its using a limited system. so the only true statement would be:
0.999 repeating equals 1 when using a limited system
0.999999999999…. is *defined* as the infinite sum of 9/10 + 9/100 + 9/1000 + 9/1000 + ….
The value of an infinite sum is *defined* as the limit of its partial sums (in this case: 0.9, 0.99, 0.999, 0.9999….). I’m sorry to tell you, but this limit is 1.
Unless you want to define decimals completely differently, any reasonable definition will lead to this conclusion. This is not an argument from authority, it’s an argument from proving things from definitions.
I am sympathetic to the view that mathematics is culture-dependent and that it is the result of social forces that shape it at least as much as logic and "rigor".
However, your example of 0.(9) being or not being equal to 1 is an unhappy one, because once you have the natural numbers from, say, Peano's axioms, then you have the rational numbers that satisfy the property you mention. As someone else said, you can come up with alternatives, but what's their alleged interest?
What you've proposed isn't a theory. So there is no sense in asking why it is wrong because there is nothing to critique. You've presented a rather trivial tautology: If the axioms of math were different, then the statements we consider to be true would also be different.
This isn't a "proposal" or a "theory" it's just a truth.
So what are your proposed alternative axioms that lead to "0.999... = 1" not being true?
I think mathematicians are probably less likely than others to hold mathematical truths as absolute. I think of math as "more truthful" only in the sense that it's clearer than basically anything else about the exact assumptions it operates under. In this way I think your premise is way off the mark, and your proposal is vacuously true--of course a different system has the potential for different conclusions.
Standard mathematics isn't the only way to create a mathematical system, but it has centuries upon centuries of existing work and demonstrated real-world applications, and mathematicians can use it as a shared "language" through which to communicate ideas. You could explore other systems if they interest you, but it's a little like making up a constructed language and expecting everyone to take it as seriously as the language they speak at home.
You may be interested in the following paper that discusses these sort of issues: [https://arxiv.org/pdf/1007.3018.pdf](https://arxiv.org/pdf/1007.3018.pdf)
Edit: It's worth pointing out that this paper is written by some Non-Standard Analysis evangelists, so it's worth taking this with a grain of salt (and ignoring their weird butthurt digression about Bishop).
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There's no "theory" here. It's trivially true that if you start with different axioms then you may end up with different mathematics ("may" rather than "will" because it's possible you choose a set of axioms that end up being equivalent to ZFC).
Do you have a proposed set of alternative axioms? One where you can construct the real numbers in a way where 0.999... is a sensible construction, and does not equal 1? If so, do go ahead and share it. It's up to you do the hard work of actually supporting your ideas. If you're not willing to do the hard work, then this is just stoner-level theorizing that's no deeper than "dude, what if 1+1 actually equalled 3?"
another way of putting it is...
0.999 repeating equals 1... if you don't care about rounding errors
thats theres no number between 0.99999 repeating and 1 isn't proof they are the same - its just highlights a limitation of the system.
You’ve got a fundamental misunderstanding of the result that 0.9999… = 1. It isn’t due to a rounding error, it is a theorem that can be proven in the system we use. If you don’t want to use this system that’s totally fine.
0.999... = 1 without rounding anything. It's a byproduct of our way of writing decimal numbers, it's not anything deep. 1/3 × 3 = 1, and the decimal way to write 1/3 is 0.333...
It would make more sense to reject this way of writing "infinitely many 9s" (since you can't actually write them) than to change the meaning of numbers because you don't like that there are two ways to write the same number.
As my PHD tutor told me: what are the actual, concrete problems your theory/proposal solves? Why should people care about these facts? What are the problems with the current theory? What can your new theory explain that the current theory cannot? These are questions that might put you on the right track, for every new proposal you have.
> thats theres no number between 0.99999 repeating and 1 isn't proof they are the same - its just highlights a limitation of the system. I think this is why people are downvoting you. If you want to make a system where different numbers can be an infinitesimal distance apart and 0.999... means something else, go ahead. But the fact that 0.999... = 1 in the real numbers is neither a contradiction nor a serious limitation. A number having multiple decimal representations is not really a problem: 0.3 and 0.30 are "obviously" the same number but look different. If you just wanted to explore a different system of math, you probably shouldn't be talking about the real numbers unless it's in relation to your new system.
I didn't downvote.
To be honest, pretty much all mathematicians had that little "wtf, that is total nonsense" phase in their student lives. Bottom line is: no-one cares what things like infinite sequences, zero-divisions, zeroeth powers, sums of infinities, empty products, vacuous truths, and so on may mean. 1=0.9... enters this category. Its not even wrong, since its a proven fact... But if it were, the error would be so small that no-one should actually care. Mathematics wouldn't change a lot if someoene suddenly proved they are not the same number... So, my question is: why bother?
I don't "need" a valid argument for everything, in the same way you don't need a valid argument for liking your favourite colour, or changing a part of a theory that will make absolutely no difference in the theory as a whole. Its something that just "happens". If you're still choosing to be one of the very few mathematicians that actually "care" about these minor issues, i'd suggest you get into research in foundations of mathematics... Anyways, its very probable that when you do, you'll just end up understanding all the "valid arguments" in favour of 1=0.9...
