Monday Math: Why 1 Is Not a Prime
Jan. 21st, 2013 05:01 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowI'm currently rereading Jim Butcher's Turn Coat. There is a scene, early on, in which Dresden has a run-in with a naagloshi, a skinwalker out of Navajo legend. The creature is so frightful that seeing it through a wizard's Sight reduces him to gibbering terror; in his struggle to regain control over himself, Dresden begins calculating prime numbers. (And pretty impressively; by the time he reaches a place of temporary sanctuary, he's well into the 2000s.) Unfortunately - to a mathematician's eye - he begins with the number 1.
There may have been a time when mathematicians regarded 1 as a prime number. That time is long past. This may seem confusing, since 1 does meet the condition we're taught in school - that of not being divisible by any number except 1 and itself. However, like much that is taught in school, that criterion has long since been discarded. I'll talk about why, and about various related things, under the cut.
The first thing to remember is that mathematical notation and terminology is designed for the convenience of mathematicians. If a different definition proves to make theorems easier to state and to prove than another, then that definition will probably be adopted. Here, it is one of the most important theorems of number theory that gets the casting vote, the Fundamental Theorem of Arithmetic. It dates back to Euclid, and it seems apropos to mention his definition, in the Heath translation: A prime number is that which is measured by an unit alone. Given the geometric orientation of Euclid's work, it is natural that he uses "measured" where we would say "divided"; but note that he does not have the number dividing itself. The definition seems to make 1 a prime number, but it doesn't. Here's Euclid's definition of a number: A number is a multitude composed of units. For Euclid, 1 is not a prime number because 1 is not a number.
Of course, modern mathematicians disagree; 1 is most certainly a number. It still isn't classed as a prime number, and the language of the Fundamental Theorem of Arithmetic may make clear why this is so. The usual version is a bit more general than this, but for our purposes this is good enough: Any positive whole number can be written as a product of prime numbers, and this representation is unique except for rearrangement of the factors.
So why do we exclude 1 from our list of prime numbers? Because if we count 1 as prime, the representation is not unique: the product 1x2x3 equals the product 2x3, and these are not rearrangements. We'd have to change the theorem: except for rearrangement of the factors and the possible inclusion of 1's. By not counting 1 as a prime number, we drop that final clause, making the theorem simpler - and that's a good enough reason. There's more to it than that, but I'll leave that to the next post on this subject, next week.
By the way, you may be bothered by the first claim in the theorem. 1 is a positive whole number; how can we write it as a product of prime numbers? Let's start by looking at some noncontroversial examples. 6 is the product of the list of prime numbers {2,3}, and 14 is the product of the list {2,7}. Now, 6x14=84 is the product of the concatenation of these lists, {2,3,2,7}. In general, if m is the product of one list and n is the product of another list, mn is the product of the concatenation. So, now, 6=6x1; hence the list yielding 6 (on the left side) should be the concatenation of the list giving 6 (on the right) and the list giving 1. Clearly, the only way to make this work is to decree that 1 is the product of the empty list {}. Similarly, we decree that a prime number like 5 is the product of the one-element list {5}. (Remember, notation is a tool, designed for our convenience; if we want to say that the product of the empty list is 1, we are free to do so, and it is sufficiently convenient to do so that pretty much all mathematicians will accept this decision.)
There may have been a time when mathematicians regarded 1 as a prime number. That time is long past. This may seem confusing, since 1 does meet the condition we're taught in school - that of not being divisible by any number except 1 and itself. However, like much that is taught in school, that criterion has long since been discarded. I'll talk about why, and about various related things, under the cut.
The first thing to remember is that mathematical notation and terminology is designed for the convenience of mathematicians. If a different definition proves to make theorems easier to state and to prove than another, then that definition will probably be adopted. Here, it is one of the most important theorems of number theory that gets the casting vote, the Fundamental Theorem of Arithmetic. It dates back to Euclid, and it seems apropos to mention his definition, in the Heath translation: A prime number is that which is measured by an unit alone. Given the geometric orientation of Euclid's work, it is natural that he uses "measured" where we would say "divided"; but note that he does not have the number dividing itself. The definition seems to make 1 a prime number, but it doesn't. Here's Euclid's definition of a number: A number is a multitude composed of units. For Euclid, 1 is not a prime number because 1 is not a number.
Of course, modern mathematicians disagree; 1 is most certainly a number. It still isn't classed as a prime number, and the language of the Fundamental Theorem of Arithmetic may make clear why this is so. The usual version is a bit more general than this, but for our purposes this is good enough: Any positive whole number can be written as a product of prime numbers, and this representation is unique except for rearrangement of the factors.
So why do we exclude 1 from our list of prime numbers? Because if we count 1 as prime, the representation is not unique: the product 1x2x3 equals the product 2x3, and these are not rearrangements. We'd have to change the theorem: except for rearrangement of the factors and the possible inclusion of 1's. By not counting 1 as a prime number, we drop that final clause, making the theorem simpler - and that's a good enough reason. There's more to it than that, but I'll leave that to the next post on this subject, next week.
By the way, you may be bothered by the first claim in the theorem. 1 is a positive whole number; how can we write it as a product of prime numbers? Let's start by looking at some noncontroversial examples. 6 is the product of the list of prime numbers {2,3}, and 14 is the product of the list {2,7}. Now, 6x14=84 is the product of the concatenation of these lists, {2,3,2,7}. In general, if m is the product of one list and n is the product of another list, mn is the product of the concatenation. So, now, 6=6x1; hence the list yielding 6 (on the left side) should be the concatenation of the list giving 6 (on the right) and the list giving 1. Clearly, the only way to make this work is to decree that 1 is the product of the empty list {}. Similarly, we decree that a prime number like 5 is the product of the one-element list {5}. (Remember, notation is a tool, designed for our convenience; if we want to say that the product of the empty list is 1, we are free to do so, and it is sufficiently convenient to do so that pretty much all mathematicians will accept this decision.)
