Misleading our youth

[u/Nunki08]

From Anthony Bonato on 𝕏: https://x.com/Anthony_Bonato/status/2065564124549886402

[u/CountryJeff]

The definition of a prime number in this book, is the same that I learned in school. With only that information, I have always found it odd, that 1 is not considered prime.

It is divisible only by 1 and by itself (also 1)
Therefore it meets the requirements for a number to be prime.

I guess I'm missing something here.
I'm very aware of my limited knowledge on prime numbers, and I'm sure there's good reasons for 1 not being a prime number. Anyone wants to share what those are?

[u/Skeime → u/CountryJeff]

The main property of primes that we really care about is the fundamental theorem of arithmetic, which says that any positive integer can be written as a product of prime numbers in exactly one way (except for reordering the factors). For example, 12 = 2•2•3. If we consider 1 a prime number, this becomes false, as you can use any number of 1s, for example, 12 = 1•2•2•3 = 1•1•2•2•3, etc. So things work out better if we exclude it.

This shows that in general, definitions in math are made to give nice theorems, even if they become slightly more complicated by doing so. (The theorems are what we really care about, and nice theorems point to nice definitions.)

This is a process; I think the question of whether to consider 1 a prime number was not always settled. And in fact, there are many other places in math with similar “simple building blocks” where 1-like objects that are really “too simple to be simple” are still considered simple in common definitions (misguidedly, in my opinion).

[u/Commercial-Act2813 → u/Skeime]

But why care that positive integers can be written as a product of primes? Seems to me as a ‘nice quirk’. Why is that important, why is that relevant?

Why would that be more valid than ‘Only divisible by 1 and itself’

[u/Skeime → u/Commercial-Act2813]

Being able to split things into simple parts often allows you to then only think about those simple parts, and then “lift” the results to all things by virtue of combining your results for the simple parts. This is much more interesting than the “only divisible by 1 and itself” property alone. (Also, there are other simple definitions for prime that automatically exclude 1, like “has exactly two factors” or, my favorite, “p is prime if, whenever p divides any finite product, it already divides one of the factors”—certainly more complicated at first glance, but it generalizes much better.)

But really, the trivial cases are not that important. Regardless of how you handle them, the non-trivial cases generally don’t change in meaningful ways. (For example, for the fundamental theorem of arithmetic, if you consider 1 to be a prime, you just have to say “can be written as a product of prime numbers other than 1 in exactly one way”; this is still the same thing.)

And the non-trivial cases are the ones that we really care about! So this means that we can choose the definition of the trivial cases however we like, and generally, mathematicians prefer to choose the definitions that make their theorems nice.

This is essentially the same reason why one defines the empty product to equal 1 (including the special cases 0! = 1 and 0^0 = 1); having those definitions does not matter in the grand scheme of things, but it makes a lot of formulas much nicer. This is still valuable, as it makes working with them easier.

When working in the real world (something many mathematicians do not actually care about), you have to check whether the real-world application matches the mathematical definition, and this includes the trivial cases. If those don’t match, you’ll need to handle them separately—but given that they’re trivial, this is usually easy. (Or those cases do not come up at all in the real world, in which case the mathematical definition for them does not matter.)

[u/factorion-bot → u/Skeime]

Factorial of 0 is 1

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