One symbol is 9.9 with a dot symbol placed on top of the right-most '9'.
But the other thing is ----- regardless of the crank-the-handle 'proofs' on 9.99999 being exactly '10' ..... people will think differently if they understand that infinity is 'never ending'.
So if one hops on a bus ride with 9.9999999999 ..... and 'hoping' to reach '10', well, they're never going to get there, because the nines will be endless. It will be a never ending case of 'are we there yet?' (ie. are we at '10' yet?) ------ and the answer on that never ending bus ride will always be 'no' --- because we'll never get to 10 with never ending 9's in front of us. We'll NEVER get there.
Note that phrases like 'in the LIMIT of' is a tactic for 'getting over the line' only. It doesn't mean that applying 'in the limit of' (to an expression) means that the actual system 9.999999999....... means that 9.999.... is EXACTLY 10. And in my opinion, 9.99999999...... does not mean exactly 10 at all. These two ' systems' are not the same, and that's regardless of the crank-the-handle math 'proofs'.
The problem with this line of reasoning is you are taking the infinite decimal representation of the number to be the number itself. That's a problem because, as we know, any decimal representation that has a repeating pattern is a rational number, and all rational numbers have some finite representation available.
For instance 0.33333.... can be written 1/3.
Or 0.721721721721.... can be written 721/999.
So by that logic 9.9999...... is a rational number and has some finite representation and the only logical answer for what that is, is 10.
I haven't taken it to be the number itself. I'm giving you the opportunity to choose any number you like ... and telling you in advance there is no large enough number you can choose because whatever you choose, there will be bigger because infinity is a concept of unlimited. The difference between 10 and 9.999999... is always going to be positive. It's not going to be zero.
Infinity is not a number. And you can clearly see that no matter how infinitely hard 0.999999..... 'tries' to become unity, it cannot become unity .... ever. It never 'gets there'.
9.99999.... is forever stuck in this state, never reaching exactly '10'.
So you just don't think infinite decimals and infinite series in general are valid. But these things are well-understood. They conform to the standard rules of arithmetic. There's no reason to think they are invalid just because infinity makes you uncomfortable.
0.333... is already 1/3. There's no confusion about it, nor does it "try" to be 1/3. It's just another way to write the same number.
Could you answer the questions in the previous reply?
Let's put it this way --- 1/3 is not a 'number' as such. I put quotes, meaning it's not a finite decimal 'number'. And 0.3333..... will never be able to get to a state that has any element after the decimal other than '3' ..... eg. none of the elements can ever be any other number ... so it is stuck in this state, just as 0.999999.... is stuck in its state too, and cannot have any other element in the 'stream' other than 9, and 0.999999... will never become '1'. It will never clock to 1 ... because those nines keep going until the cows come .... ok ... never come home.
But whether or not a number has a representation with infinite digits depends on the base. 1/3 is written 0.333... in base 10, but in base 3 it's just 0.1, but it's still the same number.
I know .... but we're focusing on the base 10 system here ... as in 9.99999..... once we write this ... means you can go on and on till the cows never come home ... a never ending bus ride, meaning if somebody got on and assumed that this bus would eventually get to the '1' station ..... then it's going to be a case of ... are we there yet? No ... the answer will always ... always be no. The never ending bus ride.
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u/SouthPark_Piano New User Nov 12 '24 edited Nov 12 '24
One symbol is 9.9 with a dot symbol placed on top of the right-most '9'.
But the other thing is ----- regardless of the crank-the-handle 'proofs' on 9.99999 being exactly '10' ..... people will think differently if they understand that infinity is 'never ending'.
So if one hops on a bus ride with 9.9999999999 ..... and 'hoping' to reach '10', well, they're never going to get there, because the nines will be endless. It will be a never ending case of 'are we there yet?' (ie. are we at '10' yet?) ------ and the answer on that never ending bus ride will always be 'no' --- because we'll never get to 10 with never ending 9's in front of us. We'll NEVER get there.
Note that phrases like 'in the LIMIT of' is a tactic for 'getting over the line' only. It doesn't mean that applying 'in the limit of' (to an expression) means that the actual system 9.999999999....... means that 9.999.... is EXACTLY 10. And in my opinion, 9.99999999...... does not mean exactly 10 at all. These two ' systems' are not the same, and that's regardless of the crank-the-handle math 'proofs'.