Does 0.9[R] Truly Equal 1? Understanding the Mathematical Debate

[MusicTheorist]

I've been debating whether 0.9[R] does in fact equal 1. I've seen a lot of mathematicians saying it does and I believe that it is true that 0.9[R]=1, however it doesn't seem logical, as in, from my very low level of math I can seemingly find a way to disprove it. That would be this:

If 0.9 [R]=1 then that means it eventually converges to 1. 0.3[R] would eventually converge to a number. If 0.3 converges to a number it would be larger than 0.3 and then that would mean it is not equal to 1/3. 3 times that would be larger than 1.

Also, 0.9[R]=1.0. That's an equivalent decimal. If you try to write an equivalent number of 0.3 [R] the only response I've seen is 1/3, which isn't a decimal, and I can't think of any other way to write 0.3[R] as an equivalent decimal.

I believe 0.9 [R] = 1 however I see problems in the logic. Could someone please help me understand this?

 

[Hurkyl]

If 0.9 [R]=1 then that means it eventually converges to 1.

I assume [R] means that the last written digit is repeated infinitely?

Then no, that means it equals 1.

What you may be thinking of is the fact the sequence of truncations^* converges to 1... and you've confused 0.9[R] with this sequence.


*: i.e. the sequence 0, 9/10, 99/100, 999/1000, ... whose n-th term is 1 - 10^-n

If 0.3 converges to a number it would be larger than 0.3 and then that would mean it is not equal to 1/3. 3 times that would be larger than 1.

Um, 0.3 is 1/30 smaller than 1/3.

If you try to write an equivalent number of 0.3 [R] the only response I've seen is 1/3, which isn't a decimal,

So? Decimal notation isn't the only notation for real numbers.

and I can't think of any other way to write 0.3[R] as an equivalent decimal.

Very few real numbers (relatively speaking) have more than one decimal notation -- the only ones are those which have a decimal notation ending in repeated nines. (or equivalently ending in repeated zeroes)

 

[Mark44]

I assume that by [R] you mean repeating decimal pattern. 0.33333... (the ellipsis means repeating in the same pattern) is certainly larger than .3, but each number in the sequence .3, .33, .333, .3333., and so on for any finite number of 3's is smaller than 1/3, not larger than as you asserted.

As it happens, there is a thread in another section on nearly this same subject. Instead of elaborating here, I'll just give the link to that thread: https://www.physicsforums.com/showthread.php?t=352353.

 

[MusicTheorist]

But, theoretically, couldn't you add .0000...1 to 0.9...?

 

[Hurkyl]

But, theoretically, couldn't you add .0000...1 to 0.9...?

What do you mean by .0000...1? If what you meant makes sense, and you're using 0.9... to mean what you were writing as 0.9[R], then of course you could add them, and the result would be greater than 1.

 

[MusicTheorist]

What do you mean by .0000...1? If what you meant makes sense, then of course you could add them, and the result would be greater than 1.

Well, isn't there some number that would have an infinite amount of 0s followed by a 1?

Or maybe not, keep in mind math isn't my strong subject. I joined this forum to ask a question about a possible science project .

 

[Hurkyl]

Well, isn't there some number that would have an infinite amount of 0s followed by a 1?

Nope.

The positions in a decimal are indexed by the integers. Each position only has finitely many digits between it and the decimal point.

 

[Hurkyl]

As an addendum, one can invent other numeral systems with different index sets, which, e.g. would permit something I would write like this:

0.000... | 1​

where the bit to the left of the pipe (|) does have infinitely many zeroes.

However, other numeral systems can't be used to talk about the real^* numbers you have learned in school. Sometimes they are useful for talking about different number systems, though.

*: "real", here, is a technical term that has nothing to do with the English word

 

[MusicTheorist]

Hmmm, it still seems to have some flaw. I can agree that it's one, but I feel that it shows some type of flaw in the way we represent numbers. Why is it that only number with a repeating 9 can be written as a decimal two ways? That puzzles me.

Thanks for the help though!

 

[Hurkyl]

Why is it that only number with a repeating 9 can be written as a decimal two ways?

I don't believe there's any deep reason to it -- it's just what needs to happen to use decimals to represent real numbers.

Or from another perspective, the "algorithms" for decimal arithmetic strongly suggest it -- e.g. otherwise in special cases it would be ambiguous whether or not you need to carry when adding.


Sometimes people use different conventions for decimals so that every real number has a unique representation -- e.g. the convention that decimals ending in repeated nines are disallowed.


Every notational method has its benefits and disadvantages. We use the decimals not because there is some deep reason that they should be the true way to write real numbers -- we use them simply because for a wide variety of purposes, their benefits outweigh their drawbacks.

 

[Mark44]

And to continue with Hurkyl's idea, fractions in other number systems can have two representations as well. For example, in base-two representations, .11111111... is the same as 1.0000000... Lest you think this is a far-fetched example, computers use a similar kind of representation to represent floating point numbers, although they of course use a finite number of places.

 

[Mark44]

Well, isn't there some number that would have an infinite amount of 0s followed by a 1?

How would you indentify where the 1 is? You can't just say "Go out an infinite number of decimal places and there you'll find it.

You can represent very small numbers with a finite number of zeroes followed by a 1:
0.000-- 10 0's --1
0.000-- 1000 0's --1
and so on, but you can't have "infinity" 0's followed by 1.