Re: .99999....=1

On Oct 26, 12:47 pm, Adam <n...@xxxxxxxx> wrote:
On Fri, 26 Oct 2007 07:14:44 -0700, Randy Poe wrote:
On Oct 26, 8:41 am, Adam <n...@xxxxxxxx> wrote:
The light didn't go on for me until someone made it clear that
0.333.... doesn't actually mean an infinite number of characters, but
that this is shorthand notation for a limit.

Far be it from me to disturb your light bulb,
but 0.333... does actually mean an infinite number
of characters, one digit "3" for each natural number.

Is that what *you* mean when you write 0.333...?

Yes.


An infinite number of characters isn't a number.

Neither is a finite number of characters. Both
can be representations of numbers.


If "0.333..." means "an infinite number of 3s after a decimal", then I
would argue that the following is false:

1/3 = 0.333...

And why would you argue that?

I would argue that the characters "1/3" are
a representation of a number, and that the
characters "0.333..." (meaning the infinite
string) are another representation of a number,
and that the statement above says they represent
the same number.

Every number has a decimal representation, consisting
of a sequence of digits 0-9 and a decimal point.
Why does the infinite nature of some sequences
cause you to say it can't be a number? That means
you are saying that many numbers don't have
decimal representations.

Does 1/3 have a decimal representation?
Is there a millionth digit in that representation?
Is there a billionth digit? A googol-th digit?
Is there any n such that there is no n-th digit?

That is, I would argue that:

1/3 is not equal to an infinite number of 3s after a decimal

because 1/3 is a number, whereas an infinite number of characters is
not a number.

No, it's a representation of a number.

The characters "0", ".", and "5" aren't numbers,
and the character string "0.5" is a string, not
a number. Would you argue that a character string
can't be a number, therefore 0.5 isn't a number,
therefore 1/2 = 0.5 is false?

The limit which the infinite number of characters represents is a
number, but the character string is not.

Yes, and the mathematical value represented
by "0.5" is a number and we know how to calculate
that number, but the string "0.5" is not.

To short-circuit the argument, one might argue that the character "2"
isn't a number. But it does have a one-to-one correspondence to a
value on a number line.

And so does the infinite character string "0.333..."

It has a one to one correspondence with the position
that is exactly 1/3 of the way between 0 and 1.

"An infinite number of 3s after a decimal" does not have a one-to-one
correspondence to a value on a number line because a string that
contains an infinite number of characters isn't _one_ thing -- it
always contains at least one more character.

One more character than what? It has infinitely
many characters. It has one for every natural number
n.

Here again you have changed from the infinite
string, the limit, to the sequence of finite
strings, and I don't think you realize you
did it.

What I think you mean by "it always contains
at least one more character" is that if you
consider any finite string 0.333...3 ending
at position n, there is another finite string
0.333....3 ending at position n+1. Sure, that
is absolutely true if that's what you mean.
The SEQUENCE of FINITE strings {0.3, 0.33,
0.333, ...} doesn't end. For every number in it,
there is a next number.

But none of those is what we mean by the
infinite string 0.333...

That string isn't in the sequence.

It has no boundary.

What is a boundary? A last digit? Why
is that a requirement?

If one isn't content to talk about the limit represented by the
string, then it can't be used it in a one-to-one correspondence and it
isn't a number.

Of course I am. The value represented by the
infinite string is the limit of the values
represented by the sequence of finite strings.

How does that stop the infinite string from
existing?

Lest someone argue that the result is irrational because the
denominator is growing without limit and it is therefore impossible to
combine all the terms into a single ratio of two integers, consider
the infinite sum:

2 = 1 + 1/2 + 1/4 + 1/8 + ...

The denominator is growing without limit in this series, as well; but
the result is rational.

The denominator of the finite sums is growing without
limit.

Okay.

I think you're not distinguishing clearly
between the partial (finite) sums and the "infinite
sum".

I thought that I had already done that in distinguishing between the
series and the sum (its limit).

Nope. Every time you find yourself talking about
"growing" or "changing" or "next", you are
thinking of a sequence, not the limit of
that sequence. Just as you don't want to
distinguish between the sequence of finite
strings {0.3, 0.33, 0.333...} and the single
infinite string which is their limit.

The denominators in the series are
growing in each successive term.

Yes, that is a true statement about the
partial sums, not of the infinite sum
which is one thing (a limit).

When you write 1 + 1/2 + 1/4 + 1/8 + ...
that is not a partial sum. It isn't changing. It
can't be said to have anything in it which is
"growing".

How would you have said more clearly that the denominators in
successive terms are (as opposed to the denominator resulting after
combining all terms in partial sum is) increasing?

I would have no problem saying the denominators
in successive terms are increasing.

I had a problem with the statement that the
denominator of the series is increasing. The
series is different from its individual terms.
It's the whole collection of terms.

- Randy


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