You CAN define 1/0, but you have to give up other nice things to do so. Consider, for example, the Riemann Sphere, which is made out of the complex plane an an additional point called "infinity" (so with the right definition of distance (not the usual Euclidean distance of complex numbers), so infinity is a "finite" distance from 0 (!!!) it is metrically equivalent to a sphere) so that functions of the form (az+b)/(cz+d) (Mobius functions), for ad-bc != 0, are defined everywhere on the sphere. You are unable to compute some things, though, like infinity - infinity.

__________________
-----
Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

No theory or assumption based on them can be suspect unless there's disagreement on what set is meant by the name and all current mathematicians
understands primes of the natural numbers to exclude one.

This is basically because it's easier to refer to "the primes and one" in the few cases where that's relevant than it is to refer to "the primes excluding one" most of the time.

Once you get it into your head that names are a matter of convenience more than a matter of fitting some arbitrary definition of simplicity, this whole mess stops being an issue.

Unless of course it's all about you being unable to back up one single step and note that 1, 4, 9 are the squares of the first 3 natural numbers, which would have made that other thread about something interesting rather than the useless quibble this one is too.

__________________
‘To those who regard “crime fiction” as some sacred icon which must follow a rigid formula, I will always be the man who writes 18-syllable haiku.’Trying to make sense of computers, The Error Log.

And that's the way it has to be - you make infinity a number, you lose some properties of numbers.

I was wondering when someone would mention this. As it was a claim in the "marking the Moon thread" it had made me think of the monoliths in "2001 a space odyssey". I assumed that was the inspiration of the idea posted. According to the Wikipedia, the 1 4 9 ratio of the monoliths is the squares of the first three "integers".

This morning I dug out my copy of the book, as I wanted to see if back when it was written Arthur C. Clarke had used "integers" or "Primes" - and in it (my printing by Arrow, dated 1978 (I think)) he does use "integer".

__________________
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.

(My underline.)

(Full Wiki text: http://en.wikipedia.org/wiki/Fundame..._of_arithmetic)

An extension is not a contradiction, nor is it circular. It's no different than sometimes allowing 1 to be used as a prime number (which I don't think anyone has disputed can be done).

Are you now wanting to re-write the fundamental theorem of arithmetic as well as the standard definition of primes?

To force 1 to be normally considered a prime number, you'd also now have to force this '1 and -1 allowed as factors' extension of the fundamental theorem of arithmetic to be "normal". (And that now contradicts your original "simpler=preferred" contention.)


Oh my goodness! If it were so clear that exclusion of 1 from the set of primes would make any theorem that uses that set "suspect", why on Earth do you think 1 would have been excluded in the first place? Do you think mathemeticians just decided not to "like" 1 and threw it out of the set?


It should be quite clear by now that the standard defintion of prime number in current use excludes 1. Thus it was a fair comment. Instead of trying to argue that 1 is a prime and the standard definition is wrong, you could simply have noted that you prefer to use the older defintion.

Post #92 sums it up very well.

__________________
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.

From Penn and Teller on astrology.

The ridiculousness of this thread has just hit me. We're being trolled, of course. But the subject of the troll is the claim that 1 is a prime number. Wow.

Where else on the internet are you gonna get trolled on THAT subject??

Rob

I too thought of 2001. But if 1, 2 and 3 are the first three integers, what about 0, 1 and 2? Or -1, 0 and 1? Or -2003, -2002 and -2001? Or... (You get the idea/)

I believe he should have said "natural numbers". There's scope for a quibble there (is 0 a natural number?) but not as much of one.

__________________
Nothing beautiful was ever made from gravel.

Giving the OP the benefit of the doubt, this thread belongs in ATM, surely?

__________________
Nothing beautiful was ever made from gravel.

Actually there seems to be more scope for a quibble than for the primes case, as natural numbers including 0 aka ℕ₀ is used about as often as natural numbers without 0 aka ℕ^*, so there's no real convention about which set is meant by "natural numbers".

__________________
‘To those who regard “crime fiction” as some sacred icon which must follow a rigid formula, I will always be the man who writes 18-syllable haiku.’Trying to make sense of computers, The Error Log.

Indeed, although of course it would be less of a problem for monolith builders as you can't build a monolith that is 0 by 1 by 4. (I was going to put an exclamation mark* at the end of the sentence but realised it could be confused with 4-factorial.)

Incidentally, I recall from the book that the monolith was not merely 3-dimensional, so the sequence continued.

*Or "slipbanger" as they are called in Australia.

