It is a very
common misconception that the number 1 is a prime number. If a prime number is
a number with factors only of itself and 1 then why is 1 not a prime number? 1x1
is the only factor of the number 1 so therefore the number 1 must be a prime
number. Many mathematicians in the past even believed 1 to be prime. This is
wrong. The real definition of a prime number is “An integer greater than one is
called a prime number if its only positive divisors (factors) are one and
itself.” (Caldwell). The key words in that phrase are “greater than one”. But
this does not answer the question why 1 is not a prime number.
There exists a
really important theorem, by Euclid, in mathematics that can help us prove that
1 is not prime, the fundamental theorem of arithmetic. It states, “every
positive whole number can be written as a unique product of primes”. (Grime)
This theorem helps us understand that primes are the building blocks for all
numbers. Lets take the number 15 as an example. 15= 3x5. 3 and 5 are prime
numbers. Now look back at the theorem, there is a key word in understanding the
theorem, and that word is unique. Since it has to be a unique product of
primes, that means there is only one way in which we can write the product. Now
if 1 was a prime number we could state that 15= 3x5x1, we could also write it
as 3x5x1x1, or we could write it as 3x5x1x1x1. If one were a prime number there
would not be unique ways in which to write positive whole numbers as products
of primes.
The number 1 is a
unit. Euclid states in his VII book of
elements definition number 1. “A unit
is that by virtue of which each of the things that exists is called one”
(Heath, 277). This definition of a unit is quite hard to comprehend. Heath
helps us with the understanding of each definition through further explanation.
“The etymological signification of the word μovás (Greek for the word unit) is
supposed by Theon of Smyrna (p. 19, 7-13) to be either (1) that it remains
unaltered if it be multiplied by itself any number of times, or (2) that it is separated
and isolated from the rest of the multitude of numbers.” (Heath, 279). Even
though many people once believed 1 to be a prime, it is the significance of
units in modern mathematics that causes mathematicians to be much more careful
with the number 1.
Caldwell, Chris K.
"Why Is the Number One Not a Prime?" Why Is the Number One Not a
Prime? N.p., n.d. Web.
Heath, Thomas L., Sir. The Thirteen Books of The
Elements. Vol. 2. Toronto: General, n.d. Print. (Books III-IX).
This is just getting really technical now, which is not a bad thing. This is a pretty big misconception everyone has because I have never heard anyone refer to 1 as a "unit." Does it really matter though that 1 is or is not a prime number? What purpose does it serve to say that 1 can't be a prime? If it's just to be technically correct, then it does its job well!
ReplyDeleteI thought that it is very interesting to read behind the theorem of this concept of 1 not being a prime number. I never knew that the number had such an effect on other numbers. Euclid's theorem is very interesting about 1 not being a prime number through this theory that states that every positive whole number can be written as a unique product of primes.
ReplyDeleteIn mathematics, 1 being a prime number is rarely discussed. I thought your summary was well explained. I also learned that number 1 is the unit of the positive integers, which has a multiplicative inverse. It never came to mind to think that 1 is not considered a prime number and why mathematicians really inspected this theorem.
ReplyDeleteIt's always been said how 1 is a prime number but after reading this, technically it isn't. It breaks it down on Euclid's theorem which makes it confusing to understand. There isn't much of a big deal of this situation as it doesn't do much in the field of mathematics unless higher up mathematicians use this theorem to prove other unsolved problems in math.
ReplyDeleteI have always been under the impression that the number 1 was prime. After reading and analyzing your post, I now recognize the number 1 not as a prime, but as a unit. I enjoyed learning how the number 1 can't be recognized as a prime due to the special characteristics it lacks that other prime numbers possess. Great Summary!
ReplyDeletei must also say that i always believed 1 to be a prime number. the concepts of euclid and caldwell make sense.
ReplyDeleteI also thought 1 was a prime number. I now know it isn't based on your topic. I liked how you explained Euclids theorem and why it is not prime. You did a good job explaining how 1 is not considered to be a prime number.
ReplyDeleteI've always thought that the number one was prime. Now I realize that it's not and know why it's not a prime. It all makes sense now.
ReplyDeleteAfter reading this post a couple times I am still a little confused. I had normally just thought of 1 as a normal number and could be treated the same as every other number. However I am not nearly as smart as the mathematicians you noted in this post, so I will have to trust what they are saying.
ReplyDeleteWhat I don't really understand is the is the notion that mathematicians are to be more careful with one since it's not a prime number. The fact that one is not a prime number doesn't change the quantitative property of it. Nevertheless, it is kind of cool to see the actual reasoning behind the fact that one isn't a prime number.
ReplyDeleteWhile I understand the reasoning behind the argument, I just don't get why the Theorem itself is so important
ReplyDeleteWow I thought one was a prime number until I read this blog. You did a good job on explaining why it wasn't in your blog.What helped me understand it was when you mentioned that 1x1 is the only factor of the number 1 so the number 1 can't be a prime number. That was pretty interesting and I thought you had a good summary as well.
ReplyDeleteAlthough I did not know that 1 was a "unit" I do believe that I have been told that 1 is not a prime number. Even though I have been told that 1 is not a prime number, I never knew why. This was a really good explanation, and I liked the mathematical definitions that you provided.
ReplyDelete