Is there something called next to 0 number?

is there something called next to 0 number?

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The idea you're thinking of is an infinitesimal. Instead of being infinitely big such as infinity, it's infinitely small, but not nothing such as zero.

Depends what universe you're working with. In the natural numbers 1 is next to 0.

>nothing such as zero
Zero is not nothing.
Zero is a number, it is represented by a point
on the real line, it is a placeholder in decimal
notation, it is the additive identity, and it is the
cardinality of the null set.
Whereas nothing is nothing.
Lrn2zero fgt pls

>Zero is not nothing
Yes it is.
How can you even possibly think that numbers are, faggot.
0 does not exist, neither do anything next to it.
we can invent, dream, postulate that "0 exists", but it doesn't mean that it does.
lmfao

there is no real number "next to 0"
this amounts to saying that for some closed subset A of the real numbers (under the usual order and topology), there is a least upper bound of A not contained in A

so we have an interval [X,Y], and we're looking for a least upper bound C of [X,Y] that is not contained in [X,Y]
because C is an upper bound of [X,Y], C is greater than or equal to k where k is an element in [X,Y]
and because it's a least upper bound, it is less than any other upper bounds

so we look at the set (Y,infinity), which is every upper bound of [X,Y] that is not contained in [X,Y]
so C must be less than every element in (Y,infinity)
we've assumed that C is not in [X,Y], so either C is smaller than X, or C is Y, both of which are contradictions

if you take Y to be 0, then you've just proven that there is no number "next to 0"

also somebody smart check this proof pls i'm taking topology next semester and i want to make sure i've got a solid foundation

>infinitesimals
REEEEEEEE

Yeah the limit as a number approaches zero
It's my favorite math trick to use :3

>where k is an element in [X,Y]
meant to say "for every k in [X,Y]"

lol uhhh... by this logic couldn't you let Y be any real number and prove there is no number "next to" any other number?

And that is true for the reals.

Exactly. What number is next to 2 for example?

I'm not really sure how it's called, but you can define a number e>0 but with e^2=0.

Yep, but that's also correct. Just like how there's no "largest integer before infinity", there's no "closest real number to 2"

Lrn2read, moron

Only works in a ring with zero divisors.

Divide both sides by e, you get that e=0.

I see, this makes sense now that I think about it. Is there a name for this property of sets?

Yes, it's called 1
:^)

Pretty much. It's because the rationals (and also irrationals) are *dense* in the reals.

We know that between any two real numbers, there lies a rational number because the rationals are dense in the reals (using this you can show that there is no number "next to" 2 by taking any number that would be next to it and showing there is a rational that is closer).

It's a proof based on typical axioms of rationals whereas an extended real number system that includes infintesimals does not use the axioms. To simply say infinitesimals do not exist is the same as saying any mathematical construction does not exist. We construct axioms based on how useful the derived properties of the system will be. As any physics student knows infinitesimals are quite useful.

Pffft such an easy question, the number is 1, see problem solved

>as any physics student knows

that doesn't mean the hyperreals aren't dense

are you a physics student?

explain the many uses of infinitesimals

>bonus points if can you do it without citing calculus, because differentials and 1-forms are not infinitesimals
>bonus BONUS points if you can do it without citing dual numbers

0+h

>bonus points if can you do it without citing calculus, because differentials and 1-forms are not infinitesimals

en.wikipedia.org/wiki/Differential_(infinitesimal)

>bonus points if you can explain something without actually being correct, therefore making me correct

epsilon

You can approximate the behavior of a physical system changing over time with great accuracy by considering infinitesimal components of the system behaving under infinitesimal time intervals and many of the times this will work despite being nonsense in a traditional mathematical framework.

i was trying to say that the ""actual"" infinitesimals aren't used in physics (except for a case where dual numbers are needed for some gauge theory wizardry), and that the constructions used in calculus are special cases of actual, mathematically precisely defined objects that we can talk about without referring to a philosopher

i'm pissed about this because "infinitesimal" has become a buzzword recently for some reason, and people who have no idea what it actually means are using it to refer to other things which they have no knowledge of,
because i am a math tutor, i hate to see such toxic misinformation being spread like this

and just to cover all bases, i do realize smooth infinitesimal analysis has been used in a topos-theoretical setting in some crazy branches of theoretical physics, though i'm not smart enough to figure out what actually goes on there

are you talking about linear approximation or something else

x ; x ---> 0

what about 0.5 thats even closer?

Don't listen to all of the brainlets in this thread OP. You certainly can find a number "next to" 0. Through use of the axiom of choice, one can well-order the set of all real numbers excluding 0 with a relation

>This ordering will have a minimal element
[citation needed]

A well order has the property that every non-empty subset has a least element. Are you daft?

So there can't be a good order on R, by Archimedean property there are no infinitesimals in R, meaning there is no number smaller than all the others, so there can't be order with least element

Look up the well-ordering theorem.