__________________
Nothing beautiful was ever made from gravel.

It is clear to me that the number 'one' is a prime number (since it is
not the product of other integers), but for convenience in certain cases
in number theory, an ad hoc addition was made to the definition to
specify that it is 'not a prime'. By nature, it is prime. By definition,
it is not prime.

-- Jeff, in Minneapolis

__________________
http://www.FreeMars.org/jeff/

"I find astronomy very interesting, but I wouldn't if I thought we
were just going to sit here and look." -- "Van Rijn"

"The other planets? Well, they just happen to be there, but the
point of rockets is to explore them!" -- Kai Yeves

I would put that as, by one definition, it is prime, by another definition (the one that most people use), it is not prime. I don't see one definition as more natural than the other.

a) Is 0.839 the product of other integers?
No.
So it's prime?
No, because by "other" you imply that the number in question is an integer.

b) So is 5 a product of other integers?
Yes. (1 and 5, and 1 isn't 5, so it's an "other" integer.)
So it's not prime?
No, because you of course expect us to already know the "1 and itself" part of the usual defintion, so "other" does not exclude 1 or itself.

c) So is -5 a product of other integers?
No (using b) ). (Just 1 and itself.)
So is -5 a prime?
No (not usually^*); you possibly meant "natural numbers" not "integers".

Now, I'm not meaning to critique your short-hand definition. I know of course you didn't intend that to be a complete or fully accurate description, and I'm not playing semantic games (though possibly showing that semantic quibbling is counter-productive).

My point here is simply that something in maths needs to be consistently understood so that all mathemeticians and ordinary folk like me can all understand the same thing from them.


Primes are a kind of number. The definition of the set of those numbers must accurately describe that set. The set does not exist because of the definition. We counted 1, 2, 3, ... long before we called them the set of "natural numbers". Currently (yes, opinion has changed) 1 is not a prime, thus the definition must exclude them.

How is 1 a prime by "nature"? By what standard is that judged? (The contention of the OP of this thread was that a definition that is "simpler" is "preferred". My counter the whole time has been that a "simpler" definition is not preferred if it is incorrect; and that besides - exclusion of 1 as a prime made more subsequent defintions "simpler" than it made more "complex", so even by the claim in the OP it is better that 1 not normally be considered prime.)

You say "for convenience" and "ad hoc addition". I would say "correction", "refinement" and "improvement".


---
A note on negative primes: http://primes.utm.edu/notes/faq/negative_primes.html

__________________
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.

But I think there are two reasonable ways to define a prime. One is a number that can only be divided by 1 and itself. The other definition could be a number that can be divided by exactly two numbers, 1 and itself. By the first definition, 1 is in, in the second definition, it is out.

__________________
As above, so below

Very nice reply. You didn't tear down my argument, but I think you
showed it to be weak. I can't tear down your argument, but I think
I can show it to be weak.

I was alluding to my post #82 in which I gave my definition:
I'm actually sufficiently ignorant of the subject not to know how
factoring of negative numbers is handled. If you can believe such
ignorance is possible. But it appears to be safely ignorable as long
as we stick strictly to the question of whether the number 'one' is
prime or not. So I recognize that it appears to be necessary to
restrict primes to positive numbers, but can't say exactly why.
My ignorance about this is obviously germaine to my argument, but
I hope it doesn't get in the way of my argument.

It seems obvious that my definition of prime numbers captures the
intended idea of what prime numbers are. If modern mathematics
chooses to tweak that definition, for whatever reason, it moves the
definition away from the intent. By eliminating 'one' from the set of
prime numbers, the modern definition deviates from the fundamental
concept of what a prime number is.

My definition bypassed the languge about "1 and itself", by design.
I consider that language superfluous. I think it was put there as a
clarification, rather than as an essential part of the definition.

When I say "other integers" I mean precisely "other integers".
Five is not a product of other integers.

If the discussion can be limited to natural numbers (positive integers,
as I said in my definition), then I will be much more comfortable in my
argument. Thank you! (This relates to my question earlier in the
thread as to whether prime numbers or negative numbers were the
first to be invented/discovered, and is suggested by a comment in
the link you provided.) I meant "integers", trying to be as inclusive
as possible. I think that "integers" works. Five is not the product
of any other integers, for example, including negative integers. But
you were responding to my shorthand, as you say, not my definition
which limited primes to positive integers.

No argument.

This really sounds like you are making my argument for me.

I think that my definition, that a prime number is a positive integer
which is not the product of other integers, captures the intended
meaning of "prime number". This definition, primitive and not the
currently universally-used definition, includes 'one' as a prime.

I suspect that the orginal poster used the idea of simplicity as the
closest idea to what he meant. I used the idea of naturalness.
Both are very subjective. Okay-- mine is horribly subjective.
I would not interpret his preference for simplicity too literally.

