+8261 The number one is far more special than a prime! It is the unit (the building block) of the positive integers, hence the only integer which merits its own existence axiom in Peano's axioms. It is the only multiplicative identity (1.a = a.1 = a for all numbers a). It is the only perfect nth power for all positive integers n. It is the only positive integer with exactly one positive divisor. But it is not a prime. So why not? Below we give four answers, each more technical than its precursor. If this question interests you, you might look at the history of the primaility of one as described in the papers "What is the smallest prime?" [CX2012] and "The History of the Primality of One: A Selection of Sources" [CRXK2012]. Answer One: By definition of prime! The definition is as follows. An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself. Clearly one is left out, but this does not really address the question "why?" Answer Two: Because of the purpose of primes. The formal notion of primes was introduced by Euclid in his study of perfect numbers (in his "geometry" classic The Elements). Euclid needed to know when an integer n factored into a product of smaller integers (a nontrivially factorization), hence he was interested in those numbers which did not factor. Using the definition above he proved: The Fundamental Theorem of Arithmetic Every positive integer greater than one can be written uniquely as a product of primes, with the prime factors in the product written in order of nondecreasing size. Here we find the most important use of primes: they are the unique building blocks of the multiplicative group of integers. In discussion of warfare you often hear the phrase "divide and conquer." The same principle holds in mathematics. Many of the properties of an integer can be traced back to the properties of its prime divisors, allowing us to divide the problem (literally) into smaller problems. The number one is useless in this regard because a = 1.a = 1.1.a = ... That is, divisibility by one fails to provide us any information about a. Answer Three: Because one is a unit. Don't go feeling sorry for one, it is part of an important class of numbers call the units (or divisors of unity). These are the elements (numbers) which have a multiplicative inverse. For example, in the usual integers there are two units {1, -1}. If we expand our purview to include the Gaussian integers {a+bi | a, b are integers}, then we have four units {1, -1, i, -i}. In some number systems there are infinitely many units. So indeed there was a time that many folks defined one to be a prime, but it is the importance of units in modern mathematics that causes us to be much more careful with the number one (and with primes). Answer Four: By the Generalized Definition of Prime. (See also the technical note in The prime Glossary' definition). There was a time that many folks defined one to be a prime, but it is the importance of units and primes in modern mathematics that causes us to be much more careful with the number one (and with primes). When we only consider the positive integers, the role of one as a unit is blurred with its role as an identity; however, as we look at other number rings (a technical term for systems in which we can add, subtract and multiply), we see that the class of units is of fundamental importance and they must be found before we can even define the notion of a prime. For example, here is how Borevich and Shafarevich define prime number in their classic text "Number Theory:" An element p of the ring D, nonzero and not a unit, is called prime if it can not be decomposed into factors p=ab, neither of which is a unit in D. Sometimes numbers with this property are called irreducible and then the name prime is reserved for those numbers which when they divide a product ab, must divide a or b (these classes are the same for the ordinary integers--but not always in more general systems). Nevertheless, the units are a necessary precursors to the primes, and one falls in the class of units, not primes.
+8261 The number one is far more special than a prime! It is the unit (the building block) of the positive integers, hence the only integer which merits its own existence axiom in Peano's axioms. It is the only multiplicative identity (1.a = a.1 = a for all numbers a). It is the only perfect nth power for all positive integers n. It is the only positive integer with exactly one positive divisor. But it is not a prime. So why not? Below we give four answers, each more technical than its precursor. If this question interests you, you might look at the history of the primaility of one as described in the papers "What is the smallest prime?" [CX2012] and "The History of the Primality of One: A Selection of Sources" [CRXK2012]. Answer One: By definition of prime! The definition is as follows. An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself. Clearly one is left out, but this does not really address the question "why?" Answer Two: Because of the purpose of primes. The formal notion of primes was introduced by Euclid in his study of perfect numbers (in his "geometry" classic The Elements). Euclid needed to know when an integer n factored into a product of smaller integers (a nontrivially factorization), hence he was interested in those numbers which did not factor. Using the definition above he proved: The Fundamental Theorem of Arithmetic Every positive integer greater than one can be written uniquely as a product of primes, with the prime factors in the product written in order of nondecreasing size. Here we find the most important use of primes: they are the unique building blocks of the multiplicative group of integers. In discussion of warfare you often hear the phrase "divide and conquer." The same principle holds in mathematics. Many of the properties of an integer can be traced back to the properties of its prime divisors, allowing us to divide the problem (literally) into smaller problems. The number one is useless in this regard because a = 1.a = 1.1.a = ... That is, divisibility by one fails to provide us any information about a. Answer Three: Because one is a unit. Don't go feeling sorry for one, it is part of an important class of numbers call the units (or divisors of unity). These are the elements (numbers) which have a multiplicative inverse. For example, in the usual integers there are two units {1, -1}. If we expand our purview to include the Gaussian integers {a+bi | a, b are integers}, then we have four units {1, -1, i, -i}. In some number systems there are infinitely many units. So indeed there was a time that many folks defined one to be a prime, but it is the importance of units in modern mathematics that causes us to be much more careful with the number one (and with primes). Answer Four: By the Generalized Definition of Prime. (See also the technical note in The prime Glossary' definition). There was a time that many folks defined one to be a prime, but it is the importance of units and primes in modern mathematics that causes us to be much more careful with the number one (and with primes). When we only consider the positive integers, the role of one as a unit is blurred with its role as an identity; however, as we look at other number rings (a technical term for systems in which we can add, subtract and multiply), we see that the class of units is of fundamental importance and they must be found before we can even define the notion of a prime. For example, here is how Borevich and Shafarevich define prime number in their classic text "Number Theory:" An element p of the ring D, nonzero and not a unit, is called prime if it can not be decomposed into factors p=ab, neither of which is a unit in D. Sometimes numbers with this property are called irreducible and then the name prime is reserved for those numbers which when they divide a product ab, must divide a or b (these classes are the same for the ordinary integers--but not always in more general systems). Nevertheless, the units are a necessary precursors to the primes, and one falls in the class of units, not primes.
