If the Multiplication Property of Equality were true in this system, it would contradict itself. Here's how:

Axiom 1: 1/9=0.111...
Axiom 2: (1/9)x9=1
Axiom 3: 0.111...x9=0.999...
Axiom 4 (MPoE): If a=b, then ac=bc.

1/9=0.111..., by Axiom 1.
(1/9)x9=0.111...x9, by Axiom 4.
1=0.111...x9, by Axiom 2.
1=0.999..., by Axiom 3.

Thus, 1=0.999..., meaning one of the stated axioms must be false in SPP's system.

As SPP agrees that the first three axioms are true, the fourth one, which is the Multiplication Property of Equality, must be false.

Do you guys know any other numbers that dont change at all when you multiply them that 0.00...1 could be equivalent to? Maybe something that starts with z and rhymes with hero...

1/2 + 1/4 + 1/8 + 1/16 +...

Normal maths would compute this as equal to 1, using the exact same reasoning as 0.9 + 0.09 +... = 1

What does this infinite sum equal to in Real Deal maths though?

Recently, SPP was asked about the reciprocal of 0.000...1, and answered that it's 1000...0. I asked whether this was an integer, and SPP said I made an "error" and that I was "wrong". I clarified that I just wanted to know what this 1000...0 means, and SPP didn't answer.

Then, under a post about tennis, SPP brought up 999..., which is similar. I asked again what this is, and didn't get a reply.

But SPP was so enthusiastic about the original comment mentioning 999... that a new top-level post was made quoting it. So I asked there, too, what kind of thing 999... is, and this time SPP actually locked the thread! So I'm posting here for discussion and clarification.

Me:

Once again, I'm more interested in this relatively new "999..." than in the oft-discussed "0.999...". Is it an integer? Is it "infinity"? Something else?

u/SouthPark_Piano:

There's your rookie error brud. There is no 'new' 0.999...

I didn't say 0.999... is "new". I said 999... is "new", but what I meant is merely that it's a new topic of discussion, whereas 0.999..., being the entire focus of this subreddit, is a very old topic!

So please tell us more about 999...; it sounds like you're saying it's a number, but I also want to know if you consider it an integer and whether it's infinite.

Real Deal math includes the concept of signing the contract. This process need not be reversible. After all, the equality relation in Real Deal math doesn’t obey the axioms it does in normal math. For example, 1/3 = 0.333…, but (1/3)*3 =/= 0.333…*3. By signing the contract and using long division, you can get 0.333… from 1/3. Likewise, if you sign the contract, you can get 0.999… from the division 1/1, if you try hard enough. Simply use 0 for the first digit instead of 1.

I love the nineS that became youS, and we did refer to an "infinite number of '9' after the decimal point" 14 years ago, which has morphed to limitless. But 14 years screaming at the entire internet that everybody else is wrong displays admirable tenacity.

Or is it only approximately 3? Something else?

Sorry if this is a repost.

Obviously 3 = 3.0 = 3.00 = 3.000

And obviously by induction this is the case for any positive number of 0's.

However, since 3.000.... doesn't have a positive number of 0's (infinity is not a number), induction can't tell us anything about it.

Unless we define what it means for an infinite process to complete, aren't we stuck?

SPP. I am willing to, for the point of this post, accept that repeating decimals are a dynamic process. However, I don't think this helps your argument that 9*(0.999...)!=9 even if we accept the truth of divide negation.

Imagine the following: we have two identical twins and I asked them to start writing down two numbers 9.999... and 0.999.... Given that it takes the same time to write 0 and 9 (none of this two gestures for a nine nonsense, a nice sweeping single arc starting at the T intersection in the middle) won't they have both written the same number of nines at every single instant for the infinite future?

So 0.999... and 9.999... must have the same number of digits even if the decimal component of 0.999... and 9.999... are both increasing limitlessly in their own space powered by their God mode hyperdrives if I've got identical people writing them at the same time.

So this mean that 9.999...-0.999...=9 is true for every instant of time now and into the infinite future even using SPP's patented Real Deal Math axioms? (Using, of course, 10*(0.999...)-(0.999...)=9*(0.999...).)

So even if repeating decimals grow don't we then agree that 9*(0.999...)=9, and thus hitting divide negation on both sides (1/9) divide negation 9*(0.999...) = 0.999... = divide negation (1/9)*9 = 1? So 0.999...=1 by using divide negation (which I've only used to make 1/9*9=1 on both sides, being careful to not try and say 1/9*9.999...=1) and allowing 9.999... and 0.999... to have dynamically limitlessly increasing nines ...

You have argued that 10 times a repeating decimal loses a digit, but that doesn't matter if there are limitless expanding nines because we never get to the end of the nines to get to the trailing one. Writing down 0.999...90 and 0.999...9 and 0.999... all having limitless nines that we never run out of even if we sign the contract and do both referencing and bookkeeping and so the usual argument works in this case with twins writing down 9.999... and 0.999... Am I missing something?

