Fundamental Theorem of Arithmetic is the basis of all of this prime and composite stuff. This states that every natural number > 1 can be expressed as a product of prime numbers in one and only one way except for order.
For example: 5 is prime

If 1 is considered a prime, then 5 = 5 x 1 and 5 = 5 x 1 x 1 which is two different ways of expressing 5 as a product of primes, and as a result, the Fundamental Theorem of Arithmetic is wrong and there is not such thing as prime or composite. If you argue that 1 is somehow special, and make special rules to make sure that 1 does not interfere with all of the theorems that depend on the Fundamental Theorem, then what you have really done is exactly what the mathemeticans did centuries ago: Define 1 as neither prime nor composite. It is called Unary.

Now, what I want to know is whether 0.99999999..... is prime or not

__________________
Time flies like an arrow.
Fruit flies like a banana.

I'm going to use their original titles; "The Preferred Definition of Primes" and "The Revised Definition of Primes".

There is nothing wrong with revising definitions. We do it all the time. It's allowed in the rules of mathematics as long as the revision follows the rules of mathematics other than the exeption defined by the revision and the revision itself doesn't produce a mathematical quandry. The exact definition of Pi is incalculable and almost useless. Approximate definitions of Pi are used all the time. Just don't make the mistake of thinking a revised definition is the preferred definition. This is not a hard and fast rule. Sometimes a revised definition by mathematical logic becomes the preferred definition. But not just because a revised definition is usefull.

Sorry, but I don't see where it was shown that the "preferred" defintion of Primes is the "original" one (where 1 is included).

__________________
Reality moves at the speed of light.
If the text of this post is blue, it's a "Moderator comment".
[ The RULES of the Forum ] [ Forum FAQs ] [ Conspiracy Theory advice ] [ Alternate Theory Advice ]
To report a post (even this one) to the moderation team - use the /!\ icon at the top-right of the post.

It's not. I just opened a copy of Euclid's Elements, from the 3rd Century BC (translation, Heath, Dover). He doesn't even consider 1 a number, he calls it a unit. Apparently, that was true of others as well, before him, like Aristotle. Some of them didn't consider 2 to be a prime number either, not sure why.

ETA: Euclid's Prop. 14, Book IX, is considered essentially the fundamental theorem of arithmetic, that "a number can be resolved into prime factors in only one way" (from the footnote).

As I demonstrated in the other thread, the definition of prime number that explicitly excludes one appears in the Chambers Dictionary, the Penguin Dictionary of Mathematics, and the material provided for Open University mathematics courses.

I have not seen any formal definitions that include one. But then, I don't have any dictionaries that were published in the 19th century.

__________________
Nothing beautiful was ever made from gravel.

It looks like we all agree then.

That is, using the revised definition of "all agree", where 1 is excluded.

__________________
Reality moves at the speed of light.
If the text of this post is blue, it's a "Moderator comment".
[ The RULES of the Forum ] [ Forum FAQs ] [ Conspiracy Theory advice ] [ Alternate Theory Advice ]
To report a post (even this one) to the moderation team - use the /!\ icon at the top-right of the post.

I think I've thought of a way to simplify this problem.

A set of numbers is defined. The definition conforms to the rules of mathematical logic. A subset is defined with a revision of the definition of the first set. The definition in the revision prohibits one of the numbers in the first set from being part of the second set. That is why the second set is a subset. All the rest of the numbers in the subset are the same as the larger set. The subset of that set is discovered to have a great deal of utility. It has so much utility that for convienence it is given the name of the larger set. The first set still exists. Nameless...homeless...for all time...Sniffle.

So this whole issue is not an issue of mathematics. It is an issue of the name. It is a semantic issue.

Close, but not quite.

In the case of Primes, the "subset" is considered to be the set. That is due to entirely mathematical reasons, as explained on page 1 (and page 2) of this thread, by multiple posters.

It's not just a case of giving the set of "Primes without 1" the name of "Primes"; it is the currently accepted meaning of Prime; again, for valid mathematical reasons.

