Is 1 = 0.999... ? Really?

As I was trying to explain to Magnus, there’s two ways to look at this:

1.) completed infinity (on this you are 100% correct)

2.) process towards infinity (on this I am 100% correct)

So… I posit this to you:

Does endlessness ever become complete?!?!

I think you’ll like that train of thought!

There is no concept of time in mathematics.

Ideas like process and completion don’t make any sense.

“T” (time) is a variable in all branches of mathematics.

I know you’re trying to sound profound —- thing is, without time we couldn’t possibly have or understand this discussion on ANY level!

That’s really careless. Dismissing the definition of “quantity”? Really? Well then, I guess we can put in whatever we want for n. How 'bout “cow”? Is “cow” > 0? If so, then I can prove that (\sum_{i=1}^{cow} \frac{9}{10^i} < 1). But I think even you can appreciate that there is a limit to what we can substitute for n.

I’m going to demonstrate how you’re confusing intuition for logic. Your argument is:

  1. (\sum_{i=1}^{n} \frac{9}{10^i} < 1) for every (n > 0)
  2. (\infty) > 0
  3. Therefore (\sum_{i=1}^{\infty} \frac{9}{10^i} < 1)

^ Seems logical, right? But you’re missing a crucial step between 2) and 3): (\infty) is a valid value for n. I know you feel intuitively that (\infty) must be a valid value for n because (\infty) is a quantity, so no need to prove it. But when you’re arguing with someone who disagrees with you on that, you do need to prove it. You can’t just run on intuition. If I argued that “cow” > 0, and therefore the formula applies to cows as well, you would insist I prove that “cow” is a valid value for n, wouldn’t you? Same onus falls on you to prove that (\infty) is a valid value for n. But so far, all I’ve seen from you is re-assertion after re-assertion that (\infty) is a number–no proof–which tells me you believe it on intuition, not logic. You need to prove this just as much as you need to prove that what applies to finite sets also applies to infinite sets (and I wonder if this is just a special case of the same thing). But seeing as how you refuse to prove your point when I ask you too, I’m guessing you’ll cower away from this one too.

Well, it applies to any real number greater than 0. But what that means is: grab any number on the number line. (\infty) is not on the number line. It’s a direction in which the number line extends.

That’s a mathematical representation of time when modeling some physical situation. There is no time within mathematics itself. Nothing within mathematics requires time to complete.

I can have a variable “U”(unicorns) but it does not mean that unicorns are a part of mathematics.

Humans exist within time and therefore need time to understand things.

The explanation I gave earlier in the thread for any “concept of time in mathematics” was that there’s a difference between quantity and representation of quantity. The former is “already there” (for infinite quantities, it’s undefined where exactly “already there” is), but representations of quantities as well as representations of how to construct/deconstruct quantities have ordered steps that occur before or after other ordered steps - according to that order and therefore implying temporality.

When I say endlessness in reference to quantities, I’m referring to the quantitative boundaries (ends) that specfically define a quantity as distinct from higher or lower quantities. Not having this, as with an undefined/indefinite/infinite, means it has no ends. Mathematical construction of such endlessness would take an endless amount of time to complete, just as its deconstruction to a definite quantity would never happen. The best you can do is imply endlessness by using a symbol/notation that only looks like it’s “bounding” boundlessness, which has to be treated very carefully and separately from defined, bounded finites, which have specific ends. Otherwise you can be fooled into thinking there’s more than one kind infinity, and/or that each one can have a different size, when in fact it’s the finite constraints around infinity only that affect anything to do with size: there’s only one way in which endlessness can be endless.

Nothing in mathematics requires time to complete ?!?!?!?!?!

Really!?!?!?!? Please explain!

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.