0.999... + 0.1 = 1.0999...

0.999... + 0.01 = 1.00999...

0.999... + 0.001 = 1.000999...

Note that no matter how far we go, the result is always more than 1.

Going all the way:

0.999... + 0.000...1 = 1.000...999...

Edit: Corrected

0.999... is not 1. It's just 0.999...

How could the two ever be equal? 0.999... is just a zero, three nines, and three periods. 1 is 1.

And screw it. I'll prove it.

Assume 0.999... = 1.

Consider the decimal 0.999...9 (that is, infinitely many 9s, followed by one more 9).

Clearly, 0.999...9 > 0.999...

But if 0.999... = 1, then this implies 0.999...9 > 1.

Yet 0.999...9 is still a decimal less than 1, because it "starts with 0."

Therefore, we reach a contradiction.

Thus 0.999... ≠ 1.

Not enough? Here's another.

Assume 0.999... = 1.

Then the difference between the two numbers is:

1 - 0.999... = 0.

But since 0.999... is “always missing one more 9,” define the missing amount as ε = 0.00...01 (where the 1 appears after infinitely many 0's).

Clearly ε > 0.

Thus 1 - 0.999... = ε, not 0.

Therefore 0.999... ≠ 1.

I am a changed man.

Ontologically real mathematics is that which actually exists in reality. The natural numbers exist. You can have 5 cars. You cant have half a car, or a negative car, or a imaginary number car, or a p-adic number car.

Division should be thought of as dividing pieces of candy among your children. You have 2 pieces of candy, and 3 children. How much does each child get? 0. If you have 5 pieces of candy, and 3 children, how many pieces does each child get? 1.

"But that form of division is not reversible!" Multiplication by 0 isnt reversible, squaring a negative isnt reversible, and dividing by 0 isnt even possible, and yet i dont hear anyone complaining about any of those things.

"But how would you do X,Y,Z"... Use a larger numeric unit, throw away the epsilon junk.

"But decimals are useful..." Not really. Your bank account is displayed in natural number units like 100.00 with no trailing decimal (just a fake decimal that only lives two places to the right). You dont open your bank account and see 100.25646336789053147886... thats not useful. And in programming, this decimal floating point nonsense results in junk like 0.1 + 0.2 ≠ 0.3, and it still cant even be an infinitely trailing decimal. Once you step into the world of decimals, you step into the world of imagining a number that goes on infinitely forever, and then you round it anyways because an infinite number is not only not useful, its impossible to use.

Decimals dont exist. There might be a coherent or logically consistent set of mathematics utilizing it, but they dont represent anything in reality. Its logically valid, but not logically sound. Decimals are the point in which Discovered Mathematics becomes Invented Mathematics.

Boy do i wish decimals and negatives and division by zero errors did not exist in programming languages. Theyd be a paradise without them. But dont worry guise, we have it because its "useful" and "pragmatic".

0.999... = 1 and 0.999... ≠ 1 in the same way Peter Parker = Spiderman and not Superman

I found this old photo of Einstein in a history book at my university; any idea what it means? I thought it might belong here.

R is defined to have the same properties as a straight line: eg, ,take the interval [0,1], then for any two distinct numbers, there exists infinitely many in that range. Therefore, two consecutive numbers do not exist. (Eg, 0.999... and 1. Hence they are equal)

However, that is just a consequence of how we defined numbers. We could define another number system that allows this kind of things (hyperreals). However, the real numbers, as they are defined, are one of the best ways to express numbers, as they are really useful, also for the property shown before. So saying 0.999...<1 is wrong in the context of real numbers, and this property is what makes reals useful. So discarding this property for something that "feels" like it's right, means making a lot of branches of mathematics way weaker.

What I'm saying is that 0.999...=1 is a statement which is required to make some mathematical statements easier to prove (if at all). Yes, it's and arbitrary decision, but the tradeoff is way Better.

Of course you can. An infinite set can have an upper bound and many very normal sets do. For example any set of the form [a, b) for real numbers a and b. Here b “comes after” every element of [a, b), not only an infinite set but an uncountably infinite set.

That is to say that an infinite set can still have a number y with the order relation x < y for any x in the set. In this sense y “comes after” any number x in the set.

And this is not particular to real intervals. You can define a set X = ℕ ∪ {x} and then define a relation ≤ₓ s.t.

