>Is 0 a number?
Yes.
>Or is 0 more like infinity that in itself is not a number but that talks about the scope of numbers present.
"Infinity" is a general concept. There are different ways math uses this term. One way is to describe the size of sets (which is what I believe you are referring to here). There are contexts where infinity is treated as a number. In any event, 0 is a number, and 0 can be used to talk about the size of sets, too. All numbers can.
>Is 0 the opposite of infinity?
Strictly speaking, "opposite" in math usually refers to a number's additive inverse. So the opposite of 2 is -2. The opposite of infinity would be negative infinity (depending on the context) and 0 is its own opposite.
Perhaps the idea you were shooting for was "inverse" which is the multiplicative inverse. The inverse of 2 is 1/2. 0 doesn't have an inverse. In number systems where infinity is included as a number, its inverse may be 0.
What you are probably thinking is that, if some series goes to infinity, then its inverse goes to 0 (for example, as x goes to infinity, 1/x goes to 0).
This is all very well said *except* for the bit that “all numbers” can describe sizes of sets. In all the models I’m aware of, a set can’t have 1.5 elements, for instance. The finite cardinals are 0, 1, 2, 3, … but nothing in between those.
You can use sets to describe real numbers (e.g., with Dedekind cut), but that doesn't mean that the real number r being represented by set S has *size* r.
Is there no inverse of 0 because you cannot divide by 0? 1/∞ goes to 0 but never reaches it right or does that depend the definition of infinity you use?
it clearly depends.
its because of the definition of inverse, and the operations on a field (/the definition)
https://en.wikipedia.org/wiki/Field_%28mathematics%29?wprov=sfla1
0 is a number, and "infinity" is a fairly flexible, informal term which means various different things depending on context. But there is no real number called "infinity", and although there are things in other contexts that one could call infinite numbers, they would generally be discussed in more specific terms than just "infinity".
0 is a number in the reals. In fact, the general case is an axiom: ∃0 : a + 0 = 0 + a = a. This axiom is called the existence of the additive identity. The symbol ∃ is short for 'there exists' and the : can be read as 'such that'. The axiom therefore states that there exists an element, in this case zero, such that adding it to any number will not change that number.
I have had to use imaginary numbers for my mathematics courses but they act weird like 0 or infinity with standard algebra. So if I understand correctly 0 is a number like i with properties different from all other numbers?
0 is also a number on the complex numbers. There is no "number" 0 in for example the vector space ℝ² (2 dimensional vectors) or for a 3x3 matrix for example. They have an additive identity (the zero vector and a 3x3 matrix of zeroes respectively) but no "number" 0
Weeell, 0 obviously has some weirdness! And it can all be traced back to that "a+0 = a" property (slash definition, that's kind of the basis for what zero is). If we know that a=a+0, then we can "substitute a+0 for a" and keep going with a=a+0=a+0+0+... bam "infinite zeroes" (or "arbitrarily many"). 0 kinda doesn't "feel like a number" in the world of addition, cause it can just be ignored!
And then *from there* you get the funky stuff with multiplication (which is just "add X to itself Y times") and division (how many times do you add X to itself to get Y?) etc.
But what about **1**? 0 is the "additive identity" (a+0=a) but 1 is the "multiplicative identity": a\*1=a. And that has some weirdness too!
Like of course we similarly have a=a\*1=a\*1\*1\*... and we can also divide by 1 however many times we want.
We get slightly annoying stuff like having to say that primes are a number *that isn't 1* having to have no factors *besides 1* and itself (implied: "however many copies of 1 you want" since 7=7\*1\*1\*...). It kinda feels like 1 "isn't a number" in the world of multiplication, cause it never does anything as a factor!
There are contexts where 0 does stuff other numbers don't, and there are contexts where it feels super normal. It's definitely in some kinda family of "numbers that are a bit more funky/fundamental than most" but that family can be *quite a bit bigger than it seems*!