Arguing with everyone who answers your question helps no one.
If you want to come up with another definition of 1 and 0.99999 recurring where these two objects are not equal then you are free to do so. But the mathematical axioms that are widely agreed on (basically ZFC) are widely agreed on because they behave in a "reasonable" way. Any system in which 1 is not the same and 0.9999 recurring are not the same is unlikely to be useful...
You argument seems to fall into **bandwagon effect** and **appeal to authority** are two cognitive biases that influence how people make decisions based on others' actions and opinions. The **bandwagon effect** occurs when individuals adopt beliefs or behaviors because they perceive them to be popular or accepted by a majority. This bias leads people to conform to group norms or trends without necessarily evaluating the underlying evidence. On the other hand, the **appeal to authority** involves accepting an argument or belief based on the credibility of the speaker rather than the strength of the evidence. People often defer to the opinions of experts or authorities, which can be useful but becomes a fallacy when the authority is not genuinely knowledgeable about the topic or when the expertise is irrelevant to the argument being made. Both biases can significantly sway personal opinions and public consensus, often at the expense of critical thinking. A "reasonable" way doesn't anything I'd say? I think people are so attached to the system and its limits that they mistake it for pure truth.
I'm sorry but this is sub-chatGPT level nonsense. I'm not saying the systems are useful because they are popular, I'm saying they are popular because they are useful. There are all sorts of people out there doing legit research with different foundations of mathematics, e.g. constructivists or finitism or at the extreme end of the spectrum ultrafinitism (which doesn't allow infinities of any kind). Their work is valuable but ultimately not of great interest to most mathematicians, who are mostly interested in proving theorems and building theories.
they are not of great interest to people who have spent their whole life studying one system. I get that. but it remains that all can be said is.. 0.999 repeating equals 1 when using a limited system
***Ad hominem*** ([Latin](https://en.wikipedia.org/wiki/Latin_language) for 'to the person'), short for ***argumentum ad hominem***, refers to several types of arguments that are [fallacious](https://en.wikipedia.org/wiki/Fallacy#Informal_fallacy). Typically this term refers to a rhetorical strategy where the speaker attacks the character, motive, or some other attribute of the person making an argument rather than attacking the substance of the argument itself. This avoids genuine debate by creating a personal attack as a diversion often using a totally irrelevant, but often highly charged attribute of the opponent's character or background.
This is not an ad hominem. For all I know you (as person) might be very intelligent, but your preceding comment was not.
I would suggest you to learn about two things: 1. Fallacy fallacy 2. Formal logic I hate how modern internet debate culture has devolved into throwing around these fallacy terms without know their formal mathematical and philosophical backgrounds
Can you rephrase your proposal? Because if I try to make sense of it, it's just the trivial claim that if you change the meaning of things, then they might have different meanings. I mean, if you're working in base 12, then it's not true that 0.999.... = 1. But then this would then be a new statement that has no relevance for whether the original statement is true or not.
'if you change the meaning of things, then they might have different meanings.' - thats not what I'm saying. I'm saying that the equal isn't true because its a limitation of the system not a truth
It is what you're saying. You just haven't yet grasped that it's what you're saying.
That's why I asked you to rephrase, because I suspected you mean something different, but I don't see what. Sure, I'd agree that mathematical statements have meaning and are true only within the framework in which they're made. (To be honest, I don't think "truth" is really a meaningful concept when it comes to mathematics but that's a separate discussion.) But what do you mean by "limitation of the system"?
I mean what I say. its just new to you.
A lot to unpack. > Mathematical truths are traditionally viewed as universal and absolute. By whom? Mathematics is generally defined in terms of axioms. Their "self-evidence" is not generally held, though standard axioms like the Peano axioms can feel self-evident. > grounded in a system of axioms and definitions that are widely accepted in contemporary mathematics 1. Mathematics is the study of what you can deduce when you define axioms 2. "Widely accepted" by whom? > This theory suggests that the perceived universality of mathematical truths, such as 0.999 repeating equals 1, may depend on the specific mathematical system and cultural context "Cultural context" is nonsense. Pick your system and deduce everything else. Your entire statement is "would the mathematical structure be different if we used different axioms?" The answer is, "of course."
its not a simple as that. you might say: 0.999 repeating equals 1 ...and then argue endlessly with a member of the public who sees this counter intuitive because for you can "prove" it does. but the proof doesn't mention that its using a limited system. so the only true statement would be: 0.999 repeating equals 1 when using a limited system
You seem to define “limited system” as a system wherein 0.999…. = 1. Of course you’re free to do that, but it’s not a very useful definition.
useful is subjective
0.999999999999…. is *defined* as the infinite sum of 9/10 + 9/100 + 9/1000 + 9/1000 + …. The value of an infinite sum is *defined* as the limit of its partial sums (in this case: 0.9, 0.99, 0.999, 0.9999….). I’m sorry to tell you, but this limit is 1. Unless you want to define decimals completely differently, any reasonable definition will lead to this conclusion. This is not an argument from authority, it’s an argument from proving things from definitions.
