No. I can put 1/9 into my calculator and obtain .1.. Give me the p/q I can put into a computation and obtain 0.9..
A calculator is not mathematics. It is a finite machine that can only handle X number of places. And it is programmed to spit out 1 for 1/1 instead of .9~ because .9~ is an obtuse way to say 1 and is never used in everyday conversation.
My point with the 1/3 example is that if you accept the convention that 1/3 = .33.. by identity, then this also should lead to accepting .99.. = 1 by identity. Not just that it's convention, so go with it. All math is a construct. Within the conventions adopted in that construct, .999 = 1 by identity. That isn't to say we can't define new constructs and new number theory.
Correct, hence the hyperreals, and other number systems. But they are not compatible with the Real number system.
And yet it's doing pretty good with pi. I want the p/q question that shows the 9s the same way 1/9 shows the 1s. This isn't a tough ask, if 0.9.. is actually rational, producing a never ending stream of 9s should be very easy.
No, I can accept that 1/3 is 0.3.. because I can do the 1/3 computation on paper and obtain as many 0.3s as I feel comfortable obtaining. I can use a computer to aid and go to 2.7 trillion decimal places if I so choose. And that for me is proof enough that 1/3 = 0.3.. Provide the same p/q construct, system, etc that I can do with 0.9.. using the rules of rational numbers.
let's start with TT's 1/3. .3333.... = x 3.3333... = 10x 3.3333... - .3333... = 10x -x 3 = 9x 3/9 = x 1/3 = x 1/3 = .3333.... .9999... = x 9.9999... = 10x 9.9999... - .9999... = 10x - x 9 = 9x 9/9 = x 1 /1 =x 1/1 = .9999... I do not think this will satisfy you as you want something that produces the .999... from p/q. And it is possible, just hard to write it out on this computer formatting that we have on this forum, but I will try. We will divide 1 by 1 using long division. .9999999999.... 1|1.00000000000000... 9 (9 x 1 gives us 9, then subtract 9 from 10) ____ 10 <- carry a zero down from above 9 ___ 10<- carry a zero down from above 9 ___ 10<- carry a zero down from above 9 __ ad infinitum Oh, but Alan, that is tricksy false mathematical mumbo-jumbo! Either you believe it or you don't, but it works, and is completely valid. So, if .9~ is NOT 1/1, what is it? I now shift the burden onto you as I have given my proof, now it is up to you. EDIT: Is there a way to keep it from removing white space?
Cannot do engineering without the mathematicians getting you all the nice tools you need to do it with, though.
I don't think 0.9~ is rational, and thus I cannot describe it as p/q. A single p/q. I can do it with 1/3, I can do it with 1/8. Show me any other rational number that I cannot express as a fraction and perform iterative calculations, that itself isn't a number like 0.9.. 1/3 is infinitely long, rational, and I can express it as a single p/q, and perform iterative math to calculate as many decimal points as I want to do. 1/9th is the same. 0.9.. is not the same as those others. Rationality requires non-reducing integers p and q. You've yet to show me a non-reducible integer p/q that produces a single 9 in the 0.99 chain. And yet, I can provide 1/3 and 1/9th, etc. So why would you conclude that it is still rational?
Because I showed the algebra to get a ratio of integers, which I will show again: Let x = .9999... multiply both sides by 10: 10x = 9.9999... Subtract the top equation from the bottom one. 10x - x = 9.999... - .9999... 9x = 9 x = 9/9 x = 1 There are also more rigorous proofs on line that show that all repeating decimal numbers are rational: http://www.stumblingrobot.com/2016/...peating-decimal-represents-a-rational-number/
I don’t dispute that 0.9.. BEHAVES like a rational number. I’m saying show me a single p/q that PRODUCES the decimals. You are going from decimal to p/q. Now go the other way, and produce the 9s. If you want me to show you how to do it with 1/3, I’ll be happy to. It’s rational. Not behaves like one, but is. I can generate 3s after the decimal point all day, in an interactive fashion. Show me for 0.9..
Suppose, suspending disbelief even momentarily, that to be rational you had to produce a p/q that could iteratively produce the exact decimal places required. So 1/3 is rational under this definition. 1/4 is rational under this definition. Etc. Now suppose that if the decimals cannot be produced, for any p/q, the number is not rational. Take 0.9.. and put in that category. What maths break?
It doesn't behave rational, it is rational. I linked a proof that shows all repeating decimals are rational. I showed earlier doing long division how to produce .999... by dividing 1 into 1. But the way this site removes white space makes it look like shit. And I don't have to show a p/q that provides .999..., I only need to provide how .999... yields 1 or 1/1 that shows it can be written as a ratio.