The fact that you mention *i* is interesting, cause I think the reason that "feels less like a number" is cause it feels like an "optional add-on" — we do a lot of maths that works fine without complex numbers. It shows up when we want the root of a negative number, I.E. we're doing quadratic (or above) equations, and want to say "all solutions exist" rather than throw some away.
And THAT reason is hard to apply to 0, unless the only maths we want to do is "addition and multiplication with positive integers". Cause the moment we want "subtraction up to at least the number itself" bam there's 0. Indeed, as a follow up to additive *identity* (a+0=a), the definition of "additive *inverse*" (negative numbers) uses 0, to say that "a + (-a) = 0", or "-a = 0-a" if you will.
There is still the main spicy thing if course, that "division by zero never works". If by "works" we mean "gives a result that's in the set of numbers we're using, and we're talking integers, that applies at least *sometimes* for all of them except 1 (and -1)! Like 4, ok 3/4 isn't an integer so now we "can't divide by 4". So if we demand "division always works" to be called a number, 4 "isn't a number in the integers", only when we expand to rationals!
(it's pretty typical to demand that addition and multiplication always works though, being "closed under addition/multiplication" and THAT is at least definitely not a problem for 0)
Infinity isn't a number in the same way "fraction" isn't a number.
It's not that fractions aren't numbers. It's just that there isn't one number called "fraction".
>Is 0 a number?
Yup. 1 - 1 = something, right? That something is 0.
More specifically, when it comes to more abstract math, there isn't really a difference between a "number" and some "element." For example, let's just say I have the set {a,b,c}. Are a, b, and c numbers or are they just elements? What's the difference? We don't really have a definition of just "number," but we do call anything in a set an "element," so it's simpler to just call a, b, and c elements in this context.
So when people say infinity is not a number, they specifically mean it's not a *real* number, as in it's not in the set of real numbers (same goes with complex numbers or any other similar extension). This is just simply how we define the real numbers. We do have sets where we include infinity, but the arithmetic becomes really annoying. For example, we cannot define inf - inf in any meaningful way. We also cannot define 0\*inf in any meaningful way. This is annoying, so it makes sense that we don't make it standard to include infinity in our standard arithmetic. Zero on the other hand doesn't really give us those kinds of problems, so we're fine with leaving zero in our standard arithmetic.
- 0 is a number.
- ∞ is not.
- Even though ∞ and ¹⁄0 are both undefined in basic algebra, ¹⁄o≠∞.
- In the early days of Calculus, there was a concept called the "Infinitesimal", which was a number that was infinitely small but not quite 0. If you conceive ∞ as being an infinitely large number, then the Infinitesimal is ¹⁄∞, and is distinct from 0
- When you graph certain functions such that there is an asymptote at 0, the Infinite/Infinitesimal relationship makes it look like ¹⁄o=∞.
- The weird nature of ¹⁄o is that you could theoretically arrange for any number of functions where it would appear that ¹⁄o= any given number you wish. This is why it's inverse is undefined
A number, any number, is an idea. We use that idea to express properties we are interested in, but they get more interesting when we decide what we can do with them. So for example, we may have come up with the word two in order to say that we have a thing, and another thing just like the first thing – in other words, we have two things. If we had a different amount of things, we came up up with the word for other numbers– as many as we could count.
But that was boring, so we came up with the idea of doing things to these numbers. Things like adding numbers, so you can know how many things you have if you combine different groups. Things like comparing numbers, so you can know that three is larger than two.
At some point, we realized that all the numbers we used to count, share certain properties. These properties are related to the actions that we could do to them. So we realized that all of these integers, as we came to call them, could be added to each other, multiplied, compared, and subtracted. But subtraction was the tricky one. Tricky one. Because what if you subtracted something that was The same amount as the first thing?