Which definition is "correct" in this case just means which definition
has been accepted by convention. So being correct doesn't mean
that it is the best definition or the one that people naturally came
up with when they started thinking about relationships between
different numbers.

If you are right that the convention of excluding '1' as prime made
more subsequent defintions "simpler" than it made more "complex",
then that shows the convention to have some utility. Perhaps
enough utility to justify it.

It departs from the essential meaning of what a prime number is.
It adds an artificial restriction to make certain theorems simpler.
'One' clearly would be considered the quintessential prime number
if not for the desire to simplify those theorems. 'One' is a positive
integer which is not the product of other integers. It is the prime
prime number in every way except by the modern definition.

-- Jeff, in Minneapolis

__________________
http://www.FreeMars.org/jeff/

"I find astronomy very interesting, but I wouldn't if I thought we
were just going to sit here and look." -- "Van Rijn"

"The other planets? Well, they just happen to be there, but the
point of rockets is to explore them!" -- Kai Yeves

To me it is obvious that the natural definition is "A positive integer
which cannot be evenly divided by any other integers."

Following your argument, a "second" would be "A number that can
be divided by exactly three numbers." A 'tertiary" must be "A number
that can be divided by exactly four numbers." And so forth.

The essential idea of "prime numbers" is not that they ARE divisible
by a limited set of numbers, but that they are NOT divisible into
other integers. It is their indivisibility which makes them prime.

-- Jeff, in Minneapolis

__________________
http://www.FreeMars.org/jeff/

"I find astronomy very interesting, but I wouldn't if I thought we
were just going to sit here and look." -- "Van Rijn"

"The other planets? Well, they just happen to be there, but the
point of rockets is to explore them!" -- Kai Yeves

Part of why the currently used definition of prime numbers excludes 1 is that it is even "deeper" than the 'the intended meaning of "prime number"'. 1 and -1 are "units" in the set of integers. You could argue that being a unit shouldn't preclude it also being called a prime, but that would still ignore the other reasons not to consider 1 a prime.


I'm still not convinced by your "essential meaning" of what a prime is. That a prime number can only be produced by multiplying 1 and the number itself is a very specific thing. Multiplying 1 by itself seems very much the odd or trivial case; the exception.

That 5, 7 or 2^43112609-1 are prime is quite different to "1 is a prime".


...it's a "unit". (insert smiley)

__________________
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.

If the pronoun 'it' refers to "the currently used definition", then you
are saying that the definition actually defines something "deeper"
than prime numbers, rather than prime numbers. And I agree.

Nobody has given any reason for a unit not to be a prime, so the
"other reasons" are the only reasons. And those other reasons are
apparently that number theorists consider it to make some of their
theorems simpler.

Multiplying a number by 1 doesn't produce anything! It leaves the
number unchanged! It is an instruction to do nothing!

Yes. Multiplying any number by 1 is trivial. It is a non-operation.

The number one is super-prime. It is more prime than any other
integer. Its primacy among numbers is matched by its indivisibility,
which makes it the first prime number.

Yes, it is a unit. So?

-- Jeff, in Minneapolis

__________________
http://www.FreeMars.org/jeff/

"I find astronomy very interesting, but I wouldn't if I thought we
were just going to sit here and look." -- "Van Rijn"

"The other planets? Well, they just happen to be there, but the
point of rockets is to explore them!" -- Kai Yeves

But multiplying 1 by another number isn't.

If you have five apples and I say, just keep them - that's like a multiply by one, essentially "do nothing". But if instead you have one apple and I say take 5 as many; then you've multiplied that 1 by 5. (And you'll find that 5 can't be factored into anything but 1 and 5 - it's prime).

If you have one apple and I say, just keep it - that's like a multiply by one, essentially "do nothing". But if instead you have... um... welll... the reverse case is just the same. Again a trivial nothing. (And yes you'll find that 1 can't be factored into anything but 1 and, ... well, 1. It's not the same case.).

So I stand by the comment that 1 as a prime is quite different to all the other primes, and that the defintion of 1 as "unit" and not "prime" has a specific relevance in maths.

__________________
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.

Jeff, you seem to be thinking of primes in terms of what they are not. Think of them instead as building blocks of other numbers.

As has already been explained, primes "make" other numbers by raising them to integer powers.

For instance, 2 makes 1, 2, 4, 8, 16, 32 and so on by raising it to the power of 0, 1, 2, 3, 4, 5 and so on respectively. 3 makes 1, 3, 9, 27 and so on. 5 makes 1, 5, 25, 125 and so on. And every number that isn't a prime is a product of one or more of these factors.

