Where is the line (mathematically) between a math progression and infinity?

I am not tutored in math well enough to know if this will even make sense. Hopefully people here will get my drift…

So, we can have a progression such as:

1+1=2

1+1+1=3

1+1+1+1=4

And so on.

But my understanding is things like 0.999…999 (infinite series) actually equals 1.

To my mind it is really, really close to “1” but not quite “1”. But mathematicians assure me it is actually the same as “1” and I won’t argue it.

So, where is the line drawn? How many 0.999… do you have to go before you say it is the same as “1”.

I guess what I am asking is when do you stop saying I am doing “+1 to anything you say” yet still be a finite number and say it is “infinity”?

Were you aware that this ranks with airplane on treadmill and Monte Hall for generating internet chaos? It’s such a well known issue that there’s a good Wiki article on it, I recommend that as a good starting point.

https://en.wikipedia.org/wiki/0.999

In brief, yes it does equal 1. But any proof involving mathematical infinities requires care and rigor, you can’t rely on ad hoc methods or “common sense”.

Thanks, I am working my way through it but does it answer where a mathematician says we will stop calling this a series of “+1” and call it the very undefined and no longer an actual number of infinity? (I see the skepticism section but I am still not sure of the answer.)

Missed the edit window for my last post.

I am asking where is the tipping point?

0.9 ≠ 1 (obviously)
0.99 ≠ 1 (obviously)
0.999 ≠ 1 (obviously)

Where is the line drawn? An infinite series = 1 but infinite is not a number so how many 0.999…do you need?

I guess what I am really asking is how to mathematicians decide to when/where to draw that line?

I am not a mathematician and I didn’t initially find the premise of this very plausible myself but I did learn a good way to prove it to almost anyone.

What does 1 divided by 3 equal in decimal terms? Use a calculator or do it by hand the long way if you like.

Now do the opposite and multiply that result by 3. Use a calculator or do it by hand the long way if you like. You just performed two opposite operations that should completely offset one another.

What do you get? Hint - that is not a design failure that means your calculator is limited somehow. They really are the same.

Well, never. That’s what infinity means, loosely - the 9s can’t ever end, or the expression does not equal one. That’s where things start getting subtle and may seem contradictory. You can give the expression arbitrarily many 9’s and it’s still not quite equal to one. But there’s a difference between “arbitrarily many” and “infinitely many”.

“Infinity” isn’t really something that corresponds to anything in the everyday world, so common sense intuition isn’t necessarily helpful. If you’re going to deal with infinities, you need to do the math rigorously and trust it. Eventually you may gain “mathematical intuition”, but that’s not the same as “common sense”. I honestly don’t think that you well get much useful insight into anything much here unless you’re interested in doing math.

You are mistaken then. There is no line or judgement for that specific question. The answer is all of them forever. If they ever end, they are not equal to 1.

This won’t get you anywhere. You’re trying to do math without actually doing math.

If you want valid proofs, with varying degrees of rigor, they are all in the Wiki article, and imo there is no worthwhile shortcut. You must learn how to do the math to understand the proofs, or you will never get to grips with mathematical infinities.

There are some scientific and mathematical ideas where it’s possible to give a good non-tehnical account that gives the layman good insight. This really isn’t one of them.

In short, any number series (or anything else) that ends is finite no matter how long it is or how big it is (15 quadrillion digits is still finite if it terminates there). Finite is the opposite of infinite so there is never any judgement at all by anyone when using the term infinite. It literally has to go on forever for the definition to apply.

The are many infinite numbers in the real world. 1/3 is one and the reason that controversial question exists. Pi is another one. There is no end and that has profound and counter-intuitive implications.

I agree with the sentiment but not for this particular question. Break out a pocket calculator. Divide 1 by 3. Stare at the result. You can see that it is 0.3333… and anyone that is decent at basic arithmetic can see why. Now multiply by 3 to try to put it back to 1. It is obvious before you even do it what the result will be before you even hit the equals key. That may not lead to a deep understanding that proofs do but it is fairly intuitive at the level that most people want when they ask this question.

I know what you mean, of course - ratios arise quite naturally for which there is no finite decimal expression. But we don’t have real world examples of “infinitely many things”, so it’s not a concept for which we have useful common sense intuition.

This question stemmed from a discussion on another message board where we were discussing getting arbitrarily close to the speed of light.

In theory you can never actually achieve light speed (if you have mass) but there is no limit to how close you can get.

It seems a manifestation of getting infinitely close but never quite reaching infinity.

To me that seems a practical application. We can’t go light speed (infinite) but we can get ever closer without limit yet still be finite.

That is what I am trying to get a grip on.

Honestly, I’m not trying to be difficult, but I can’t see what insight that you think this gives? Could you explain? There’s nothing self-evident here to me.

Honest question…

Why is that wrong?

0.9 ≠ 1
0.99 ≠ 1
0.999 ≠ 1

Where is the line drawn that it does equal one?

Why as a student of math is this a "bad"question to ask?

It is just the simplest way I know of to demonstrate this to non-mathematicians. They aren’t going to study proofs or listen to lengthy mathematical theory. However, almost everyone can do basic arithmetic.

It is dead simple:

  1. Take 1
  2. Divide it by 3
  3. Multiply it by 3 to reverse the step above

Whether you do it by hand, on a calculator or a computer, your final answer will be 0.9999… and it is easy for most people to see why. They cover repeating decimals in lower levels of school. Most people don’t understand infinities well but they can understand that simple demonstration if you express it in terms they are familiar with.

Yes, it’s called an asymptote.

It’s sounds like you already pretty much have the idea, except that light speed is not infinite, it is finite. There are no actual real world infinite quantities involved. The math just says that you would require infinite energy to reach the speed of light - so you can’t do that.

1 is the limit of that series as it tends to infinity.

Certain series have the property that successive terms become smaller and smaller, in such a way that the sum of an infinite amount of terms equals a finite number. Nothing short of an infinite number of terms gets you exactly at the limit, but you can get arbitrarily close. The ellipses (…) is meant to indicate that 1 is the limiting value. In a similar way,

1 + 1/2 + 1/4 + 1/8 + 1/16 + … = 2.

With all due, respect, two people including myself have taken the time to respond to this question already upthread. You need to go back and reread it if you don’t understand it because it is a simple but profound concept. In short, numbers (or anything else) that end, no matter how far out, aren’t infinite. There is no line and nobody decides it. It is either infinite or not.

Honestly, I still don’t get it. What do you think this shows?

I don’t think it’s a “bad” question. It’s just that the answer is “go learn the math that’s in the Wikipedia article”. I think you’re looking for a shortcut to doing math without doing math, and there isn’t one.