Now we have 0.123456789CD And the number between 0.C.. and 1 is 0.D.., and now 0.D.. is special because it is itself and the number between itself and 1. Now we do this infinite times Alan. Infinitely adding a new designator for the terminal digit. And what do we conclude? That there are infinite numbers between the TERMINAL digit and 1. Infinite. Between. That simple.
Stopped reading at "most of a Ph.D". Can't trust a quitter. Also, [uck fay] Florida. I ride with fl0at_
You are changing the rules mid game. You cannot have a discussion about the base 10 number system and suddenly add new digit into the mix. It would be like saying you move your king to the 9th row on the chess board. I am not claiming the .999... is 1 in all base systems. Just the standard conventional base 10 that we all use every day. In binary .111111... is equal to 1. In hex .fffffff... is equal to 1 but .999... is definitely less than 1. This is just a quirk of the decimal notation of real numbers.
I am not changing anything. I am showing the 0.9.. IS NOT EQUAL TO 1. This is truth vs taught bullshit. I told TennTra earlier in this thread that I’m not arguing convention, because mathematics absolutely teaches that 0.9.. = 1. I do not disput that that is convention. It is still wrong. Here is why. There are infinite digits between any digit and the next whole number. Any. Every. Period. That is truth. If you want to say “well I know it’s true, but convention says, so it is” we have nothing to say. Because you aren’t being correct. You are being dogmatic.
This same argument of operations could be used to argue that 1/3 does not equal .3.., and that no number lies between them, yet you've already accepted that is true. So how does that serve as an argument that something must lie between .9.. and 1?
I completely get the thought and how as you do that you could see the number (as the value is your y axis and decimal places or iterations is your x) that the number continually converges toward 1 but doesn't reach it. However I also see that there is no next step in the iteration. 0.9.. is in itself an infinite string of 9s. It's not a always getting continually longer. But I know you got that. The mathematicians likely fisagree with me when I say it's convention. But by convention I Don't just mean it's close enough so we will just call it 1. But within the real number system we use they are equal. Maybe you or someone else can tell me the error in the following. But once we accept that 1/3 = .3.. then I jut see it as... 1/3 + 1/3 + 1/3 = 1 .3.. + .3.. + .3.. = 1 .9.. = 1
There are infinite points between any two points on the Real number line, this is true. But because of a quirk of the decimal notation of those points, .999... and 1.000... both are labels for the same point on the Real number line. It isn't dogma. It is provable, but because you have a preconceived idea that they must be different you are not accepting any proofs that are given to you.
I begin with 4 pieces of candy corn. I press two onto my canines to make vampire teeth. The tip breaks off another piece, falls to the floor and finds its way under the refrigerator. The dog jumps up and devours a piece but leaves a small slobbery fragment on the floor. How much candy corn do I have remaining to the nearest thousandth?
If I sit down and do long division of 3 into 1, I will get 0.3.. I will get no other number. If I sit down and do long divion of 1 into 1, I will get 1.0.. I will get no other number.
Because that’s flawed. You cannot sum infinite numbers the same as finite. There is a natural remainder inside of 1/3 (when you flip it to 0.3.. there is always that small piece left that is producing the infinite repeat) So it’s actually 0.3.. + Remainder + 0.3.. + Remainder + 0.3.. + Remainder = 1 Or 0.9.. + sigma Remainder (which is 0.[0..]1) = 1 There, I give the first mathematical operation of infinite maths freely.
Hell, with the news that has come out today, 1=2, oranges divide into apples and candy corn flows from the heavens like manna.
No, you are too accepting of “proof,” and unable to see logic. Thus, dogma. It has nothing to do with preconceived notions, I am literally, post after post, showing you. All of your proofs are invalid, and cannot stand up to simple tests. Your most often cannot handle simple precision. I am showing you this, and giving the logic behind it. I’m showing you an alternative math notation to see it. I am showing you over and over again. What is preconceived is the recycled previously conceived things that were thrown up all over you in the past.
But that's a different argument. You were using an argument against .9.. equaling 1 that you don't apply to 1/3. If there is always a next iteration possible then you are always just approaching 1/3 with each new 3.
No, 1/3 = 0.3.. Exactly equal. But when manipulating that number as a decimal, the infinite repeating remainder must be considered.
Are you just asserting that I can't sum them? What if I wrote it as: 1/3 = Sum(From 1 to infinity) the quantity (3 times 10 raised to the negative i) Now if I multiply both sides by 3, I don't know if any rule that says I can't put that 3 inside the summation to give me 9. Now I get 1 = .9.. Also - why can't I sum a string of infinite repeating numbers? Is it just coinincdence that .3.. + .3.. = .6..?
If 3 * 1/3 = 1 And 1/3 = 0.3....... Then 3 * 1/3 = 0.9..... Which means 0.9..... = 1 What am I missing?
I agree that they are exactly equal, but only if you don't apply the idea of operations or next step of the iteration. As soon as you do that it becomes something less than 1/3.
Is 1/3 = 0.3 or is it equal to 0.3 remainder 1/3? So why then do you drop the remainder when considering 0.3..? There are two infinitely repeating operations, the expansion of the decimal places and the treatment of the remainder. You are missing the treatment of the remainder.
