1 is not prime. Really?

Retro

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I have never agreed with the definition that 1 is not prime as the reason seems arbitrary to me, done for convenience, nothing more.

Before we get into it, here's a quick clarification: an integer is the same thing as a whole number and hence the terms are used interchangeably in this post.


The core characteristic of any prime number is that it's an integer which divides evenly by the integers of 1 and itself. That's it. That's true for primes 2, 3, 5, 7, 11 etc and that's true for 1 as well: 1 x 1 = 1.

The article below explains why 1 is not defined as a prime since defining it as a prime leads to an infinite number of ways to define numbers as the result of multiplying smaller integers together. Not defining it as prime prevents this. So, with 1 not prime, the divisors of 6, for example, are 2 x 3. Make 1 prime and it could be 2 x 3 x 1, 2 x 3 x 1 x 1, 2 x 3 x 1 x 1 x 1 and so on forever, all with the same result of 6. Hence, from the article, the formal definition is:

A prime number is a positive integer that has exactly two distinct whole number factors (or divisors), namely 1 and the number itself.

However, I think this argument is flawed since one can make similar arguments in other cases. For example:

Zero can't be a number, because adding zero to a number can be done an infinite number of times without changing the result. So, for 8, we have 8 + 0, 8 + 0 + 0, 8 + 0 + 0 + 0...

Similarly, prefixing any number with zero doesn't change its value. So, prefixing 8 leads to 08, 008, 0008... again with all these having the same value of 8.

Hence saying zero can't be a number would be absurd.

We can pick on 1 again for the same reason in multiplication, since 8 x 1, 8 x 1 x 1, 8 x 1 x 1 x 1... all have the same result of 8. So, to prevent this, 1 cannot be a number. This is clearly absurd too.


So, back to the reason that 1 is not officially defined as prime. Since there are an infinite number of ways to write a number as the product of three or more whole numbers when 1 as prime is included, simply leave that result out. This is done in many areas of maths where infinity pops up with recurring numbers, so why should it be any different here?

 

TheURL

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It's a bit arbitrary - there are only no negative prime numbers because we say there aren't. But in principle -43 is no more or less "prime" than +43. The 5 constants in Maths (and yes there's an S in the word Maths because it's an abbreviation of Mathematics, which is plural! :)) - 0, 1, e, pi and i. So we have to accept that 0 and 1 will present us with problems that can only be settled by consensus.
 

Retro

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Arbitrary indeed, nice example with the negatives.

(and yes there's an S in the word Maths because it's an abbreviation of Mathematics, which is plural! :))
You've opened up a can of worms there! :p

It seems to mainly be a cultural thing between the UK and America. Also, the argument that the S is there since mathematics is plural unfortunately doesn't work as the Numberphile video explains. Having said that, I prefer saying maths too.

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Here's some other takes on this:


 

TheURL

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OK I accept that "mathematics" is not a plural noun. But another argument for using "maths" that the video did not touch on was "liaison". The word "maths" is usually followed by other words in a sentence. In French, "liaison" refers to the phonetic phenomenon where a normally silent final consonant is pronounced when followed by a word that begins with a vowel sound. This results in a smooth transition between words in spoken French. The word "math" ends abruptly and clumsily, and independent of technical correctness, "maths" is the convention of choice. When I did maths at university no lecturer ever said "math" throughout the whole three years I was there. And this was a British university in London, about a mile from where the Queen lived. The Queen spoke the Queen's English (though even she made the odd mistake in one or two of her Christmas speeches). So in conclusion, there is an argument for it being math, but it's not it's maths. :)
 

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I agree that maths is smoother and the convention used in the UK as per your example, but to say that one is wrong and the other right is incorrect since they have the exact same meaning.
 

TheURL

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Well I didn't say that "Math" was "wrong" but that in the UK "Maths" was "the preferred".
 

TheURL

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Now then - what about people who pronounce it "medcin" when it's spelt "medicin"?!
 
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