1=.999999...?

[Juggernaut]

I agree with you that a number should not be a process or change over time, the way .9[bar] appears to do. But like I said, it's useful to make .9[bar] a number. So logically, if we wish to make .9[bar] a number, we should not think of it as changing "as the 9's increase to infinity". Instead .9[bar] must be thought of as the destination being approached by the sequence of "partial decimals" (0.9, 0.99, 0.999, ...). Imagine a grasshopper which starts 1 foot from a castle wall. He hops 9/10 of the way to the wall, then hops 9/10 of the remaining distance, then 9/10 again, and so on. His distance from the wall goes like (0.9, 0.99, 0.999...) as his hops increase over time. He never gets to the wall, but he is always approaching the wall. 0.9[bar] is a representation of where he would be after infinitely many jumps. But we know that that's the wall, so 0.9[bar] = 1.

Forms do not work because they are imaginary, but work because we can think of them as real... The work is what we do having the form set in our minds...The work should not be the form itself...I can thread a needle as a form with a thread as a form, but if the thread was always changing its form, why would I, and how could I???I first must nail down what the form is, and for that reason all forms are conserved...That is the meaning of identity...

[aporia]

I repeat, the number 0.9[bar] is not changing. It is always the same single number. It always holds the same single place on the number line. It is the number representing the unchanging singular destination of the grasshopper. Even though the grasshopper moves, his destination is always the same. And if you apply some logical thinking as illustrated in my previous post, the only possible destination of the grasshopper is 1. Hence 0.9[bar] = 1.

The work we do with 0.9[bar] = 1 set in our minds is useful scientific modeling work, as I discussed and illustrated earlier with the radioactive decay example. Certainly nothing in the ordinary world will be usefully represented directly by 0.9[bar]. For anything like that, we'd just use 1 as you have said many times. But 0.9[bar] = 1 comes about as a consequence of certain natural, simple models of common physical processes. In these cases, 0.9[bar] represents a result from an idealized model that approximates a real-world event or process.

[Juggernaut]

I repeat, the number 0.9[bar] is not changing. It is always the same single number. It always holds the same single place on the number line. It is the number representing the unchanging singular destination of the grasshopper. Even though the grasshopper moves, his destination is always the same. And if you apply some logical thinking as illustrated in my previous post, the only possible destination of the grasshopper is 1. Hence 0.9[bar] = 1.

The work we do with 0.9[bar] = 1 set in our minds is useful scientific modeling work, as I discussed and illustrated earlier with the radioactive decay example. Certainly nothing in the ordinary world will be usefully represented directly by 0.9[bar]. For anything like that, we'd just use 1 as you have said many times. But 0.9[bar] = 1 comes about as a consequence of certain natural, simple models of common physical processes. In these cases, 0.9[bar] represents a result from an idealized model that approximates a real-world event or process.

.999... is not even a number.. fractions are not numbers...Ones are numbers...Two ones are two...Three ones are three...You might represent fractions, but you can hardly show what they represent, so 1/3 is the problem, and not the solution; and 1/2 is the problem, and not the solution... The more these problems are changed in form the less the reveal any truth, that is: As a concept that can be compared to a reality...It is easy to represent one unit as one on a page, and notch in a gun, or a finger on a hand...In that instance we are not making a statement of equality, but of identity...One is one...If .9[bar]could represent one, then the world would spare me, and write one in place of it... We reduce our fractions... We would not try to say 3/3, or 5/5 were one, or put them needlessly in any equasion, let alone as an answer...We know it is one,and write it so... Don't try to represent reality as so many fractions...First of all, it is not... It adds complexity without end...How many ones do your have in your pocket if you count every coin as one??? If you count every coin as two halves, or as three thirds or as ten tenths, does it show any better the reality of the situation, or does it work only to confuse reality by making an unnatural division of nature??? We simplify our nature, our reality with forms/concepts, of which math is only one example... Cut everything with Occam's razor...

[aporia]

If the only thing you ever want to do with numbers is count coins in your pocket, then I agree with you -- there'd be no point in using 0.9[bar]. But there's more to numbers than counting things! Even in the grocery store things are measured in finite decimals, like 1.73 pounds, and 1.73 is just a way of writing the fraction 173/100. If 1.73 isn't a number, then what is it, and why can we add and subtract and multiply and compare with it just like a number?

You might represent fractions, but you can hardly show what they represent, so 1/3 is the problem, and not the solution; and 1/2 is the problem, and not the solution... The more these problems are changed in form the less the reveal any truth, that is: As a concept that can be compared to a reality

1/3 cup of water is an amount of water such that, when you put 3 of those amounts together, you get a cup of water. I can do that in my kitchen. I think that shows exactly what 1/3 represents. It's a little more complicated to represent than 1, and the representation depends on the idea of 1 cup for its existence; but the representation of 1/3 cup is just as precise and just as connected to reality as that of 1 cup of water.

One is one...If .9[bar]could represent one, then the world would spare me, and write one in place of it... We reduce our fractions... We would not try to say 3/3, or 5/5 were one, or put them needlessly in any equasion, let alone as an answer

I agree that we would not put them in an equation needlessly. The point is that sometimes there is a need. For example, suppose we wish to add 1/3 + 1/5. To do this we write

1/3 + 1/5 = 1*1/3 + 1*1/5 = (5/5)*(1/3) + (3/3)*(1/5) = 5/15 + 3/15 = 8/15.

In the middle steps of this solution, we represent 1 as both 3/3 and 5/5 because doing so allows us to create a common denominator for the fraction. The "reduced" form of 1 will not help us solve this problem -- a more complex representation of 1 is needed. Also, "answers" have no special status or special need to be reduced in math, because answers don't always answer all questions. Sometimes your answer becomes a tool to solve a new problem. But solving the new problem may require representing your old answer in a more complicated way. For example, our answer from the previous problem is 8/15. Suppose we want to sum 1/3 + 1/5 + 1/30. It will save us some work to use our answer from the previous problem: 1/3 + 1/5 = 8/15. But we need to turn our answer into 16/30 so that we can get it into a common denominator with 1/30 and get the final result, 17/30.

Therefore, the "simplest" representation of a number may not be the most useful or valuable one. It depends on what you're using the number for.

It's the same way with 0.9[bar]. For example, suppose you need to subtract 0.173 from 1. Is it easier to do the subtraction as 0.9999... - 0.173, or as 1.000... - 0.173? Most people would find the first subtraction easier to do, because the second would require several carries. And it would be much worse if you had to subtract an infinite decimal from 1. For example, suppose you wanted to subtract 0.12356[bar] from 1. Is that easier as 0.9[bar] - 0.12356[bar] or as 1 - 0.12356[bar]? Again, the first way requires no carries.

Numbers have many equivalent representations in math, and each representation has its own uses. Often (as with 1/3 or .9[bar]) each of those uses are good for different problems. But those problems are all real and important. They come from physics, finance, or even basic cooking. I agree that it's good to keep things simple, but we will not oversimplify to the point where all we can do with our math is count coins.

[derleydoo]

It took 35 pages of debate, but I think it's safe to say, we finally settled the argument!