Whereas 1 makes 1 and nothing else, no matter what positive power you raise it to. In other words, it's the number that doesn't do the building that is required of primes.

1 is often described as "neither prime nor composite".

I liked an earlier post which likened prime numbers to the periodic table of elements. Bear in mind that "elements" used to mean earth, air, fire and water. Now it doesn't, and water does not appear on the periodic table of elements. There may well have been people moaning about the wrongness of water being excluded - "Come on, water is the element! It's the basis of life and seven tenths of the world's surface is covered in it! What could be more elemental than that? Your exclusion of it is just semantic."

But there's no benefit to undoing progress.

__________________
Nothing beautiful was ever made from gravel.

Multiplying by 5 increases the number of apples. But it does not
produce a number. Multiplying any number by 1 does not produce
a number, and multiplying the number '1' by any number does not
produce a number. Multiplying the number '1' by the number '5'
results in the number '5', which is one of the two numbers you
started with, so nothing has been produced. You cannot produce
the number '5' or any other prime number by multiplying two integers
together. That is the quality that makes them prime.

I question that the status of the number '1' as a 'unit' has any
relevance to its status as a prime. Okay, '1' is a 'unit'. What does
that have to do with it being or not being a prime?

-- Jeff, in Minneapolis

__________________
http://www.FreeMars.org/jeff/

"I find astronomy very interesting, but I wouldn't if I thought we
were just going to sit here and look." -- "Van Rijn"

"The other planets? Well, they just happen to be there, but the
point of rockets is to explore them!" -- Kai Yeves

Er, yes it does, Jeff.

Yes it does.

Not sure what you're trying to argue here.

__________________
Nothing beautiful was ever made from gravel.

This whole discussion is yet another example of people confusing the simplified explanation they where taught in school¹ to introduce them to the subject, with understanding the subject in depth.

The set of units is the generator of all numbers by addition, the set of primes is the generator of all numbers by multiplication.

This is a deeper definition of primes that steps away from looking at natural numbers or integers and instead looks at the deeper concepts behind what numbers, addition and multiplication "really" is.

In this definition, 1 isn't prime.

1) I'm partial to the lies-to-children description

__________________
‘To those who regard “crime fiction” as some sacred icon which must follow a rigid formula, I will always be the man who writes 18-syllable haiku.’Trying to make sense of computers, The Error Log.

I prescribe reading a book on, or taking a course in, basic factorisation theory in rings. You'll soon have a very strong sense that 1 isn't prime.

That is a concept which can be applied to almost any subject.

The math, chemistry, physics, history, language, etc. majors I knew at the university all said they had learn how to abandon the 'facts' they had been taught at school, the recitation of which would invariably earn them a weary look from their professor.

We always got a kick out of listening to the theologians talk about Hebrew grammar. After spending years doing comparative Semitic linguistics, we knew that what we called 'theologian grammar' was sooooo off base. It was overly simplified and had answers for everything, even if there weren't any.

__________________
Ach, mein Sinn, wo willst du endlich hin, wo soll ich mich erquicken? Bleib' ich hier, oder wünsch' ich mir Berg und Hügel auf den Rücken?
Bei der Welt ist gar kein Rat, und im Herzen steh'n die Schmerzen meiner Missetat, weil der Knecht den Herrn verleugnet hat.

Multiplying positive integers together produces composite numbers.
No composite number is produced by multiplying 1 and 5 together.

-- Jeff, in Minneapolis

__________________
http://www.FreeMars.org/jeff/

"I find astronomy very interesting, but I wouldn't if I thought we
were just going to sit here and look." -- "Van Rijn"

"The other planets? Well, they just happen to be there, but the
point of rockets is to explore them!" -- Kai Yeves

No.

1 x 5 = 5

1 and 5 are both integers.


Not always.

1 x 5 = 5

1 and 5 are both integers, 5 is not a composite number.


True.

But 5 is a number. Not a composite number, but it is a number.

__________________
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.

No number was produced. You just kept the number you already had.

Again, no number was produced.

Yes, but it isn't produced by the multiplication. It is a number that
you already had and knew before the multiplication.

-- Jeff, in Minneapolis

__________________
http://www.FreeMars.org/jeff/

"I find astronomy very interesting, but I wouldn't if I thought we
were just going to sit here and look." -- "Van Rijn"

"The other planets? Well, they just happen to be there, but the
point of rockets is to explore them!" -- Kai Yeves

If you have 1 apple and I multiply that by 5 for you, you end up with 5 apples.

Why would it matter you knew I had 5 (or more) apples to use?

Do we need a new thread to argue what multiplication is?

__________________
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.