0.999... ? 1

[Tales]

Use this simple example to visualize how these .9999 type numbers actually equal a whole integer.

Take a loaf of bread, it is 1 loaf.  Cut the loaf in half, take the right part and cut it in half again.  Take the right part of that and cut it in half again.  Continue to infinite, then add up every crumb of bread you have, in total you have 1 loaf of bread.  This is 1/2 + 1/4 + 1/8 + 1/16 + 1/32 +... to infinite = 1

If you specifically wish to see this with .9999 then take the 1 loaf, cut it into 9/10 and 1/10.  Then take that 1/10 and again cut it just as before, continue to infinite.  This will be .9 + .09 + .009 + .0009 + ... to infinite = 1.

Easy.

[Daniel]

Use this simple example to visualize how these .9999 type numbers actually equal a whole integer.

Take a loaf of bread, it is 1 loaf.  Cut the loaf in half, take the right part and cut it in half again.  Take the right part of that and cut it in half again.  Continue to infinite, then add up every crumb of bread you have, in total you have 1 loaf of bread.  This is 1/2 + 1/4 + 1/8 + 1/16 + 1/32 +... to infinite = 1

If you specifically wish to see this with .9999 then take the 1 loaf, cut it into 9/10 and 1/10.  Then take that 1/10 and again cut it just as before, continue to infinite.  This will be .9 + .09 + .009 + .0009 + ... to infinite = 1.

Easy.

Yes, but there's two problems in the "to infinity" part.

First, (in our actual world) matter is discrete. After so many divisions you will reach an indivisible particle. Meaning, you cannot repeat the process "to infinity". You repeat the process until it is no longer physically possible, and you are left with a tiny, indivisible piece. And then what? If you add up all the pieces, and you choose to include the indivisible piece, then of course you end up with 1, since you had 1 loaf to begin with. But if you exclude the indivisible piece, and add up all the pieces, you will end up with something less than 1. So the answer is 1 if you round up, or 1 minus an indivisible particle if you round down, but without rounding there can be no answer at all.

Second, (in abstraction) "addition" is the taking of one definite number and combining it with another definite number in a certain way, and getting a definite answer. This cannot be done "to infinity" using the numbers in this sequence because one of the numbers is always going to be indefinite, and so answer too will always be indefinite.
First cut: 9/10 + some number which won't be determined until the second cut = indefinite answer
Second cut: 9/10 + 90/100 + some number which won't be determined until the third cut = 99/100 + indefinite number = indefinite answer
Third cut: 9/10 + 90/100 + 900/1000 + some number which won't be determined until the fourth cut = 999/1000 + indefinite number = indefinite answer
Fourth cut: 9/10 + 90/100 + 900/1000 + 9000/10000 + some number which won't be determined until the fifth cut = 9999/10000 + indefinite number = indefinite answer
and so on. The sequence never terminates so there is never a definite answer.
But if somehow you could get a definite answer, then what would that answer be? Some say 1. But at the moment I am of the opinion that the whole idea of being able to get a definite answer is absurd.

[Tales]

1) Use a loaf of bread in your mind rather than in the physical world.  There your crumbs can go on forever.  If you want to stick to the real world, take a room's area and subdivide it into 9/10s just as with the bread.

2) Why is that the definition of addition and what is "definite"?  Is 1/3 definite or not?  1/3 is .3 repeating and 1/3 + 1/3 + 1/3 is 1, as I'm sure you've already seen this proof ages ago.

[Daniel]

You're right, the first problem is not a problem at all if you do it only in the abstract. But the second problem is still a problem.

By "definite value", I mean that the sum is an actual value rather than a potential value. The sum has no actual value until the process terminates and the addition is actually performed.

Moreover, the very notion of cutting something in two an "infinity" amount of times, or of repeating this sort of process "to infinity", is incomprehensible. Even if it does somehow work out arithmetically (which I'm still not seeing), it has no basis in reality nor in geometry.

Speaking of which, if you regard 1/3 as an infinite sum (point-three-repeating = 3/10 + 3/100 + 3/1000 + ...), then it presents this exact same problem. And for this reason the finitists hold that the fraction 1/3 is not the same thing as point-three-repeating.
I really don't know enough about finitism to pass a judgement on whether it's a good philosophy or not, but I am pretty sure that this particular argument (against infinite sums) is sound.

[Tales]

Who says the infinite sum is a process, as if this were an event processing through time?

Take the numbers 1 and 2, there are an infinite number of numbers between them, yet the difference between 2 and 1 is a finite 1.  Similarly take any room and recognize that there are an infinite number of locations within said room, yet the room totals to just that one room.

You are thinking too hard and trying to apply irrelevant complex philosophies to what is rather simple.

[Daniel]

The process is essentially-ordered, not temporally-ordered. The algorithm we're talking about is basically this: f(x) = 9x/10+f(x/10). Which means that f(x)'s value is dependent upon f(x/10)'s value, which in turn is dependent upon f(x/100)'s value, which is dependent upon f(x/1000)'s value, and so on. You end up in an infinite regress, and so f(x) never evaluates to anything.