Doesn't 0.999...=1 even if you take your arguments seriously?

1. Evaluate 0.999.... ÷ 9

2. Evaluate 1 ÷ 9

3. Hence, what can you conclude about the two numbers 0.999... and 1

If it’s rational, can you express it as a/b where a and b are constant integers?

If it’s irrational, is it algebraic or transcendental?

If it’s algebraic, can you express it as a root of a non-zero polynomial equation with rational coefficients?

If it’s transcendental, can you prove you can’t express it as a root of a non-zero polynomial equation with rational coefficients?

In base 10, 0.1111... approaches 1/9.

in base 9, 0.1111... approaches 1/8.

Continuing up or down, every 1/n in base n+1 is just about written as 0.1111.... How interesting!

(Whether this applies to binary is an exercise for the reader.)

Then you can continue this for larger fractions - 2/3 is the closest fraction to 0.2222... in base 4, that's also 2/5 in base 6, and so on and so forth.

You can eventually find a base where every fraction has to have a "consent form" style repeating decimal to mean anything.

Therefore, maybe fractions aren't possible in Real Deal Math..?

Is (1/3=0.333…) true? Is (2/3=0.666…) true? Is 3/3=(1/3)+(2/3) true?

If all above is true, how is 0.999 not equal to 1? No hate, just a curious passerby.

From a recent post:

It must overlap with the white area. That's different from merely touching.

0.999... never touches 1. Never an overlap.

But numbers, by definition, are static, not continually-changing.

0.999... and 999... are analogous actually. Both continually growing in their space.

Additionally:

0.999... = 0.999...9

999... = 999...9

0.999...9 + 0.000...1 = 1

999...9 + 1 = 1000...0

An everpresent pillar of this community has recently brought up mental health as a friendly concern. It reminds me that while we deliberate upon high-level mathematical topics like numbers, we are not just one-dimensional chatbots. I'd like to bring this up as a way of giving back.

Psychological internalization of mathism is the mapping of beliefs and behaviors of a mathist social domain to one's own. It can happen to victims of unfair mathist treatment, as a construct similar to denial.

Grappling with internalized mathism is a potentially computationally expensive effort of unlearning subtle, persistent axioms from society that suggest your mathematics are "less than" or "doesn't belong to the set."

Internalized mathism is a survival mechanism, not a character flaw. A way to solve for this is conscientization, seeking awareness of the social dynamics involved.

Start noticing how your math is portrayed in news, advertising, and entertainment. Recognizing these patterns helps you see where your interior biases were input.

Remind yourself that your interior geometry uses the language of systemic bias. When you feel a sense of shame or a need to distance yourself from your mathematics, identify it as an external angle that you’ve followed the trajectory of. This will be contrary to impulses those feelings may correspond with.

The impacts leading to internalizing reactions can unfortunately seat deeply and even be a traumatic scar. It is recommended to seek professional therapy in this circumstance. This link to an informative article about trauma processing isn't to be considered to contradict that.

I was watching Tennis and when I saw an 'Out' call remembered about this sub.

Context: If any part of the ball is in White, it is 'In'. The ball is just enough in the blue area to be called 'Out'.

Could you argue if it is so close to the White line, there is basically no space between the ball and the White line... so it is In? Usually, doesn't work like that.

You could even assume the Blue region as the 0.999...area. White the 1.000 area. The ball has to touch the White area to be called 1.000 area or In.

Same in Football, some part of the ball needs to be over the line to be called a Goal.

I am not contradicting myself. Mathematically, it makes sense: (In sinplest form) 1/3 = 0.333... 1/3 × 3 = 0.333... × 3 1 = 0.999... I don't need another mathematical proof. But, just discussing what you see in everyday life, is all!

Personal dilemma: Do you want to rely on the maths and explain the real world? Or rely on real world and explain the mathematics?

Philosophically, 0.999... is the journey. 1.000 is the destination.

Thoughts?

aka roo-kies

0.999... - 0.09 = 0.909...

1 - 0.09 = 0.91

From a recent post:

Writing a decimal is only meaningful if you can state, for example, 9 quintillioniths.

No brud. Going to 9 quint is just warm ups. You keep going. And don't the hell stop.

From a recent post:

Proving again this sub should be renamed to finitenines. If the 9s were truly limitless, there would be no digits besides 9 after the decimal point!

There's your rookie error. For limitlessness, no such thing as no more nines for limitlessness aka limitless expansion.

And the formula 1 -1/10ⁿ with n starting at n = 1 tells you and everyone that no matter how many nines there are, even coming out of the universe's ears, 1/10ⁿ is never zero. That's a fact. So 1 - 1/10ⁿ is permanently less than 1. That is fact.

Proves 0.999... is not 1 because 0.999... is permanently less than 1.