However in pure maths... the matter is not quite so simple.

For starters if you have an irrational number like the square root of 2, then if it is rounded off then

it strictly speaking is being given as a rational numbered approximation.

By the same principle is 0.9 recurring the same as one to any finite conventionally rounded approximation,

but if you could have, in theory, literally every 9 listed, then it is not quite the same in pure maths. (???)

I have seen a theorem in a Foundations of Maths book written by a Mathematics professor (Warwick I think)

which states with proof that between any two distinct rationals there is an irrational number,

and between any two distinct irrational numbers there exists a rational number, but they

should not be thought of as alternating along the number line.

The number 0.9 recurring is a rational number because it has a repeating sequence.

So therefore we have to think of it as exactly one, otherwise it has to have a number inbetween it and one.

It certainly converges to one as the number of digits tends to infinity....

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0.999... is only a theoretical number.

So you think some numbers are only theoretical. So what are the non-theoretical ones?

Bob

]]>.99999... is just another way to write 1

]]>I said this way back in post 2.

Try 0.99999

Bob

]]>Well I used sound, thought-out search criteria. I found 9 (or 8.99999... if you prefer)

I'm talking about threads not posts. The latter may well be un-countable.

Bob

Which search criteria was it?

]]>I'm talking about threads not posts. The latter may well be un-countable.

Bob

]]>hi ke

Welcome to the forum.

Your new thread brings the number on this topic to 9. Search on 0.99999 if you want to explore the many comments others have made.

Bob

I would say 99.999... is the number.

]]>Those extra zeros make it much more accurate!

Bob

]]>I have had a change of heart and no longer think .999999... = 1. I think

.999999... =1.00000000...

]]>Your new thread brings the number on this topic to 9.

Just as long as it is not 8.9999999... That would be intolerable.

ke wrote:That is only true for a finite series.

I like to think of it like this

.99999999... - .9999...

.9999... = .99999999... ( I just took a few measly nines out of those 3 dots) I call it 9 extraction from dots. After all there are gazillions of them in there and they will not be missed. Now it is easy to see

.99999999... - .99999999... = 0. You just have to remember ... - ... = 0. I call it m's law of the dots.

Using this great idea the proof you gave by multiplying by 10 works fine.

True.

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