How to avoid the regress...
Since point-nine-repeating is presumably 1, then we can just say that f(1) = 1, and if that's the case then f(x) = x, and we can simply use substitution. 9x/10+f(x/10) = 9x/10+x/10 = 10x/10, which, when x is 1, does come out to 1.
But this does not seem to be valid, since in doing this, we've completely changed the sort of problem we're dealing with. In this new version it's just a matter of simple addition. We cut the loaf into two pieces: and we then add the one actual value (the larger piece) with the other actual value (the smaller piece), and we end up with an actual value (the entire loaf). In the original problem, however, we seek to add an actual value (the larger piece) with a potential value (the sum total of all the larger pieces over the course of infinitely many cuts... potential insofar as the loaf has not yet been cut an infinite number of times). Even if we assume that this second value could somehow be actualized (allowing the two values to be added together), it doesn't follow that sum will be 1, since by definition we are only ever assimilating the larger pieces (never the smaller piece, which, though it gets smaller and smaller, never just magically disappears).

[GloriaPatri]

Let's go with your algorithm:

Let x/10 be defined as the variable y

f(x) = 9x/10 + f(y)

In turn f(y) = 9y/10+f(y/10)

Let y/10 be defined as the variable z

f(y) = 9y/10+f(z)

And so forth. You've shown that f(x) = 9x/10+9y/10+9z/10+.....+9n₁/10+....9n_infinity/10. Since every variable can be defined as 1/10 of the variable before it this evaluates to f(x) = 9x/10+9x/100+9x/1000+....+9x/10ⁿ, where n goes to infinity.

Thus f(x) = 0.9 repeating.

Your point, Daniel?

Besides, the proper algorithm you're looking for is the summation of 9x/10^n for n = 1 to n = infinity.

[Daniel]

As far as I can tell, both algorithms are equivalent. The summation formula is just more abstracted, which hides the underlying math and makes the infinite regression a lot harder to see. My point is that using the summation on an infinite series is question-begging (says the finitist), since it presupposes that all the terms in an infinite series can in fact be summed.

Another way to visualize this point-nine-repeating number would be to look at its partial sum:
s(n) = 1 - 1/(10ⁿ)
With that formula, you hypothetically could just plug in +? for n and get point-nine-repeating (apart from the fact that 10^+? is not a valid mathematical operation...)
But as can be seen, if you did that then s(+?) would not be equal to 1 but rather to 1 - 1/(10^+?). That second part is not 0 and can't just be discarded. Though it doesn't really evaluate to anything (since again, 10^+? isn't a number, and you cannot divide 1 by something that's not a number), if it could be evaluated then it would evaluate not to 0 but to some positive non-zero number (since the numerator is a positive non-zero number, and the denominator would also be a positive non-zero number if only it were a number in the first place), so you can't just ignore that part.

[Tales]

I again state that you are bringing irrelevant philosophical concepts into this analysis which is why you are confusing yourself on what is otherwise very easy.

You agree it is not temporal but then go on to say that this output depends on the function's previous output, and so on so forth.  Temporal matters are those of change, change is time.  If you are talking about function outputs as if they are changing with each iteration then you are talking about time.  As our temporal minds analyze the function we perceive it as if it were changing, but it is not.  As soon as that function is called into existence it at that same moment already and forever equals its final output.

There is no time in math for it is supernatural.  The infinite series simply is equal to what it is equal to.  There is no + 9 and then + another 9 and another 9 forever.  It simply is 1.

Similarly, again I bring up the example of a room, it has an infinite number of locations within it, yet summed together they equal that room.  These two things simply are equal, they are two ways to say the same thing.  You can look at it one way and say "an infinite number of locations" and the other and say "a finite room."  If you analyze it from infinite to finite it seems that these are unequal unless you do some complex mathematical analysis to convince you.  But if you analyze it from finite to infinite it is easy to understand - take a finite room and keep chopping it up forever, you will discover an infinite number of new locations in that room, hence indeed these two are equal.

You are tripping yourself up because:

1) you analyze it from infinite to finite and that is tricky and nonintuitive

2) are bringing in irrelevant philosophical matters which make you distrust the simple mathematical proofs like 1/3 x 3 = 1.

[cgraye]

As far as I can tell, both algorithms are equivalent. The summation formula is just more abstracted, which hides the underlying math and makes the infinite regression a lot harder to see. My point is that using the summation on an infinite series is question-begging (says the finitist), since it presupposes that all the terms in an infinite series can in fact be summed.

Remember, the value of a series is not the result of an algorithm or a procedure that can be completed, it is by definition the limit of the nth partial sum of the series as n increases without bound.  You do not have to assume you can actually add an infinite number of terms as you would add a finite number of terms.  You clearly cannot do that.