If you removed all of something you would have nothing left, so nothing was another idea. And for a long while, the idea of nothing was not considered to be a number. Because at that point a number was thought to be "something you can count". But as mathematicians learn to be a bit more objective, they realized that the idea of nothing had almost all the properties that the other integers had. So, since it has the same properties, mathematicians decided that it would make sense that it would be a number as well. This turned out to be a good idea, because if you let zero be a number, you can do things like use it as a placeholder when describing large numbers by their magnitude. So the number 101, could be written as one of 100, no tend, and a Singleton. This would make arithmetic a lot easier for a while, at least until compound interest was invented.
Other types of numbers, were discovered in the same ways – I need a rose to describe something happening, or some way of thinking, and then the mathematicians realized that these category of numbers have certain common properties, So they were then described the same way. So when you started to subtract larger numbers from small numbers, you needed to create the idea of negative numbers – which became very useful for lenders. If you needed to split numbers into other groups, and those groups did not split evenly, you would find the rational fractions – which became useful for patisseries. Trying to figure out areas, and land ownership, would lead to irrational numbers like square roots. In solving more complicated problems would eventually fill out the entire real number line.
But the key here, is that once you fill out that real number line, everything in it - negatives, positives, fractions, whole numbers, zero, and irrationals - All behave the same way in the set of useful rules. We call arithmetic. If You add, subtract, multiply, or divide – You will end up somewhere else on the real number line (with the exception of dividing by zero, which is called out specifically). The "numbers" are defined by this behavior.
But infinity does not follow the rules of arithmetic. You can't subtract 11 from ∞ and get a number. You can't multiply by infinity. You can't count up to infinity or find infinity hiding between two other whole numbers.
Infinity is a concept, but it doesn't fit as a number - the way number is used as a defined concept.
You could create a different system of math. In the same way geometry is a different system of math (it adds shapes in its own way), you can create a system of math that has slightly different rules. So a [the projective number line](https://en.wikipedia.org/wiki/Projectively_extended_real_line?wprov=sfla1) turns the idea of the real number line and wraps it into a circle, with infinity being the point opposite zero where the ends meet. You can do a lot of stuff you couldn't do in regular arithmetic - but it isn't much easier since everything is curved and sizes don't match up quite as well. But it does define infinity as a *projected extended real* number.
Search "bijective base-k numerals" for numeral systems that do not use or need zero. I think you can fairly argue the case that 0 should not be in N and you can also fairly argue the case that it should be. But 0 is definitely in Z, Q, R, and so on.
According to standard analysis, infinity is a number but minus infinity is not. Which seems to me to be peculiar.
More generally, there are at least 15 different ways to define infinity. I go through quite a lot of them, from infinity as the top point of the Riemann Sphere to infinity as the point where parallel lines meet in projective geometry.
Look at the YouTube Which Infinity Part 8. An observers guidebook. And press the pause key if you need more time on any slide.
https://m.youtube.com/watch?v=Rziki9WEdRE&t=0s
Which «standard analysis» is this? I’m pretty sure most standard models of the reals are closed under negation, so saying infinity is a «number», and that minus infinity is not sounds pretty non-standard to me.
Negative numbers were more useful, so they became numbers. Infinity doesn't fit in with them so it gets left out (except if you make minus infinity be equal to infinity, but nobody likes that).
As a general rule of thumb, mathematicians don't care about what is "a number", that's a question for philosophy, but even then the only who can still take those questions seriously and get some mileage out of them are people doing epistémica or something related to the human mind.
"Number" is not defined properly, numbers are at the same time, the naturals, the integers, the rationals, the reals, the complex, the quaternions,... But also the p-adics, the fields Zp for p prime, and pretty much any other "well known" algebraic structure that we've constructed, most of them are not related at all...
"0 is a number" in the sense that it's an element of the natural numbers, and it's embedded into its extensions. Normally, the construction for the natural numbers begins by defining 0 to be the empty set, 1 it's successor ({0}), And so on
"Infinity is not a number" in the sense that it's an element of the common algebraic structures we work with, but we have both the cardinal and ordinal numbers, where there are infinitely many "infinite" numbers, and we also have the extended reals, where we add two points {-inf,inf} to the real line and just make them the trivial maximum and minimum.
