
Re: It is a very bad idea and nothing less than stupid to define 1/3 = 0.333...
Posted:
Oct 8, 2017 7:22 AM


It happens that netzweltler formulated : > Am Samstag, 7. Oktober 2017 14:18:44 UTC+2 schrieb FromTheRafters: >> netzweltler wrote : >>> Am Samstag, 7. Oktober 2017 01:42:15 UTC+2 schrieb Jim Burns: >>>> On 10/6/2017 6:03 AM, netzweltler wrote: >>>>> Am Freitag, 6. Oktober 2017 02:54:40 UTC+2 >>>>> schrieb Jim Burns: >>>>>> On 10/5/2017 3:12 PM, netzweltler wrote: >>>>>>> Am Donnerstag, 5. Oktober 2017 17:59:25 UTC+2 >>>>>>> schrieb Jim Burns: >>>>>>>> On 10/5/2017 10:00 AM, netzweltler wrote: >>>>>>>>> Am Donnerstag, 5. Oktober 2017 15:22:35 UTC+2 >>>>>>>>> schrieb Jim Burns: >>>> >>>>>>>> [...] >>>>>>>>>> _We don't do what you're describing_ >>>>>>>>> >>>>>>>>> Nevertheless, >>>>>>>> >>>>>>>> "Nevertheless"? >>>>>>>> Do you agree that what you're describing >>>>>>>> is not what we're doing? >>>>>> >>>>>> *NETZWELTLER* >>>>>> DO YOU AGREE THAT WHAT YOU'RE DOING >>>>>> IS NOT WHAT WE'RE DOING? >>>>> >>>>> Let's say I agree. Doesn't mean that it is obvious to me >>>>> what *you* are doing. >>>> >>>> Great. Let's say you agree. Will you stop saying that >>>> "0.999... means infinitely many commands"? >>> >>> No. Because it is not obvious to me why the equation >>> 0.999... = 0.9 + 0.09 + 0.009 + ... >>> should be wrong. >> >> It's not wrong. The first one is a representation of the number one. >> The second is a representation of the number one. Two things equal to >> the same thing are equal to each other. >> >> [...] > > I'd have to look it up: Did you say that 0.999... IS the result of infinitely > many addition operations or IS NOT the result of infinitely many addition > operations?
If you had an oracle with enough time which could do the arithmetic and hand you an answer, then yes. Without such an oracle, then I'd have to say no. That's why I said "after" doing infinitely many steps you would have that number exactly. John Conway used language similar to 'after infinitely many of these steps, there is an explosion of sets...' to describe a similar notion in describing his construction of the surreals, so I'm not exactly breaking any new ground here.
I'm saying that that representation (or rather the infinite sum version which you previously laid out) is a result of taking the infitite string of nines (or rather terms) and repeatedly truncating them to a usable (printable) length so as to be defined as a textual representation of that number. Since the number of truncations are not finite the ellipsis is used to indicate this fact. This doesn't mean that the number itself is reliant upon the ability to do infinite arithmetic to have anyones permission to exist.
As someone else has already pointed out, e and ? are numbers and can be used in mathematics perfectly, especially in polar notation. Sometimes they effectively cancel each other out such as in Euler's Identity. A 'problem' exists when attempting to express either of these existing numbers in decimal expansion representation (to write them down) and it looks as if it can't be done exactly if you look at the representation as if it were arithmetic defining a number rather than a representation built from that number and defined as representing that number.
Euler solved the Basel Problem *exactly*, despite the fact that ? appears in the numerator and ? can't be 'reached' by doing the arithmetic inherent in any of its representations in less than infinite time.