[Daniel]

You agree it is not temporal but then go on to say that this output depends on the function's previous output, and so on so forth.  Temporal matters are those of change, change is time.  If you are talking about function outputs as if they are changing with each iteration then you are talking about time.  As our temporal minds analyze the function we perceive it as if it were changing, but it is not.  As soon as that function is called into existence it at that same moment already and forever equals its final output.

It's not that the one number "changes" as the others change, as if in time. It's that the one number is logically contingent upon the other numbers.
Suppose I said that I am thinking of a number a, whose value is defined as being equal to that of some other number b, plus one. And the value of b is defined as being equal c + 1. And so forth, ad infinitum. And then I ask myself, what is the value of a? The answer is, it has no value. Because it only has value insofar as b has value, and b only has value insofar as c has value. Each number is relative to something else, so without that absolute starting point then nothing else has any value. And if the chain goes on forever (which it does, since it's an infinite series) then there is no starting point. Thus, if I know in advance that a has a value, then I also know in advance that the chain must terminate at some starting point (and that the series must be finite). No starting point means no value.
Now suppose it was not an infinite chain but it was just three numbers: a, b, and c. If c was defined as 10, then a would be 12, by definition. Not because of the passage of time (as if, in a temporal succession, b first becomes 11 in response to c's value, and afterwards a becomes 12 in response to b's value). Rather, a would simply be 12 by the very fact that b is 11, by the very fact that c is 10.

There is no time in math for it is supernatural.  The infinite series simply is equal to what it is equal to.  There is no + 9 and then + another 9 and another 9 forever.  It simply is 1.

Remember, the value of a series is not the result of an algorithm or a procedure that can be completed, it is by definition the limit of the nth partial sum of the series as n increases without bound.  You do not have to assume you can actually add an infinite number of terms as you would add a finite number of terms.  You clearly cannot do that.

To clarify, it seems that both of you are saying that a summation is not the same thing as repeated addition? (Or, at least not always the same thing?)

[GloriaPatri]

You agree it is not temporal but then go on to say that this output depends on the function's previous output, and so on so forth.  Temporal matters are those of change, change is time.  If you are talking about function outputs as if they are changing with each iteration then you are talking about time.  As our temporal minds analyze the function we perceive it as if it were changing, but it is not.  As soon as that function is called into existence it at that same moment already and forever equals its final output.

It's not that the one number "changes" as the others change, as if in time. It's that the one number is logically contingent upon the other numbers.
Suppose I said that I am thinking of a number a, whose value is defined as being equal to that of some other number b, plus one. And the value of b is defined as being equal c + 1. And so forth, ad infinitum. And then I ask myself, what is the value of a? The answer is, it has no value. Because it only has value insofar as b has value, and b only has value insofar as c has value. Each number is relative to something else, so without that absolute starting point then nothing else has any value. And if the chain goes on forever (which it does, since it's an infinite series) then there is no starting point. Thus, if I know in advance that a has a value, then I also know in advance that the chain must terminate at some starting point (and that the series must be finite). No starting point means no value.
Now suppose it was not an infinite chain but it was just three numbers: a, b, and c. If c was defined as 10, then a would be 12, by definition. Not because of the passage of time (as if, in a temporal succession, b first becomes 11 in response to c's value, and afterwards a becomes 12 in response to b's value). Rather, a would simply be 12 by the very fact that b is 11, by the very fact that c is 10.

There is no time in math for it is supernatural.  The infinite series simply is equal to what it is equal to.  There is no + 9 and then + another 9 and another 9 forever.  It simply is 1.

Remember, the value of a series is not the result of an algorithm or a procedure that can be completed, it is by definition the limit of the nth partial sum of the series as n increases without bound.  You do not have to assume you can actually add an infinite number of terms as you would add a finite number of terms.  You clearly cannot do that.

To clarify, it seems that both of you are saying that a summation is not the same thing as repeated addition? (Or, at least not always the same thing?)

The value zero is the starting point, for both portions of the number line. Hence why zero is the additive identity.

I suggest you look up the axioms that are utilized in the creation of the real number system.

[Quaremerepulisti]

My point is that using the summation on an infinite series is question-begging (says the finitist), since it presupposes that all the terms in an infinite series can in fact be summed.

What in the world are you talking about?  The "value" of summation of an infinite series, or if you prefer, the "value" of a function with infinity as an argument, are defined in terms of limits.  And there is no question that the limit of 1 - 10^(-n) (or however you want to write it down) as n goes to infinity is well-defined and is 1.

[Daniel]

My point is that using the summation on an infinite series is question-begging (says the finitist), since it presupposes that all the terms in an infinite series can in fact be summed.

What in the world are you talking about?  The "value" of summation of an infinite series, or if you prefer, the "value" of a function with infinity as an argument, are defined in terms of limits.  And there is no question that the limit of 1 - 10^(-n) (or however you want to write it down) as n goes to infinity is well-defined and is 1.

Oh, my mistake. I didn't realize that. Guess that answers the question.

[Daniel]

.