Why 0.9999999...=1???

[crimson rivet]

Someone explain

[real tundra]

Any questions?

[desert gulch]

you can prove this problem on the number line

There is no number on the number line between 0.999.. and 1 so we can prove that

but the best way to solve this problem is algebra

like him @real tundra

[real tundra]

I think you can do it this way for other bases too like 0.11111...=1 (base two) or 0.666...=1 (base seven). Although can probably do same with the number line thingy now that i think about it.

[desert gulch]

yesss
but i think with this type of this problems use your way is the most suitable

especially in test

[real tundra]

Idk I just find the usual explanation of multiplying and dividing 0.999... a bit hand wavy because I'm not entirely sure you're allowed to do those operations... But idk

[real tundra]

I've never seen this question in a test but do you mean like it's more convincing than the typical one where people like multiply 0.999... by ten then do some stuff.

I think the number line method sounds about as convincing too tho.

But Idk

[desert gulch]

i think we can do with many way but just discuss

share your idea

[real tundra]

Idk I don't have any ideas 😭😭

[abstract fossil]

Depends

For real numbers it’s true

So go back to the definition of a real number

If that’s what you’re wondering about

[desert gulch]

I'm not sure about my way either. but i think your way is good

[abstract fossil]

There is a real way of doing it

[real tundra]

Don't you have to prove that 0.999... is a real number in the first place doing it that way?

Or is it sufficient to say like "assume 0.999... not a real number -> ... ->" and "assume 0.999... is a real number -> ... ->" like both cases

[civic cliff]

An alternative simple explanation is to use algebraic manipulation:

A/9 = 0.AAAAAAAAA...
1/3 = 3/9 = 0.333333...
3 • 3/9 = 3 • 0.3333... = 0.9999...
But then 0.999.. = 9/9 = 1

[desert gulch]

with your idea but this depends about how many A

[real tundra]

How do you prove that the normal laws of multiplication would hold for such an object though? I mean isn't this assuming 0.999... exists and is a real number in the first place.

[civic cliff]

i mean, i used A cuz usually capitalized letters are used as algarisms

[abstract fossil]

You have to first explain what that short hand “…” is defined as, if it’s for the number that is defined by the sequence a_1 =0.9, a_2 = 0.99, etc then use the definition of real numbers for say Cauchy sequences of rationals it will be a real number

[civic cliff]

as it is a periodic dizim, its not only real but its racional

[abstract fossil]

The thing we want to show is that sequence and the constant one defined by 1, are Cauchy equivalents

[civic cliff]

Or you can do:
x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x = 9/9 = 1

0.999... = 1

[real tundra]

Ah okay thanks that's what I was looking for that should show it's real then right?

[abstract fossil]

Yeah if we’re explicit about what … repeating means and we use say the definition of real numbers by Cauchy sequences then it’s just some real number

And we can show that it has the same property and indeed moreover is the same number as 1

This could also be done by showing that 0.99999…. Is a multiplicative identity for the Cauchy representatives

[real tundra]

Yeah well once you've shown it's a real number then it's okay to use like the 10*(0.999...) way isn't it?

[abstract fossil]

Well sort of ish, you have be precise and show that each thing you get is still the same real number

That’s why it’s just easier to directly show its defined by the same sequence for 1

[real tundra]

Why? Isn't a*b still real if they're both real I don't understand

[abstract fossil]

Yeah you have to show that

But moreover that your representation of that multiplication is the same as you’re claiming it to be

[real tundra]

Huh? Isn't closed under multiplication assumed

[abstract fossil]

No

That’s part of the construction

[real tundra]

But if it's all real it should be closed i dont see why you need to prove it again i thought that's just a property of R

[abstract fossil]

That’s a proven property of R

Depending on ur definition

[real tundra]

Yeah i think proving multiplication is closed is a bit too detailed

[civic cliff]

i think you guys are.. Overcomplicating things

[abstract fossil]

If ur definition is “the unique ordered field with the least upper bound property” then sure

[abstract fossil]

Don’t have to do it by multiplication

[real tundra]

I think that's mainly what I've seen online where u do 10(0.999...) and then some addition and other stuff

[abstract fossil]

Just show that a Cauchy sequence that defines 0.99999 and 1 are Cauchy equivalent

It’s very direct that way

[real tundra]

Is that any more direct doing it that way, than say; showing 0.999... is defined by a cauchy sequence and so belongs to R and so the usual online method follows?