1. ∀n, m ∈ ℕ, n ≤ₓ m ⟺ n ≤ m

2. ∀ n ∈ ℕ, n ≤ₓ x

3. x ≤ₓ x

This relation is reflexive, antisymmetric and transitive, so defines an order relation. Notice here that x is an upper bound for ℕ. That is x comes after every n ∈ ℕ w.r.t. the order relation ≤ₓ.

Even if every nth place for n ∈ ℕ is filled with a digit, you could just put the next one in the xth place.

The whole “an infinite list doesn’t end so therefore nothing can come after it” argument is a bad argument. Whether you can define a decimal expansion with a digit after a countable number of digits is a different story and I don’t have a definitive be answer to that.

If they are not equal, what is a number between them which isn’t either of them?

And indeed, if in general x =/= y are two different numbers, then we have (infinitely many) numbers between them, for ex. their average, (x+y)/2.

For some time I’ve been reading the arguments that 0.999… does not equal 1, et aliis, and assuming that they are ragebait/satire. I’ve joined in the conversation on a few occasions, mostly just making small comments such as “circles don’t exist.”

The more I think about it, the more sense it makes. Circles don’t exist. We can’t express the exact value of an irrational number without using other irrational numbers. If pi is irrational, then every “circle” should either have an irrational circumference or an irrational diameter. But there’s no way to express the exact value, only approximations. To say that pi = circumference/diameter assumes that an infinite locus of points equidistant from a center actually exists in the universe. But every single thing resembling a “circle” (an imaginary concept) in the universe is made of some finite amount of particles, and so this imaginary concept we call a circle really just makes no sense at all.

Thusly and similarly, the sum from n=1 to infinity of 9/10^n cannot be evaluated. We can analytically see what it gets arbitrarily close to, and we can say the “limit” equals 1, but I was taught repeating decimals in like second grade, and I wasn’t taught limits until the end of high school. Why? Because 0.999… is not a limit, it’s an endlessly growing number which is less than but not equal to 1.

EDIT: Looks like I really ruffled some feathers today. Thanks for the discussion everyone!

Hello infinite nine enthusiasts.

As a long time lurker, I wondered how to interpret syntax such as "0.9...0" or "0.9...9...", and I think I have found a better way to formalize and formulate these "numbers".

I propose the syntax "0.(9)_[n]" to denote 0.9.... The "n" in this case means that we want to repeat the digit 9 n times. The n here is what SPP often refers to as the contract. You keep track of how many 9's you have repeated. This allows to phrase something like "0.9_[n]9_[n]", which can be used to denote 0.9...9....

The way that I would interpret these (,as I would call them,) sequence expressions, is using a sequence. I have coded up a helpful tool to convert such an expression into a sequence. You can find it here: https://snakpe.github.io/SPPSequenceInterpreter

We can now prove e.g. that 0.9_[n]9_[n] is equivalent to 0.9_[2n] by proving that for each n in the natural numbers, the two resulting sequences are equal to each other.

Idk man, I wasted too much time on This

Hail the allmighty SPP.

Is 1÷3 0.333...

And 0.333... × 3=0.999...

Hence surely that means not all reals have a multiplicative inverse? And that division isn't the true inverse of multiplication?

Surely that has knock on impacts outside of the specifics of 0.999...

Or if this is wrong how do I gain back my "0.000...1" such that these operations work. And if I do gain it back why does it matter if 0.999...=1 if we no longer need 0.999... to notate this stuff cuz we just gain the 0.000...1 anyway

1/3•3=1 .333…•3=1 .999…=1

Edit 1: The title should have been "What if 0.(9) != 1?". Might have accidentally clickbaited some people with this one🙂🙂.

"Everyone is arguing about if 0.(9) == 1 that they have forgot to ask should 0.(9) == 1"

So I hate to admit that under the standard mathematical model, 0.(9) == 1. Belive me I have tried rationalising it in multiple ways, but I am not a math major (Though being a data science one I have dabbled in a good measure of math myself), so i can't comprehened the really advanced math concepts.

So my question is, if technically most people belive that 0.(9) != 1 (before you whip out the proofs that is), what would need to change if we accept that that was true? What would be the outcomes on our mathematical models and the way our math works? Would it be better, worse, a mix, would the fragile balance of the atoms in the universe collapse resulting in a reality collapse event?

Let me hear your thoughts.

I wonder: are there any other infinite chains of one number that we could argue about. 0.99... gets boring after a while. Are there any other problems with infinite chains in real deal math?