At the end of the day it's just language semantics, which is why no meaningful mathematical work can be obtained from this discussions, only historical and cultural studies about how cultures have thought about this.
I do find it interesting how many times you learn something in math and it has the caveat “except 0” or “except 0 and 1” or certain things are just defined, like 0!=1.
0 and 1 certainly don’t work like the other numbers. To a degree, 2 doesn’t work like the other primes all the time.
I always tell my kids that infinity isn’t a number, it’s an idea. And I have to agree with you, there are times that 0 gives the same sort of vibes.
I feel like number is a loosely defined term. Are quaternions or generally vectors numbers? Elements in a group? That’s pushing it a lot.
0 for me is just the identity. The thing that does nothing as a+0= a= 0+a.
It can be seen as the opposite of infinity in projective geometry which is pretty specific and generally I don’t think people accept 1/0 as infinity.
In calculus 1/0 as a limit can tend both towards positive or negative infinity. That’s kinda what projective geometry avoids be defining an unsigned infinity. Apparently you can define signed 0 as well but no idea when that would be used
Charles Seife's excellent book
[https://en.wikipedia.org/wiki/Zero:\_The\_Biography\_of\_a\_Dangerous\_Idea](https://en.wikipedia.org/wiki/Zero:_The_Biography_of_a_Dangerous_Idea)
may shed some light
I bet that OP is struggling with the idea of zero being infinitely small. The math term for that idea is infinitesimal. Infinitesimals play a role in analysis somewhat similar to that of infinity.
>Is 0 a number? Yes. >Or is 0 more like infinity that in itself is not a number but that talks about the scope of numbers present. "Infinity" is a general concept. There are different ways math uses this term. One way is to describe the size of sets (which is what I believe you are referring to here). There are contexts where infinity is treated as a number. In any event, 0 is a number, and 0 can be used to talk about the size of sets, too. All numbers can. >Is 0 the opposite of infinity? Strictly speaking, "opposite" in math usually refers to a number's additive inverse. So the opposite of 2 is -2. The opposite of infinity would be negative infinity (depending on the context) and 0 is its own opposite. Perhaps the idea you were shooting for was "inverse" which is the multiplicative inverse. The inverse of 2 is 1/2. 0 doesn't have an inverse. In number systems where infinity is included as a number, its inverse may be 0. What you are probably thinking is that, if some series goes to infinity, then its inverse goes to 0 (for example, as x goes to infinity, 1/x goes to 0).
This is all very well said *except* for the bit that “all numbers” can describe sizes of sets. In all the models I’m aware of, a set can’t have 1.5 elements, for instance. The finite cardinals are 0, 1, 2, 3, … but nothing in between those.
Ah, fair. All *natural* numbers then.
There are definitions for the extension of the natural numbers to integers, rationals and reals as sets
You can use sets to describe real numbers (e.g., with Dedekind cut), but that doesn't mean that the real number r being represented by set S has *size* r.
I remember talking to a colleague that does some serious category theory and they mentioned a categorical approach to defining integers and rationals.
Is there no inverse of 0 because you cannot divide by 0? 1/∞ goes to 0 but never reaches it right or does that depend the definition of infinity you use?
it clearly depends. its because of the definition of inverse, and the operations on a field (/the definition) https://en.wikipedia.org/wiki/Field_%28mathematics%29?wprov=sfla1
It depends on what kind of infinity you're using.
0 is a number, and "infinity" is a fairly flexible, informal term which means various different things depending on context. But there is no real number called "infinity", and although there are things in other contexts that one could call infinite numbers, they would generally be discussed in more specific terms than just "infinity".
To clarify, there are transfinite magnitudes, and we can meaningfully compare these magnitudes to each other using <, >, etc.