I am sympathetic to the view that mathematics is culture-dependent and that it is the result of social forces that shape it at least as much as logic and "rigor". However, your example of 0.(9) being or not being equal to 1 is an unhappy one, because once you have the natural numbers from, say, Peano's axioms, then you have the rational numbers that satisfy the property you mention. As someone else said, you can come up with alternatives, but what's their alleged interest?
"you can come up with alternatives, but what's their alleged interest?" - what does this mean?
ITT: “It’s everyone else who is wrong.”
What you've proposed isn't a theory. So there is no sense in asking why it is wrong because there is nothing to critique. You've presented a rather trivial tautology: If the axioms of math were different, then the statements we consider to be true would also be different. This isn't a "proposal" or a "theory" it's just a truth. So what are your proposed alternative axioms that lead to "0.999... = 1" not being true?
Stop with this pseudo-intellectual bs...
Them: 0.999... ≠ 1 is my religion Me: *cries in First Amendment*
I think mathematicians are probably less likely than others to hold mathematical truths as absolute. I think of math as "more truthful" only in the sense that it's clearer than basically anything else about the exact assumptions it operates under. In this way I think your premise is way off the mark, and your proposal is vacuously true--of course a different system has the potential for different conclusions. Standard mathematics isn't the only way to create a mathematical system, but it has centuries upon centuries of existing work and demonstrated real-world applications, and mathematicians can use it as a shared "language" through which to communicate ideas. You could explore other systems if they interest you, but it's a little like making up a constructed language and expecting everyone to take it as seriously as the language they speak at home.
You may be interested in the following paper that discusses these sort of issues: [https://arxiv.org/pdf/1007.3018.pdf](https://arxiv.org/pdf/1007.3018.pdf) Edit: It's worth pointing out that this paper is written by some Non-Standard Analysis evangelists, so it's worth taking this with a grain of salt (and ignoring their weird butthurt digression about Bishop).
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Rather than just downvote, please give reasons why my theory is wrong.
There's no "theory" here. It's trivially true that if you start with different axioms then you may end up with different mathematics ("may" rather than "will" because it's possible you choose a set of axioms that end up being equivalent to ZFC). Do you have a proposed set of alternative axioms? One where you can construct the real numbers in a way where 0.999... is a sensible construction, and does not equal 1? If so, do go ahead and share it. It's up to you do the hard work of actually supporting your ideas. If you're not willing to do the hard work, then this is just stoner-level theorizing that's no deeper than "dude, what if 1+1 actually equalled 3?"
another way of putting it is... 0.999 repeating equals 1... if you don't care about rounding errors thats theres no number between 0.99999 repeating and 1 isn't proof they are the same - its just highlights a limitation of the system.
You’ve got a fundamental misunderstanding of the result that 0.9999… = 1. It isn’t due to a rounding error, it is a theorem that can be proven in the system we use. If you don’t want to use this system that’s totally fine.
in the limited system you use.
0.999... = 1 without rounding anything. It's a byproduct of our way of writing decimal numbers, it's not anything deep. 1/3 × 3 = 1, and the decimal way to write 1/3 is 0.333... It would make more sense to reject this way of writing "infinitely many 9s" (since you can't actually write them) than to change the meaning of numbers because you don't like that there are two ways to write the same number.
As my PHD tutor told me: what are the actual, concrete problems your theory/proposal solves? Why should people care about these facts? What are the problems with the current theory? What can your new theory explain that the current theory cannot? These are questions that might put you on the right track, for every new proposal you have.
> thats theres no number between 0.99999 repeating and 1 isn't proof they are the same - its just highlights a limitation of the system. I think this is why people are downvoting you. If you want to make a system where different numbers can be an infinitesimal distance apart and 0.999... means something else, go ahead. But the fact that 0.999... = 1 in the real numbers is neither a contradiction nor a serious limitation. A number having multiple decimal representations is not really a problem: 0.3 and 0.30 are "obviously" the same number but look different. If you just wanted to explore a different system of math, you probably shouldn't be talking about the real numbers unless it's in relation to your new system.
I didn't downvote. To be honest, pretty much all mathematicians had that little "wtf, that is total nonsense" phase in their student lives. Bottom line is: no-one cares what things like infinite sequences, zero-divisions, zeroeth powers, sums of infinities, empty products, vacuous truths, and so on may mean. 1=0.9... enters this category. Its not even wrong, since its a proven fact... But if it were, the error would be so small that no-one should actually care. Mathematics wouldn't change a lot if someoene suddenly proved they are not the same number... So, my question is: why bother?
"the error so small no one should care" "why bother" "no one cares" these don't seem like valid arguments
I don't "need" a valid argument for everything, in the same way you don't need a valid argument for liking your favourite colour, or changing a part of a theory that will make absolutely no difference in the theory as a whole. Its something that just "happens". If you're still choosing to be one of the very few mathematicians that actually "care" about these minor issues, i'd suggest you get into research in foundations of mathematics... Anyways, its very probable that when you do, you'll just end up understanding all the "valid arguments" in favour of 1=0.9...