Is 1 = 0.999... ? Really?

You’re not making any sense.

I agree that (1/9)*9 = 1.

I agree that 0.9… = 3*(1/3) & 3*0.3…

I agree that 10-3 = 7

I agree that 1.0-0.3 = 0.7

I agree that 1.00-0.33 = 0.77

I don’t see the problem.

But I guess if calculators were programmed by nice guys, we’d get different results.

How the hell did I do that! I started with 10-3 because because the 1 carries over to a ten when you do 3/1… then I quickly typed 1-0.3 which was 0.7, then I extrapolated infinite 7’s, then I typed 1-1/3 and got 6 repeating. Had I typed 1-.33 I’d have seen what you just corrected.

I’m not seeing where that train of thought is going now … I just assumed a factor of 10 for each place would always be 10-3=7 right down the line…


Silly, thx for pointing that out!

It also was occurring to me that 1/9 is 0.1…

As it expands, the ones keep getting smaller and smaller until at this hypothetical convergence it becomes zero …

And if you have 9 of these going at the same time, all nine of them converge at zero (like if you plotted the regress on a graph with a zero axis).

This means that 9/9 = 0

Hmm… strange.

Back to the mistake you pointed out phyllo…

Read above post as well!

That creates a last digit if you reverse engineer it, with an infinity in front of it… which causes pretty serious issues through dimensional flooding (infinite 6’s before a 7 + infinite 3’s before a 4)

I take it that your point is that if you go “in reverse” and you subtract 1- 0.333 all the digits are 6s except the ‘last’ digit which is a 7. But is you “go forward” and divide 2/3 all the digits are 6s. So it appears that you can see the missing 0.000…0001 which is the difference between 0.999… and 1.

It’s remotely possible in the case of 0.9…

That would be intense mental gymnastics which strike me initially as resulting perhaps in gibberish…

The biggest problem I see with reverse engineering is when you add multiple reverse engineered numbers together; which causes a solution greater than 1.

The problem I see as a whole, is that in the easier to understand instance of 1/9, you have a situation where the regress approaches zero… which under convergence hypothesis means that 1/9*9=0 rather than 1!

This topic has been done to death, but ah, what the hell *shrugs.

I would say 0.999> is as close to being 1 as you can possibly be, without being 1.
Why?
Simply because they aren’t exactly the same, only 1 can be 1, only x can be x.
Maybe 0.999> is so close to being 1 that for all intensive purposes it might as well be 1, but strictly speaking it’s not.

Maybe it would help to visualize what 0.999> would even look like in nature?
Could 0.999> even exist in nature?
Can matter and space be infinitely small/big?
Perhaps we will never know.

Assuming they can, what would a domino that’s 0.999> centimeters in height look like, and is it equivalent to a domino that’s 1 cm in height?
Place the two dominos on a perfectly flat, perfectly level table.
You should be able to wave your hand 1 cm above and over the table without touching the 0.999> cm domino and knocking it over, but you shouldn’t be able to wave your hand 1 cm above and over the table without touching the 1 cm domino and knocking it over.

That would be the real world empirical, tangible difference between something that’s 1 cm, versus something that’s anything less, whether it’s 0.1 cm less, or 0.01 cm less, or even 0.000> cm less than 1.
However, if your hand is any closer to the table than 1 cm away from it, even just the teeniest, tiniest bit, you will touch both the 1 cm domino, and the 0.99> cm domino, as you wave your hand over the table, knocking them both over, because 0.999> is as close to being 1 as you possibly can come without being 1, so anything closer than 1 cm away, even if it’s only 0.0000000001 cm closer less is going to be within its range to interact with it.

…Or maybe at that distance your hand will pass over both of them, without touching them, but it should be possible to place your hand at a distance from the table, so the bottom of it occupies the same space as the top of the 1 cm domino, but not the 0.999> domino, so you’ll be able to knock the former over at that distance, but not the latter.

Suppose you’re looking at a 0.999~cm high domino from a certain distance. You measure it and you get a result like 0.9cm. Of course, this is not true, but that’s what you can see from that distance; you can’t see the remainder, it’s too small. You decide to come closer, just to be sure, and now you measure the domino to be 0.99cm high. You keep coming closer, and each time you come closer, you get a result that is closer to, but not quite equal to, 1.00cm. Now, if you kept doing this for eternity (assuming you had an eternal life to live) and got a sequence of measurements that goes something like 0.9cm, 0.99cm, 0.999cm, 0.9999cm and so on, that would be a domino that is 0.999~cm high. If, on the other hand, you got a sequence of measurements such as 0.99cm, 0.999cm, 1.000cm, 1.000cm, 1.000cm and so on (basically stablizing at 1.00cm high at some point) that would be a domino that is 1.00cm high. Clearly, they are not the same height.

There are objects with a height that is not equal to a finite number of equally-sized units (such as centimeters) which means there is no combination of an integer and a unit that can represent them exactly. Finite decimals are more expressive but they have their own limitations. Most importantly, they can only represent finite quantities. Hence, infinite decimals.

Infinity might be a difficult concept to grasp for some. Mostly, it’s the kind of infinity that Aristotle called “actual (or completed) infinity” that they have problem with. It’s difficult to understand that height, width, depth, length and distance can be infinite.

The gap of a trillion to the trillionth power surprise You?

Regarding the bolded part, one must ask: is that true? Let’s not be bound by our number system. There are number systems other than base-10 and these other number systems can represent numbers that our number system cannot, such as numbers that are larger than 0.999~ but smaller than 1. Consider base-16 number system a.k.a. hexadecimal number system. In hexadecimal number system, 0.999~ is represented the same way, as 0.999~. But since we’re dealing with hexadecimals, which work with more than 10 digits, there are numbers higher than it but lower than 1 e.g. 0.AAA~, 0.BBB~, 0.CCC~ and so on. Of course, the basic idea of the quoted part of your post is correct, but I think it’s important to note that 0.999~ is not the largest number that is lower than 1. Rather, it is the largest number lower than 1 in decimal number system. This means it’s possible for your hand to be closer to the table than 1cm away from it and still not knock the shorter domino over.

It can, but the mathematical use of a recurring number, in industry, would lose its utility at a certain point.

I made a mistake above. 0.999~ is represented differently in base-16 systems. Nonetheless, I am sure the point remains.

…and the point is?

The point is that there are numbers larger than 0.999~ but smaller than 1. For example, 0.FFF~ (in hexadecimal system) is larger than 0.999~ but smaller than 1. In base-20 system, we have 0.JJJ~ which is larger than both hexadecimal 0.FFF~ and decimal 0.999~.

…and the solution would be?

Solution to what?

I thought you posed a dilemma of sorts?

This thread is only about base I0 Magnus and so talking about other bases or systems is not relevant here
Also all the relevant arguments have already been made which is why the thread stopped two years ago

Yes, but bases can be consolidated, no?

I don’t agree.

I don’t agree either. Other bases are fair game in mathematical debates. (Btw… Magnus is right)