It's not just semantics. The number 1 is no longer considered to be a Prime number - whatever we call the set of Primes.

__________________
Reality moves at the speed of light.
If the text of this post is blue, it's a "Moderator comment".
[ The RULES of the Forum ] [ Forum FAQs ] [ Conspiracy Theory advice ] [ Alternate Theory Advice ]
To report a post (even this one) to the moderation team - use the /!\ icon at the top-right of the post.

Actually, I think that it is really a question of semantics. Because the issue is how we define something. But I don't see any reason to sniff about it. In Greek the word "aster" was (I think) used to refer to all heavenly bodies. Now the word "star" in English is restricted to what we now understand to be a unified set, luminous stars. So the term is no longer used that way, and if want to refer to everything outside of the atmosphere we can say "celestrial objects" or something like that. So similarly, the word for the set with the primes and 1 was moved to the set without the 1 for a similar reason, only there is no real need to refer to the set plus 1. But it's nothing to mourn, it's just a word changing meaning.

Take a word like "salary," which was originally used to refer to the salt that soldiers were paid. Now we use it to mean any payment in general. I don't think anybody mourns the loss of the other meaning.

__________________
As above, so below

I don't think anyone dissagrees that the set of Primes used to include 1 but now doesn't; so that what specific meaning is conveyed by "Prime" has changed.

But post #1 of this thread tried to argue that the "simpler" definition^* of "Prime" (that includes 1) was "preferred". By extension, this implies that Specifically, post #1 argued that 1 should still be considered a Prime (my underline):

So it (this thread) is not just a matter of semantics, it's about what a Prime is.

Note aastrotech still (post #37) argues (my underline)...:

(^* In contrast, the consensus is that this "simpler" definition is wrong anyway - as it includes 1; which isn't a Prime.)

__________________
Reality moves at the speed of light.
If the text of this post is blue, it's a "Moderator comment".
[ The RULES of the Forum ] [ Forum FAQs ] [ Conspiracy Theory advice ] [ Alternate Theory Advice ]
To report a post (even this one) to the moderation team - use the /!\ icon at the top-right of the post.

I realise you were being funny, but in case anyone else is wondering, it isn't a prime. There are no primes in the real numbers, because anything factors anything else in the real numbers. The concept of primality of numbers exists in integer arithmetic. There are no fractions or decimals in the integers, so we don't have to worry about them. The rationals and reals respectively form a field - in a field every non-zero element has a multiplicative inverse, so every non-zero element is a unit, and none of them are primes.

For amusement, observe that 2 is neither prime nor irreducible in the complex integers:
2 = (1+i)(1-i) = (-i)(1+i)(1+i)

Actually, I offered evidence that the set of primes originally didn't include 1, going all the way back to Aristotle.

I gotta tell you, the humor in this post is downright beautiful.

__________________
SeanF

"Ask to understand, but don't challenge unless you have the knowledge."--NEOWatcher

The contents of this post are ©2009 by SeanF and may not be copied or retransmitted in any form without the express written consent of SeanF

I think it would make a good skit for Family Guy.

Where it ends with Chris calling everyone a poopy-head and running upstairs?

__________________
If they can get you asking the wrong questions, they don't have to worry about the answers.

oooo can I be Chris??

Incomplete. I said that by "the simple definition" under the terms of mathematical logic 1 is prime whatever "the consensus" currently, (fashionably) calls it.
I don't see how it's not an issue of semantics.

Does the simple definition for the set of numbers formerly called prince oops I mean prime, create a mathematical quandry that renders it mathematicly wrong to define a set of numbers that way?

Hey, wait a minute, have I discovered a name for the set of numbers formerly called prince? We could call them "princely numbers". Hey, I call dibs. I named it.

So would you agree that by the definition of numbers called prince that 1 is a princely number?

Hey, wait a minute, maybe that symbol Prince carried and wanted to be known as somehow means prime.

This question has been answered repeatedly.

__________________
Nothing beautiful was ever made from gravel.

The wiki page on primes, down to the paragraph quoted.

Hmmm, maybe this wiki needs some edits.... I wonder why it doesn't mention Aristotle?