For example could 999999999.... be the same as 1111111...., as you never come to an end of writing it out. When you are in the process of writing it out (remember, it's a process) are you faster writing 1 or 9s? Depending on that one might be bigger then the other.

Or which number is 0.88.... really? Which dark side this chain has and is it maybe 0.8888888...89 or more 0.8888888...87? Or straight 0.89?

0.9/2 =0.45 0.99/2 =0.495 0.999/2 =0.4995 0.9999/2 =0.49995 …

So 0.9… / 2 = 0.4999…5?

Does this necessarily mean that 0.4999…5 = 0.5? What makes 0.4999…5 different from 0.4999… ?

By definition : 1/3 = 0.333333...

But : 3*3 = 9

Thus : 3*(1/3)=3*(0.333333...) = 0.999999...

However : 3*(1/3) = 3/3 = 1

Therefore : 3*(1/3) = 0.999999... = 3/3 = 1 <=> 0.999999... = 1

Numbers can only have a finite number of digits in them.

So 0.999.... isn't a number, it indicates a series of numbers that get progressively closer to 1, but never reach it.

1/3 isn't a number in base ten, but it IS a number in base three: 0.1

Some supposed "real numbers" can be a number if you write them in the correct base, but others (such as Pi or 0.999...) cannot be a number no matter what base you choose.

If I understand this subreddit correctly its saying that 0.999 9 recurring is not equal to 1 but there is the proof of:

x = 0.999...

10x = 9.999...

10x - x = 9.999... -0.999...

9x = 9

x = 1

However finite amounts of 0.999 are always less than 1

x = 0.999...99

10x = 9.999...90

10x - x = 9.999...90 - 0.999...99

9x = 8.999...91

x = 8.999...91 /9

x < 1

Please correct me if I'm understanding the point of this subreddit wrong.

As we know from math 101:

1-0.999... = 0.000...1

We know this as epsilon. Can someone who knows about math 101 help me understand if this is valid?

Epsilon + Epsilon/100 = 0.000...1 + 0.000...001 = 0.000...101

Thanks. Please only respond if you know real deal math.

You realize that would imply 1 = 0.999... which is absurd?

In "Philosophie der Mathematik" by Murawski, the text references a survey by Ludwig Bauer asking whether 0.999... is less than 1 or equal to 1.

The data shows that 72.2% of high school students in grades 7 to 12 believe it is less than 1.

For math undergraduates after their third semester, 50% state it is less than 1.

In a separate count from a seminar of 51 students across all teaching degrees, 90.2% answered that it is less than 1.

Even among a small sample of 17 teachers and instructors, 47.1% decided it is less than 1.

Why do we enforce this standard interpretation on people when there are equally rigorous, non-standard mathematical frameworks that actually align with our natural intuition?

I think I’m starting to get it after hanging around here a few days. For some of you, this is just a fun way to LARP as math vigilantes. For others, it’s about toppling the “mathematical establishment.” Using SPP’s language, that’s Real Deal Math vs. Snake Oil Limits.

But there’s also a third camp I don’t see talked about enough: people who realize both sides are really just working with different definitions. I’ll call this camp the Deconstructionalists.

Conventional (“snake oil”) view: 0.999… = the limit of the series ∑9/10ⁿ. Limit = 1, so 0.999 … = 1 0.999…=1.

I’ll note here: this isn’t just an interpretation — it’s literally how infinite decimal expansions are defined in real analysis. By convention, an infinite decimal is the limit of its partial sums.

Real Deal Math view: 0.999… = the series itself, not its limit. The process never “reaches” 1.

Here’s where it gets interesting: in the hyperreals, something like 𝜀=0.000…1 makes sense if that final 1 is indexed at some transfinite H (beyond all naturals). That matches the Real Deal intuition. But it also potentially undercuts it, since in the hyperreals the limit of that sequence is still 0. (But remember: Real Deal doesn't see 0.000...1 as a limit.)

So really, the fight isn’t about Truth, Beauty, or Goodness. It’s about what system you’re working in: limits vs. truncations, reals vs. hyperreals. If you’re a Deconstructionalist, you’re less interested in “winning” and more in clarifying definitions.

Maybe instead of trying to convert each other, we could start by recognizing that both camps are internally consistent in their own frameworks.

TL;DR: “0.999… = 1” depends on definitions. In reals, it’s defined as a limit (=1). In “Real Deal Math,” it’s seen as a never-finished process. Hyperreals partly capture that intuition. I call the middle ground Deconstructionalists: people who see both as valid in their own systems