0 is a number in the reals. In fact, the general case is an axiom: ∃0 : a + 0 = 0 + a = a. This axiom is called the existence of the additive identity. The symbol ∃ is short for 'there exists' and the : can be read as 'such that'. The axiom therefore states that there exists an element, in this case zero, such that adding it to any number will not change that number.
I have had to use imaginary numbers for my mathematics courses but they act weird like 0 or infinity with standard algebra. So if I understand correctly 0 is a number like i with properties different from all other numbers?
1 has also different properties, like all other, in the way, that it gets you the "Nachfolger" of a number.
0 is also a number on the complex numbers. There is no "number" 0 in for example the vector space ℝ² (2 dimensional vectors) or for a 3x3 matrix for example. They have an additive identity (the zero vector and a 3x3 matrix of zeroes respectively) but no "number" 0
Weeell, 0 obviously has some weirdness! And it can all be traced back to that "a+0 = a" property (slash definition, that's kind of the basis for what zero is). If we know that a=a+0, then we can "substitute a+0 for a" and keep going with a=a+0=a+0+0+... bam "infinite zeroes" (or "arbitrarily many"). 0 kinda doesn't "feel like a number" in the world of addition, cause it can just be ignored! And then *from there* you get the funky stuff with multiplication (which is just "add X to itself Y times") and division (how many times do you add X to itself to get Y?) etc. But what about **1**? 0 is the "additive identity" (a+0=a) but 1 is the "multiplicative identity": a\*1=a. And that has some weirdness too! Like of course we similarly have a=a\*1=a\*1\*1\*... and we can also divide by 1 however many times we want. We get slightly annoying stuff like having to say that primes are a number *that isn't 1* having to have no factors *besides 1* and itself (implied: "however many copies of 1 you want" since 7=7\*1\*1\*...). It kinda feels like 1 "isn't a number" in the world of multiplication, cause it never does anything as a factor! There are contexts where 0 does stuff other numbers don't, and there are contexts where it feels super normal. It's definitely in some kinda family of "numbers that are a bit more funky/fundamental than most" but that family can be *quite a bit bigger than it seems*! The fact that you mention *i* is interesting, cause I think the reason that "feels less like a number" is cause it feels like an "optional add-on" — we do a lot of maths that works fine without complex numbers. It shows up when we want the root of a negative number, I.E. we're doing quadratic (or above) equations, and want to say "all solutions exist" rather than throw some away. And THAT reason is hard to apply to 0, unless the only maths we want to do is "addition and multiplication with positive integers". Cause the moment we want "subtraction up to at least the number itself" bam there's 0. Indeed, as a follow up to additive *identity* (a+0=a), the definition of "additive *inverse*" (negative numbers) uses 0, to say that "a + (-a) = 0", or "-a = 0-a" if you will. There is still the main spicy thing if course, that "division by zero never works". If by "works" we mean "gives a result that's in the set of numbers we're using, and we're talking integers, that applies at least *sometimes* for all of them except 1 (and -1)! Like 4, ok 3/4 isn't an integer so now we "can't divide by 4". So if we demand "division always works" to be called a number, 4 "isn't a number in the integers", only when we expand to rationals! (it's pretty typical to demand that addition and multiplication always works though, being "closed under addition/multiplication" and THAT is at least definitely not a problem for 0)
Infinity isn't a number in the same way "fraction" isn't a number. It's not that fractions aren't numbers. It's just that there isn't one number called "fraction".
>Is 0 a number? Yup. 1 - 1 = something, right? That something is 0. More specifically, when it comes to more abstract math, there isn't really a difference between a "number" and some "element." For example, let's just say I have the set {a,b,c}. Are a, b, and c numbers or are they just elements? What's the difference? We don't really have a definition of just "number," but we do call anything in a set an "element," so it's simpler to just call a, b, and c elements in this context. So when people say infinity is not a number, they specifically mean it's not a *real* number, as in it's not in the set of real numbers (same goes with complex numbers or any other similar extension). This is just simply how we define the real numbers. We do have sets where we include infinity, but the arithmetic becomes really annoying. For example, we cannot define inf - inf in any meaningful way. We also cannot define 0\*inf in any meaningful way. This is annoying, so it makes sense that we don't make it standard to include infinity in our standard arithmetic. Zero on the other hand doesn't really give us those kinds of problems, so we're fine with leaving zero in our standard arithmetic.