Presumably because it was widely held that 3 was in fact the smallest actual number, 'the first one to have hight, width and length' (Theon Phil. util. math. 46.14.) The author of my quotation is as as late as 2nd Century AD. I have been trying, and failing, to understand the Greek concept of numbers for some time.

And apparently Nicomachus. They both seemed to have held 3 as the first prime number. Aristotle, 2 though.

Yes, I've just found the reference in Aristotle (Topica 157b1) and a page from Heath, History of Greek mathematics here which summarizes nicely.

But the "simple definition", if it includes 1, is wrong.

I understand why you prefer that defintion, and your point about 'simpler being better'... but a defintion is of no use if it gives the wrong result.

For valid mathematical reasons (explained in this thread) 1 is not considered to be a Prime number.

That is not "fashion", any more than the currently best-known value for pi. That is the current "new improved" mathematical understanding. So your simpler definition is incorrect.

In the end, your only argument so far for 1 being a Prime seems to be so that the definition of Prime can be simpler. That's circular.

[In fact, you essentially pointed out (post #23) that having 1 not be a Prime is useful in more "types of primes" than if 1 were Prime. So even by your "simpler is best" concept - 1 should not be considered Prime.]


If it were only semantics, we'd not be arguing whether 1 is a Prime or not. I don't care if you call them Primes, Prince or Apple pie. They don't include 1, and your "simpler defintion" does not work.

__________________
Reality moves at the speed of light.
If the text of this post is blue, it's a "Moderator comment".
[ The RULES of the Forum ] [ Forum FAQs ] [ Conspiracy Theory advice ] [ Alternate Theory Advice ]
To report a post (even this one) to the moderation team - use the /!\ icon at the top-right of the post.

pzkpfw,

Are you saying that the definition of a set of numbers "all whole numbers divisible only by the whole numbers of themselves and one to yield a product of whole numbers" doesn't contain in the set the number one? That the number 1 cannot be divided by itself? That defining a set of numbers this way will be called circular reasoning and fouled by a moderator?

No, I'm saying that 1 is not a Prime.

You can come up with what you think is a definition of Primes, and that definition may allow 1; but that is not a definition of Primes, because 1 is not a Prime.

For example, you can say that the definition of Prime numbers is:

...and yes, that set would contain 1.

But 1 is not a Prime, so that definition is flawed. (It may have been considered correct in, say, 1950, but not now.)

You can certainly define a set of numbers as you do above, and that set will contain 1 because 1 is divisible by 1. Just don't call that the set of Primes. (Or try to argue that because of your defintion that 1 is a Prime).

_{(A pedant might argue that the word "and" in your definition implies exclusivity, so that the number and 1 must be distinct values, but I'll give you this one.)}

----

I really don't get what it is that you are so nervous about here. If you think I'm wrong, say so, and try to prove it. No moderator will foul you for that. I've been telling you that for days now.

__________________
Reality moves at the speed of light.
If the text of this post is blue, it's a "Moderator comment".
[ The RULES of the Forum ] [ Forum FAQs ] [ Conspiracy Theory advice ] [ Alternate Theory Advice ]
To report a post (even this one) to the moderation team - use the /!\ icon at the top-right of the post.

Well, 0.99999999... = 1, so by definition, 0.99999999... is not prime. QED.

Rob

Well I think this discussion has reached a conclusion. I'm satisfied with the results. I think the issue has been clearly delineated. I don't know of anything more to add.

Thanks again, where warranted, for participating...reasonably.

Except the apology you owe me.

__________________
Nothing beautiful was ever made from gravel.

Maybe we should create another thread to discuss this in greater detail

(Runs and hides before anyone can throw a tomato).

__________________
Time flies like an arrow.
Fruit flies like a banana.

What is there to discuss? Are you claiming there is some debate on this point?

Let x = .999...

Then 10x = 9.999...

Remembering it is legal to subtract the same value from both sides of an equation:

10x = 9.999...
- x = .999...
-------------------
9x = 9

x = 1, QED.

What did you want to discuss about this?

Rob