One and Prime Numbers.

[Jim_B]

Why isn’t the number one (1) considered a prime number?

Prime numbers are any number not perfectly divisible by any other number, except themselves, and one.

I get the idea. One is perfectly divisible by one. But isn’t that just unavoidable? It has to be divisible by one, because it is one.

Also, are prime numbers universal? I mean, would an alien on another planet think one is not a prime number? Or would he?

:):)

[Andy_L]

It’s somewhat arbitrary - there have been mathematicians who considered 1 to be prime. However, it’s convenient to be able to state that all integers are the product of a unique set of prime factors - and if you allow 1 to be prime, you can’t say that (6 would equal 321 or 32111 or an infinite number of other decompositions). See here for more Fundamental theorem of arithmetic - Wikipedia

[Thudlow_Boink]

To expand on what Andy L said, from Wolfram Mathworld:

The number 1 is a special case which is considered neither prime nor composite (Wells 1986, p. 31). Although the number 1 used to be considered a prime (Goldbach 1742; Lehmer 1909, 1914; Hardy and Wright 1979, p. 11; Gardner 1984, pp. 86-87; Sloane and Plouffe 1995, p. 33; Hardy 1999, p. 46), it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own. A good reason not to call 1 a prime number is that if 1 were prime, then the statement of the fundamental theorem of arithmetic would have to be modified since “in exactly one way” would be false because any n=n·1. In other words, unique factorization into a product of primes would fail if the primes included 1. A slightly less illuminating but mathematically correct reason is noted by Tietze (1965, p. 2), who states “Why is the number 1 made an exception? This is a problem that schoolboys often argue about, but since it is a question of definition, it is not arguable.” As more simply noted by Derbyshire (2004, p. 33), “2 pays its way [as a prime] on balance; 1 doesn’t.”

Primes are the building blocks of numbers in much the way elements are the building blocks of chemical compounds: just as water is H[sub]2[/sub]O, 50 is 2*5[sup]2[/sup].

If it helps, you can think of 1 as “invisible,” a “do-nothing” when it comes to multiplication and division (which is what primes have to do with). It’s not a prime; it’s not a composite; it’s nothing. (Mathematically speaking, it’s the identity.)

[ftg]

@Jim_B

Also, are prime numbers universal? I mean, would an alien on another planet think one is not a prime number?

The beauty and simplicity of excluding 1 as a prime for such things as the Fundamental Theorem of Arithmetic strongly suggests that anyone with a human-like intelligence would favor 1 is not a prime. It really helps a lot.

Otherwise you’d be saying “All integers 2 or greater have a unique factorization into primes other than 1.” And this “other than 1” would be repeated on page after page of their version of Hardy and Wright.

But maybe some reasonably civilized group of aliens just didn’t care about number theory and primes. They focused on real numbers, calculus, etc. So they never really thought about it. Or maybe they considered 1 a prime so early on that it would have been culturally difficult to change things when they realize later it was a mistake. (E.g., we really, really should have had zero as a natural number right away. It would have made a lot of things easier.)

Note that there are other algebras where there’s a concept of primeness. What if the aliens had quickly jumped to using one of these? But even then for anything useful in general, 1 shouldn’t be a prime. (Although several primes become composite.)

[Hari_Seldon]

It is also related to the fact that the product of the empty set of numbers is 1 and that the zeroth power of every positive number is 1. That convention is required for various formulas to hold. There are deeper reasons, but they all point in the same direction: 1 is not prime.

[Blue_Blistering_Barnacle]

This has been helpful to me. Like OP, I’ve often felt that excluding “1” from primes seemed rather arbitrary.

[Chronos]

Or maybe they considered 1 a prime so early on that it would have been culturally difficult to change things when they realize later it was a mistake.

Sort of like our situation with pi. The ratio of circumference to radius (sometimes called “tau”, 6.283…) is a much more natural quantity to work with, since the radius is a much more natural description of a circle than diameter. But pi is so firmly established now that it’s going to be awfully hard to change.

[friedo]

@Hari_Seldon

[T]he product of the empty set of numbers is 1 and… the zeroth power of every positive number is 1.

Aren’t those two statements equivalent?

[markn_1]

@Chronos

Sort of like our situation with pi. The ratio of circumference to radius (sometimes called “tau”, 6.283…) is a much more natural quantity to work with, since the radius is a much more natural description of a circle than diameter.

Eh, there are arguments for tau and there are equally (or to my mind, more) compelling arguments for pi, especially when you leave the highly restricted topic of geometric circles and look at how pi (or tau) is used elsewhere in mathematics.

Of course, the logical solution is a compromise.

[glowacks]

@markn_1

Eh, there are arguments for tau and there are equally (or to my mind, more) compelling arguments for pi, especially when you leave the highly restricted topic of geometric circles and look at how pi (or tau) is used elsewhere in mathematics.

Of course, the logical solution is a compromise.

I don’t know if you intended this, but that URL attempting to explain the pi side of the argument doesn’t lead anywhere interesting. I might say that such a fact sums up most of the “arguments” in favor of pi as far as I’m concerned. That’s not a debate that should be in this thread though, but I would be interested if people have attempted to logically defend pi and pry it off that multiple of 2 that it’s so attracted it.

[markn_1]

I’m not sure what you mean. This is the URL:
http://www.thepimanifesto.com/

Are you saying you can’t see that page, or you just don’t agree with the arguments?

[Chronos]

Half of the Pi Manifesto is devoted to the fact that the area formula for a circle, using pi, is A = pir^2. But as soon as you state that formula, you’ve already conceded the argument, because from a pi-ist point of view, the area formula is instead A = piD^2/4. At first blush, this looks about as bad as the tau-ist formula A = tau*r^2/2. But the tau-ist formula is superior, because the factor of 1/2 actually makes sense in that formula: It’s the same factor of 1/2 that shows up in the formula for the area of a triangle.

[Trinopus]

Also, it means there would be lots of ways of factoring any number…

5 = 51 = 511 = 5111 etc.

[Hari_Seldon]

@friedo

Aren’t those two statements equivalent?

Of course.

[ftg]

@Trinopus

Also, it means there would be lots of ways of factoring any number…

5 = 51 = 511 = 5111 etc.

See the second post.

[Trinopus]

@ftg

See the second post.

I missed that. (Why do you bother posting something like this?)

[ftg]

@Trinopus

I missed that. (Why do you bother posting something like this?)

A lot of people here point out these sort of things. I’ve been on the receiving several times myself. I find it to be a useful reminder to try and be more careful in the future.