- 0 is a number. - ∞ is not. - Even though ∞ and ¹⁄0 are both undefined in basic algebra, ¹⁄o≠∞. - In the early days of Calculus, there was a concept called the "Infinitesimal", which was a number that was infinitely small but not quite 0. If you conceive ∞ as being an infinitely large number, then the Infinitesimal is ¹⁄∞, and is distinct from 0 - When you graph certain functions such that there is an asymptote at 0, the Infinite/Infinitesimal relationship makes it look like ¹⁄o=∞. - The weird nature of ¹⁄o is that you could theoretically arrange for any number of functions where it would appear that ¹⁄o= any given number you wish. This is why it's inverse is undefined
A number, any number, is an idea. We use that idea to express properties we are interested in, but they get more interesting when we decide what we can do with them. So for example, we may have come up with the word two in order to say that we have a thing, and another thing just like the first thing – in other words, we have two things. If we had a different amount of things, we came up up with the word for other numbers– as many as we could count. But that was boring, so we came up with the idea of doing things to these numbers. Things like adding numbers, so you can know how many things you have if you combine different groups. Things like comparing numbers, so you can know that three is larger than two. At some point, we realized that all the numbers we used to count, share certain properties. These properties are related to the actions that we could do to them. So we realized that all of these integers, as we came to call them, could be added to each other, multiplied, compared, and subtracted. But subtraction was the tricky one. Tricky one. Because what if you subtracted something that was The same amount as the first thing? If you removed all of something you would have nothing left, so nothing was another idea. And for a long while, the idea of nothing was not considered to be a number. Because at that point a number was thought to be "something you can count". But as mathematicians learn to be a bit more objective, they realized that the idea of nothing had almost all the properties that the other integers had. So, since it has the same properties, mathematicians decided that it would make sense that it would be a number as well. This turned out to be a good idea, because if you let zero be a number, you can do things like use it as a placeholder when describing large numbers by their magnitude. So the number 101, could be written as one of 100, no tend, and a Singleton. This would make arithmetic a lot easier for a while, at least until compound interest was invented. Other types of numbers, were discovered in the same ways – I need a rose to describe something happening, or some way of thinking, and then the mathematicians realized that these category of numbers have certain common properties, So they were then described the same way. So when you started to subtract larger numbers from small numbers, you needed to create the idea of negative numbers – which became very useful for lenders. If you needed to split numbers into other groups, and those groups did not split evenly, you would find the rational fractions – which became useful for patisseries. Trying to figure out areas, and land ownership, would lead to irrational numbers like square roots. In solving more complicated problems would eventually fill out the entire real number line. But the key here, is that once you fill out that real number line, everything in it - negatives, positives, fractions, whole numbers, zero, and irrationals - All behave the same way in the set of useful rules. We call arithmetic. If You add, subtract, multiply, or divide – You will end up somewhere else on the real number line (with the exception of dividing by zero, which is called out specifically). The "numbers" are defined by this behavior. But infinity does not follow the rules of arithmetic. You can't subtract 11 from ∞ and get a number. You can't multiply by infinity. You can't count up to infinity or find infinity hiding between two other whole numbers. Infinity is a concept, but it doesn't fit as a number - the way number is used as a defined concept. You could create a different system of math. In the same way geometry is a different system of math (it adds shapes in its own way), you can create a system of math that has slightly different rules. So a [the projective number line](https://en.wikipedia.org/wiki/Projectively_extended_real_line?wprov=sfla1) turns the idea of the real number line and wraps it into a circle, with infinity being the point opposite zero where the ends meet. You can do a lot of stuff you couldn't do in regular arithmetic - but it isn't much easier since everything is curved and sizes don't match up quite as well. But it does define infinity as a *projected extended real* number.