[real tundra]

I think if so then it's better

[abstract fossil]

There’s a disconnect here. 0.999… and 1 are representatives for the same number

By showing 0.999… is a real number you’d almost automatically show it’s 1

[real tundra]

That's what we're trying to show yeah

[real tundra]

Oh is the way that you show 0.999... is a cauchy sequence involves a step where you equate it to 1 you mean? Idk what is cauchy sequence things tbh

[abstract fossil]

Yeah sort of. Let me try and explain it.

If we go by the definition of real numbers with Cauchy sequences, then what that basically means is that a real number is just some equivalence class, a set where all elements in this class is considered the “same” by some relation (I’ll get to which one soon)

So two distinct real numbers (which we remember can be seen as two distinct classes) have elements in them that are not considered the “same”.

0.999… and 1 are representations of that same class. Moreover, 1 is uniquely defined as the multiplicative identity. So showing 0.999… has that property and in so in particular is a real number; then 0.999… and 1 are the same.

A Cauchy sequence is a sequence of numbers which eventually clump together. The sequence doesn’t have to have a limit.

We define a relation on Cauchy sequences of rational numbers, by saying that two Cauchy sequences are equivalent iff their difference converges to 0. Note that this doesn’t necessarily mean they convergence and have the same limit, but just that their difference converges to 0.

This relation is a so called equivalence relation (which means the relation has certain properties akin to the = relation).
And so we can define the class of all elements which are the same under this relation.
This class defines some real number.

[real tundra]

Are you saying if you show 0.999... is cauchy equivalent to 1, that shows they are in the same equivalence class. then that means they share properties of multiplicative identity? So you like show they are of a particular equivalence class within the whole of R rather than just living inside R?

Sorry if its a stupid question this is all new 😭

[abstract fossil]

No that’s not a stupid question at all, that’s precisely what I mean, so yes that would in particular mean they represent the same real number

I understand it’s a lot

The way we see R as a set of numbers is a useful way of describing them, the structure is the same as the collection of all Cauchy classes

They’re different objects

But behave in the same way

[real tundra]

Wait isn't that really useful tho like u can say even more about two numbers than just living in R

U can say they behave same way no?

If I understand correctly 😭😭

[real tundra]

So are like different subsets of R categorised by different cauchy classes?

[abstract fossil]

So if we want to show something rigorously about real numbers, say that two representations of a real number is the “same”, then we have to use the more complicated object (the collection of Cauchy classes)

But if we just want to show properties about real numbers then we can forget about the small details and use properties we’ve shown using the complicated model and just treat the classes as numbers as how we would usually do

But the latter model here is not detailed enough to show if say two representatives of a real number is the same, makes sense?

[real tundra]

The latter model only shows they satisfy some properties of R, but cauchy classes shows that the numbers are equivalent ? Idk

Sorry im slow 😭😭

[abstract fossil]

Don’t worry, you’re doing very well; I’m honestly a bit surprised this is digestible.

Let me maybe create a concrete example

[real tundra]

You don't have to teach me the whole concept dww 😭

[abstract fossil]

Okay, well the main point I’m trying convey is that the construction of the real numbers gives an object that behaves the same way we’d expect our intuitive notion of real numbers to behave.

So once we’ve shown enough properties about them using the complicated model, then we can sort of forget that say the real number 3, is “the class of rational Cauchy sequences whose difference to the constant sequence (3,3,….) converges to 0” and instead treat it with respect to the structure of the real numbers, abstracting away stuff about Cauchy sequences etc. and focusing on the behaviour instead.

Now this isn’t anything new really, R is just built upon Q, where we did the same for Q.

Q is also constructed from equivalence classes, but upon Z (or N depending on how u do it)

In short:
When you want to show stuff about representations of real numbers, ur best bet is to go back to the construction.

When you want to show stuff about properties of real numbers, ur best bet is to forget about Cauchy sequences and use the properties that have already been proven.

Notice that the question, 0.9999… = 1? doesn’t fit within the second one

[real tundra]

Ok i think this answers my question thx

[real tundra]

Yeah i see now

[abstract fossil]

I’m sorry for the lecture

[real tundra]

Better to do construction than properties in this particular case basically

[abstract fossil]

Yeah

Also this question isn’t even necessarily true for other number systems that “contain R”

Like hyperreals for example

[real tundra]

Oh okay cool

[blazing ocean]

i think thats the easiest way to prove this

[civic cliff]

☝️

[urban ermine]

but floor(0.9999999...)=0
floor(1)=1

[civic cliff]

nope, floor(0.9999...) = 1

[real tundra]

Floor(0.999)=0 floor(0.999...)=floor(1)=1 . The ellipsis means that it's infinite series which happens to converge to 1 if you check it.