Is 1 = 0.999... ? Really?

[James S Saint]

This is one of those issues that display the clear distinction between a good philosopher and a expert mathematician. The good philosopher will tell that they cannot be equal and the modern day mathematician will tell you that they are declared to be equal by mathematicians.

Currently Wiki, and a great many mathematicians will tell you that 1 = 0.999.... Many "proofs" are displayed to show how wrong those are who disagree. The "good philosopher" can display how every one of those proofs are fallacious.

What is your belief?

And WHY?

Note that you can change your poll-vote later if you wish.

[phyllo]

This is one of those issues that display the clear distinction between a good philosopher and a expert mathematician.

A good engineer will tell them that they are wasting their time.

Reminds me of this joke:

A mathematician and an engineer are sitting at a table drinking when a very beautiful woman walks in and sits down at the bar.

The mathematician sighs. "I'd like to talk to her, but first I have to cover half the distance between where we are and where she is, then half of the distance that remains, then half of that distance, and so on. The series is infinite. There will always be some finite distance between us."

The engineer gets up and starts walking. "Ah, well, I figure I can get close enough for all practical purposes."

[MagsJ]

0.999 is very close to 1, but it ain't no 1 Mister

Don't you just love a recurring number

[Harbal]

Currently Wiki, and a great many mathematicians will tell you that 1 = 0.999.... Many "proofs" are displayed to show how wrong those are who disagree. The "good philosopher" can display how every one of those proofs are fallacious.

What is your belief?

I believe that it isn't worth arguing over such a tiny amount.

[MagsJ]

Let's decimate this discussion.

[James S Saint]

From Wikipedia, the free encyclopedia

In mathematics, the repeating decimal 0.999… (sometimes written with more or fewer 9s before the final ellipsis, for example as 0.9…, or in a variety of other variants, denotes a real number that can be shown to be the number one.


The repeating decimal continues with an infinite number of nines.

In other words, the symbols "0.999…" and "1" represent the same number. Proofs of this equality have been formulated with varying degrees of mathematical rigor, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.
.
.






Although these proofs demonstrate that 0.999… = 1, the extent to which they explain the equation depends on the audience.

No one is going to support Wiki and the mathematicians???

When I came here years ago, anything said by Wiki, mathematicians, and certainly physicists (if there is any longer a difference), was holy doctrine. And now you cast stones???

Such heresy.

[surreptitious57]

I do not think that they are the same for they occupy different places on the number
line but they are however as close to each other as any two numbers can possibly be

[James S Saint]

So how do you account for Wiki's proofs?

[surreptitious57]

I cannot fault them. But now that I think about it it is wrong to place an irrational number on a specific place on the number line because if it has
an infinite number of decimal places then one cannot be absolutely certain where it goes. One can only be probably certain. Incidentally whoever
told you that mathematicians and physicists are perfect is wrong because this is not a quality human beings possess at all. And while mathematics
[ but not physics as it is a science and therefore inductive ] may be a perfect discipline those who practice it are just as fallible as everybody else

[James S Saint]

But their claim is that 0.999... is not an irrational number.

[Amorphos]

To have either 0.9r or 1, you have to have something exact as the standard. As there are no exact standards [improbability etc] then neither case is exactly true to begin with.

The maths has to rely on a metaphoric pretext where numbers aren't real and present their own logic standards. math is of course metaphor, but unless it marries to what reality is then its conclusions are irrelevant. as that's impossible then maths/science will just have to remain as part of philosophy.

basically there is a degree of ambiguity in any case.

reality does not divide perfectly, not in shape/geometry nor numbers/patterns imho.

[surreptitious57]

But their claim is that 0.999 ... is not an irrational number

I thought it was because it has infinite decimal places. However that alone does not make it irrational
according to Wikipedia because it has to be random or non repeatable too such as with pi for example

[James S Saint]

But their claim is that 0.999 ... is not an irrational number

I thought it was because it has infinite decimal places. However that alone does not make it irrational
according to Wikipedia because it has to be random or non repeatable too such as with pi for example

Their definition of a "rational number" is "any number that can be represented by a fraction formed of integers".

And if they declare that 0.999... is equal to 1, then they are declaring that 0.999... is equal to 1/1 = rational number.

[surreptitious57]

Even if it was not equal to one according to that proof it would still be a rational number since it is not random or non repeatable. Even though
it has an infinite number of decimal places the same as any irrational number. Any number whether rational or irrational can be expressed as a
fraction but only rational ones technically count as fractions. Because neither the numerator or denominator can tend to infinity. Interestingly
though most numbers are actually irrational as Wikipedia correctly says

[James S Saint]

In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.[1] Since q may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted by a boldface Q (or blackboard bold Q {\displaystyle \mathbb {Q} } \mathbb {Q} , Unicode ℚ);[2] it was thus denoted in 1895 by Peano after quoziente, Italian for "quotient".

The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for any other integer base (e.g. binary, hexadecimal).

So 0.999... is technically a "rational number".

But can't anyone explain why it cannot be equal to 1?

[phyllo]

So 0.999... is technically a "rational number".

It can be expressed as the fraction 9/9

[surreptitious57]

http://atheistforums.org/thread-9338.html

http://atheistforums.org/thread-29842.html

http://atheistforums.org/thread-11303.html

http://atheistforums.org/thread-16855.html

[surreptitious57]

I have to correct myself when I said I did not think 0.999 ... is equal to I because they are the same number just expressed differently
The Wiki proof is correct and is accepted by mathematicians. It does appear counter intuitive but that does not actually make it false

[phyllo]

The question demonstrates a quirk of human perception and the limitations of the decimal system. It's similar to an optical illusion but it's a mental illusion.

[James S Saint]

So 0.999... is technically a "rational number".

It can be expressed as the fraction 9/9

Emmmm....
.. no.

I have to correct myself when I said I did not think 0.999 ... is equal to I because they are the same number just expressed differently
The Wiki proof is correct and is accepted by mathematicians. It does appear counter intuitive but that does not actually make it false

Well, that is what I was expecting for at least someone to say. That was Wiki's take on it.

But happens to be wrong.

[phyllo]

Emmmm....
.. no.

If you say so then it must be no.

[James S Saint]

I'll give you a hint:

0.999... ≡ ∑[from 1 to ∞} of [9 * 10^-n]
That is:
0.9 +
0.09 +
0.009 +
0.0009 +
.
.
.

And for that reason, you know that 0.999... is NOT equal to 1.000...

Just think about what "..." actually means.

[phyllo]

Why sum an infinite number of numbers when you only need to sum 9?

1/9+1/9+1/9+1/9+1/9+1/9+1/9+1/9+1/9=1

Represented in decimal form :

.111_ + .111_ ... = .999_ and also = 1

That's it in a nutshell. No infinite series required. KISS

[phyllo]

Feels somehow unsatisfying?

Well, that's life.

[James S Saint]

Why sum an infinite number of numbers when you only need to sum 9?

1/9+1/9+1/9+1/9+1/9+1/9+1/9+1/9+1/9=1

Represented in decimal form :

.111_ + .111_ ... = .999_ and also = 1

That's it in a nutshell. No infinite series required. KISS

But is it really?
... not really.