10c - c

Interesting challenge. I'll have to chew on it.

 

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Actually, I went awry on step 1. I'm making an intuitionist argument, and Brouwer's intuitionism rejected that you can define something by defining what it can't be or can't not be. He rejects excluded middle arguments. You need to define an object affirmatively. I'm not saying your points aren't valid. They're great. I just need to work on how to confront them.

 

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Repeating 0.9 equals to 1.

 

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My initial response is that I'm not trying to prove anything about "the largest number less than 1". I was only trying to illustrate that "there is no number you can't keep adding to" is no different than "there is no number of iterations such that you can't do one more", to demonstrate that you can't really operate long division to the point where there is no remainder.

I'm trying to argue that a decimal point followed by an actual infinite number of 9's isn't real. It's not really an argument about the largest number less than 1. It's that there is no actual infinity that you can incorporate into a real number. By itself, a decimal point followed by an infinite number of 9's is not a real number. As the product of a mathematical operation, an operation performed an infinite number of times isn't real. And as the finishing member in an infinite set it isn't real. That the largest real number less that 1 is .9r is a poor argument that I shouldn't have made.

 

Decimal expansions are just a convenient (ish) way of defining numbers. An infinite decimal expansion is defined as the limit of a sequence of partial sums - and the limit is well defined.

 

x = 0.9r
10x = 9.9r

no assumptions made that 0.9r = 1 yet

10x - x = 9x
9.9r - 0.9r = 9

still no assumptions

9.9r - 0.9r = 10x - x = 9
9x = 9
x = 1 = 0.9r

Nowhere do we have to assume 0.9r = 1 to prove that it does.
Which part do you think otherwise about?

 

My argument is .9r isn't a real number.

10(infinity) - (infinity) = 9(infinity) isn't a real expression. In a similar way, you can't plug in .9r and think it's giving you a real result.

 

Sorry, I don't know what you're on about. Please explain what is wrong with the proof I posted.

 

I think in this type of situation it's best to be blunt and to the point:

0.9r is the limit of a Cauchy sequence in R, and is therefore also in R.

 

Is 10(infinity) - (infinity) = 9(infinity)?

10(infinity) - (infinity) might equal zero, instead.

But, really it's just nonsense. It makes no sense to ponder 10(infinity) distinct from infinity.

I'm saying if .9r isn't a real number, then the 10c - c = 9c operation isn't guaranteed to work.

 

Noone claiming that.

0.9r is obviously between 2 and 0. And, as Guest points out, since it's the sum of real numbers, which is between 0 and 2, it's a real number.

Remember, 'real' doesn't mean 'it's valid', it's got a specific mathematical meaning. That which Guest gave.

 

9.9r = 9.9999999999999...
0.9r = 0.9999999999999...

Is it so hard to see that the first minus the latter = 9?

Where has this 10(infinity) nonsense come from?
Recurring decimals are not infinitely large numbers. It is easy to see that 0.9r is less than 2, and easy to see that 2 is less than infinity.

 

Yeah, I'll have to get my head around the intuitionist argument on continuity. My post count is too low to post links, but I'm going to read up on the google results for "cauchy intuitionist" later.

 

1-0.9r
What is the answer BrJLa?

 

To be fair, it matters not one jot what argument you are hoping for! The construction of the real numbers exists without a person's preference for intuitive mathematics.

 

No, it's not obvious. It's not obvious it even exists. It can't be constructed without infinity. It's either got to be a fraction, expressed as a decimal to an actual infinity number of places, then multiplied by its reciprocal. Or it's got to be the sum of a sequence of numbers that includes as its actual last member "... + 9/10^(infinity)", or without some other recourse to an actual infinity.

 

As inexpressable as infinity minus 1.

 

More precisely, as meaningless as 1/infinity.

 

Why are you bringing infinity into the argument again? We are speaking of recurring decimals, not infinitely large numbers.

The answer is 0

Most of the time when I ask someone that they will answer 0.0r1
makes sense right?
The thing is the 1 does not exist as the 0s go on forever.

Any number - itself = 0
1 = 0.9r.

Also, what is you answer to the above?

 

Please don't tell people what does and doesn't exist within the realm of the real numbers until you've taken the time out to understand their construction. I realise you find the result you're speaking of odd, but don't presume that this feeling of unwarinesses puts you in a position to tell others what is correct or not.

Some might construe such actions as presumptuous, naive and a little bit arrogant.