Is 1 = 0.999... ? Really?

What is there to explain?

If you are presented with the fraction 1/3 , then it takes some time for you to do the division. It might take you 10 minutes, it might take you 5 seconds to realize that 1/3=0.333… (Or you might never realize it.)

It takes you time because time applies to you. Mathematics is not a “being” who has to “do” something. It doesn’t “need” to “get to the end”. It doesn’t need a process to “do” the calculation. Mathematically 1/3=0.333… - that’s it.

Time is just not applicable to some stuff.

Phyllo,

Even if 1/3 is equal to 0.333… that means that 0.333… *3 is equal to 0.999…, which looks ALOT different than 1!

If you say they are equal, you’re making the claim that EVERY counting number is equal to an infinity of infinities (contradiction)

If you say they’re not equal, then you are drawing a line which states “counting numbers are finite” (which is correct)

1/3=0.333…
3*(1/3)=1
3*(0.333…)=0.999…
Therefore 1=0.999…

If that’s not true, then simple division and multiplication don’t work.

You can even take out the multiplication: 1/3+1/3+1/3= 0.333… + 0.333… + 0.333… = 0.999… = 1

It’s not rocket science.

Phyllo, you don’t understand what I’m saying when I say the implication is that all finite numbers are infinite (in fact you avoided it)

Let’s say we hypothetically live in a world where fractions don’t exist… it would be unfathomable that 0.999… = 1.

In a decimal world. 0.111… * 9 equaling 1 is impossible.

The problem is not with my logic, the problem is how operators work with 1 minus base, to make them ‘appear’ equal… but then again, they don’t appear equal at all do they !!!

There seems to be a lot of confusion here about what “infinite” means. “All finite numbers are infinite” doesn’t make any sense as a statement.

That’s simple enough. All you need to do is to restrict yourself to whole numbers, natural numbers or integers.
There is no integer which represents 1/3 or 1/2 or 8/9. When evaluated, those fractions are equal to :
1/3=0 in integer
1/2=0 or 1/2=1 in integer
8/9=0 or 8/9=1 in integer

The two values given for 1/2 and 8/9 depend on whether truncation or rounding(up/down) is the standard procedure when evaluating the results.

A similar thing happens with 0.999… when using real numbers. It “jumps” up to 1.

Yes but in as far as it pertains to quantity, its pertaining makes quantity indefinite.

We agree on that.

Thanks gib.

Yes and I have noticed during the years that the main problem when discussion infinity is its indefinite-ness.
This is the hot core all these debates centre around and it not being made explicit perpetuates the confusion.

How are you not going to be confused if you want definite results out of something that is by definition indefinite?

That seems legit.

Phyllo,

I’ll press you on this for now.

Phyllo wrote: “all finite numbers are infinite doesn’t make any sense as a statement”

EXACTLY!!!

That’s my whole point. It makes no sense!

It makes no sense that 0.999… EQUALS 1!!!

In this formulation, 1 by definition is an INFINITE number!! By equality!!!

Who the heck knows what you mean by “INFINITE number”. I certainly don’t.

Infinite digits doesn’t mean infinite number.

Doesn’t matter how you word it… you’re like a squirrel running from slingshots right now. You’ll dodge for a while, but, more likely than not, one will eventually connect! To make this rated g, the pebble never hurts the squirrel

You’re still making the claim that 1 is …

EQUAL!!!

To “infinite digits”

EQUAL!!!

Sure. It looks counter-intuitive but the math equations show that it must be true.

What’s wrong with the argument using 1/3 fractions? Nothing. Unless you want to argue that dividing 1 by 3 doesn’t work.

I am not saying “This is what infinity is”. I am saying “If this is what infinity means, then this is what follows”. My point being that if (\infty) refers to a specific number that is greater than every integer, then (\infty - \infty = 0) is true but (\infty + 1 = \infty) is not; and if it refers to a non-specific number that is greater than every integer then (\infty + 1 = \infty) is true but (\infty - \infty = 0) is not.

Yes, but in such a case, (\infty + 1 = \infty) is not true.

$$ e^{i \pi}=-1$$

Go figure :open_mouth:

Is 0.999…=1 so strange in comparison?

If (\infty) refers to a non-specific number greater than every integer (in the same way that “A number greater than (3)” refers to a non-specific number since it can be (4), (5), (100) or (1000)), then (\infty + 1) equals to (\infty) (because “A number greater than every integer + 1 = a number greater than every integer”) but (\infty - \infty) is indeterminate.

Could you make up your mind what it means instead of using multiple meanings?

Another question:

In regards to 1/3 necessarily having no way of being expressed as a finite sum of rationals, that depends on the base of the number system one is using. Suppose our number system was base 3. That means our number line would look like this:

0 1 2 10 11 12 20 21 22 100 101 …

Here’s what counting from 0 to 1 would look like if went by increments of two decimal places:

0.00 0.01 0.02 0.10 0.11 0.12 0.20 0.21 0.22 1.00

Note the part in bold. This is exactly one third the way to 1.00. Note that it does not require an infinite decimal expansion. It’s just:

0.1

The quantity hasn’t changed. It still represents a third of whatever. The only thing that’s changed is the notation. We use a different notation to represent one third because we are using a different base.

^ This shows, I would hope, that the problem of an infinite decimal expansion, and therefore the problem of an infinite sum of rationals, is a superficial problem having to do only with the notation system in use. Use a different notation system and the problem goes away. It’s not a problem with the quantity itself. That remains the same regardless of the notation system being used.

So a third is not somehow outside the categories of “finite” and “infinite” (what I dubbed “semi-finite” on Magnus’s behalf), and it is not an irrational number, and it is not something that can never be complete, never quite attain it’s limit, it’s just, well, a third.

Now unfortunately, it gets a bit more complicated in the case of (0.\dot9), and I’m not going to tackle that in this post, but I hope the above sheds some light on the difference between the notation and what the notation represents, and how sometimes the problem is only a problem for notation and not for the quantity the notation represents.

Not really. The notion of time is completely irrelevant to this thread.

Phyllo, did it ever occur to you that people’s intuition that finite numbers are not “infinite digits” (the most blatant contradiction in this whole discussion) means that there’s something wrong with how humans conceive of and understand math?

I mean, if you can really stand there and say “well it makes NO sense but it’s true”

What’s to stop me from saying “well it makes no sense but it’s NOT true”?

I mean, if you’re going with a contradiction, be prepared to have all the nonsense in the world thrown at you as well.

Oh sure… Magnus is a 23105th dimensional being

Magnus, in this thread, I wonder if you even begin to hear yourself talk sometimes.

Then find the flaws in the math instead of proposing some bizarre definitions of your own. Start by writing down the definition of “INFINITE number”.

My own solution to the 0.999…=1 question does not depend on infinities at all. I didn’t use an infinite series. I didn’t use infinite sets. I used strictly multiplication and division and alternatively addition and division.

You can say anything that you like.

That has already happened in this thread.

Phyllo, you’re being disingenuous.

You know that when you divide 1 into decimals by 3 or 9 and then multiply the decimal by 3 or 9 that it NEVER equals 1!!

I know you know this.