+3453 A prime number is a number greater than one that has no positive divisors other than 1 and itself.
Although you can't divide 1 by anthing but 1, the number 1 doesn't meet the second qualifiation "a number that is greater than 1"
That's why 1 isn't a prime number.
A little off topic of the actual question, here's a chart of all the prime numbers 1-100. Highlighted means it's a prime number. :)
Definition paraphrased from Wikipedia: http://en.wikipedia.org/wiki/Prime_number
Picture source: https://missmanganssciencesite.pbworks.com/w/page/32589311/Prime%20Number%20Determination%20Method
+11912 DragonSlayer you seem to be a pretty good typer!Thumbs up from me!and for u too ND!i'll help u get back ur position by a point!lol!
+26367 why is 1 not a prime number? why?
Prime numbers are never divisible by other prime numbers. If was 1 a prime number, would be valid this no more, because every number is divisible by 1
+394 WOW Dragonslayer your vocabulary just jumped to college level. How did you do that?
+11912 Let me guess!,.......................................
He copied the entire thing from a website!
See i just guessed it all right!But copying this much is also not a easy task!So indirectly , DS did hard work in answering this!
+394 IDK Rosala, right click select copy. Then right click paste.
It is only hard work if you have arthritis.
Also, if you want to use someones work you should say so!
7up
+11912 I dont know why copying doesnt happen with me !ive tried all the methids!
+394 Hi Rosala
If you have a regular computer this might help.
https://www.youtube.com/watch?v=RhXg4tvIft4
On a phone it would be different. Maybe you can search for youtube vids on how to do it or ask someone who has one.
7up
+11912 thank you 7up for those nice advices!ive never tried copying on a phone, i'll try once but i dont think it can be done!but though thank u very much!thumbs up from me!
+118609 Thanks you SevenUP. I added the address to information for new people.
Did that help Rosala.
Maybe if you say what devise you are using someone can help more?
+11912 Melody , I just found out by myself how to copy on a computer and its pretty easy!
But thanks for all the help!
+104 This is because 1 has just one factor, 1. It also has a square root, cube root, 4th root, etc. meaning it cannot fit into any group
+1310 hahahahahah watch this
The number one is far more special than a prime! It is the unit (the building block) of the positive integers, hence the only integer which merits its own existence axiom in Peano's axioms. It is the only multiplicative identity (1.a = a.1 = a for all numbers a). It is the only perfect nth power for all positive integers n. It is the only positive integer with exactly one positive divisor. But it is not a prime. So why not? Below we give four answers, each more technical than its precursor. If this question interests you, you might look at the history of the primaility of one as described in the papers "What is the smallest prime?" [CX2012] and "The History of the Primality of One: A Selection of Sources" [CRXK2012]. Answer One: By definition of prime! The definition is as follows. An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself. Clearly one is left out, but this does not really address the question "why?" Answer Two: Because of the purpose of primes. The formal notion of primes was introduced by Euclid in his study of perfect numbers (in his "geometry" classic The Elements). Euclid needed to know when an integer n factored into a product of smaller integers (a nontrivially factorization), hence he was interested in those numbers which did not factor. Using the definition above he proved: The Fundamental Theorem of Arithmetic Every positive integer greater than one can be written uniquely as a product of primes, with the prime factors in the product written in order of nondecreasing size. Here we find the most important use of primes: they are the unique building blocks of the multiplicative group of integers. In discussion of warfare you often hear the phrase "divide and conquer." The same principle holds in mathematics. Many of the properties of an integer can be traced back to the properties of its prime divisors, allowing us to divide the problem (literally) into smaller problems. The number one is useless in this regard because a = 1.a = 1.1.a = ... That is, divisibility by one fails to provide us any information about a. Answer Three: Because one is a unit. Don't go feeling sorry for one, it is part of an important class of numbers call the units (or divisors of unity). These are the elements (numbers) which have a multiplicative inverse. For example, in the usual integers there are two units {1, -1}. If we expand our purview to include the Gaussian integers {a+bi | a, b are integers}, then we have four units {1, -1, i, -i}. In some number systems there are infinitely many units. So indeed there was a time that many folks defined one to be a prime, but it is the importance of units in modern mathematics that causes us to be much more careful with the number one (and with primes). Answer Four: By the Generalized Definition of Prime. (See also the technical note in The prime Glossary' definition). There was a time that many folks defined one to be a prime, but it is the importance of units and primes in modern mathematics that causes us to be much more careful with the number one (and with primes). When we only consider the positive integers, the role of one as a unit is blurred with its role as an identity; however, as we look at other number rings (a technical term for systems in which we can add, subtract and multiply), we see that the class of units is of fundamental importance and they must be found before we can even define the notion of a prime. For example, here is how Borevich and Shafarevich define prime number in their classic text "Number Theory:" An element p of the ring D, nonzero and not a unit, is called prime if it can not be decomposed into factors p=ab, neither of which is a unit in D. Sometimes numbers with this property are called irreducible and then the name prime is reserved for those numbers which when they divide a product ab, must divide a or b (these classes are the same for the ordinary integers--but not always in more general systems). Nevertheless, the units are a necessary precursors to the primes, and one falls in the class of units, not primes.
+2752 Because it only has one factor, to be prime the number must have two factors.