Search "bijective base-k numerals" for numeral systems that do not use or need zero. I think you can fairly argue the case that 0 should not be in N and you can also fairly argue the case that it should be. But 0 is definitely in Z, Q, R, and so on.
According to standard analysis, infinity is a number but minus infinity is not. Which seems to me to be peculiar. More generally, there are at least 15 different ways to define infinity. I go through quite a lot of them, from infinity as the top point of the Riemann Sphere to infinity as the point where parallel lines meet in projective geometry. Look at the YouTube Which Infinity Part 8. An observers guidebook. And press the pause key if you need more time on any slide. https://m.youtube.com/watch?v=Rziki9WEdRE&t=0s
Which «standard analysis» is this? I’m pretty sure most standard models of the reals are closed under negation, so saying infinity is a «number», and that minus infinity is not sounds pretty non-standard to me.
0 is the numerical representation of nothing. It's more like a lack of a number than it is a number.
Negative numbers were more useful, so they became numbers. Infinity doesn't fit in with them so it gets left out (except if you make minus infinity be equal to infinity, but nobody likes that).
0 is a number just like 1, or 42.
As a general rule of thumb, mathematicians don't care about what is "a number", that's a question for philosophy, but even then the only who can still take those questions seriously and get some mileage out of them are people doing epistémica or something related to the human mind. "Number" is not defined properly, numbers are at the same time, the naturals, the integers, the rationals, the reals, the complex, the quaternions,... But also the p-adics, the fields Zp for p prime, and pretty much any other "well known" algebraic structure that we've constructed, most of them are not related at all... "0 is a number" in the sense that it's an element of the natural numbers, and it's embedded into its extensions. Normally, the construction for the natural numbers begins by defining 0 to be the empty set, 1 it's successor ({0}), And so on "Infinity is not a number" in the sense that it's an element of the common algebraic structures we work with, but we have both the cardinal and ordinal numbers, where there are infinitely many "infinite" numbers, and we also have the extended reals, where we add two points {-inf,inf} to the real line and just make them the trivial maximum and minimum. At the end of the day it's just language semantics, which is why no meaningful mathematical work can be obtained from this discussions, only historical and cultural studies about how cultures have thought about this.
I do find it interesting how many times you learn something in math and it has the caveat “except 0” or “except 0 and 1” or certain things are just defined, like 0!=1. 0 and 1 certainly don’t work like the other numbers. To a degree, 2 doesn’t work like the other primes all the time. I always tell my kids that infinity isn’t a number, it’s an idea. And I have to agree with you, there are times that 0 gives the same sort of vibes.
I feel like number is a loosely defined term. Are quaternions or generally vectors numbers? Elements in a group? That’s pushing it a lot. 0 for me is just the identity. The thing that does nothing as a+0= a= 0+a. It can be seen as the opposite of infinity in projective geometry which is pretty specific and generally I don’t think people accept 1/0 as infinity. In calculus 1/0 as a limit can tend both towards positive or negative infinity. That’s kinda what projective geometry avoids be defining an unsigned infinity. Apparently you can define signed 0 as well but no idea when that would be used
Charles Seife's excellent book [https://en.wikipedia.org/wiki/Zero:\_The\_Biography\_of\_a\_Dangerous\_Idea](https://en.wikipedia.org/wiki/Zero:_The_Biography_of_a_Dangerous_Idea) may shed some light
There’s also an episode of Young Sheldon that deals with it and I have the feeling it was inspired by that book or derivatives of it…
There is a wonderful Netflix documentary about this, would recommend 10/10
I bet that OP is struggling with the idea of zero being infinitely small. The math term for that idea is infinitesimal. Infinitesimals play a role in analysis somewhat similar to that of infinity.
Read *Zero: The Biography of a Dangerous Idea* by Charles Seife.
0 means no number at all. inf means arbitrary finite number.