How come that ''A prime number itself is a product of one prime'' ?

Question

Every positive integer greater than 1 is equal to a ''product'' of prime numbers.
i.e. 12 = 2.2.3, 35 = 5.7, 30 = 2.3.5 etc...all holds true, but it doesn't make sense if the number chosen is p, a prime unless the number of primes in a ''product'' is allowed to be 1; i.e. 13 = 13 ?, p = p ? but this implies that ''a prime number is a product of one prime''; If so isn't that contrary to the definition of PRODUCT ?
P.S. a product is the result obtained from one or more binary ops. as i remembered. i.e. must exist more than one elements; in this case at least two natural numbers.

Re: TheMathemagician
This is still a touch bewildering;
Useful this definition may be; but don't you agree that it would be very naughty to randomly define new concepts just because it's helpful.
Besides, this contradicts the original definiton of a product .

Re: TheMathemagician, mm, Steiner:

1. S(t): t is a product of prime numbers
S(t'): t' is a product of prime numbers only.
Could we deduce that t = t' ?

From what mm had said: ''a prime is a product of one prime, e.g. 13 is 13 x 1, the only prime number there is 13. So it is really that a prime is the product of one prime times 1. In case you are wondering, 1 is not a prime number.''
S(t') seems to demand more than the statement S(t); e.g. from S(t) it suffices to say that t = (p_1) x...x (p_i) x (anything that doesn't have to be prime) for p_i prime & i >= 1. i.e. S(t) is true as long as there is one prime number.

2. Since 1 is a product of ''zero'' prime numbers hence it is a product of prime numbers. (allowing the number of prime to be zero)

Answer 1

When arithmetic is first taught in elementary school, a product is usually thought of as requiring two or more inputs. For example, the product of {2 3 4} is 2 x 3 x 4 = 24, and the product of {7 8} is 7 x 8 = 56.

However, in more advanced mathematics, a product can have only one input (or even zero inputs sometimes). The product of {12} is 12, and the product of {17} is 17, and so on.

The reason for this generalization is that it makes the concept of a product more powerful (for example, we can say that every integer greater than 1 is a product of primes). We do not gain anything by saying "Products of one number are stupid, so we just won't define them"; defining products of one to be just the single number you've got works rather nicely and is helpful in many situations, so that's how it's defined.

Think also about exponents:
3^3 = 3 x 3 x 3
3^2 = 3 x 3
3^1 = 3

Now, 3^3 is the product of three 3's, and 3^2 is the product of two threes. So 3^1 should be the same thing as the product of one 3. So there is some consistency here.

EDIT:
I don't think that this definition qualifies as "randomly defined." I think that it is the only definition that makes any sense. The only two reasonable options are "products of one are absurd and should be undefined" and "a product of one number should be just that number." Adopting the latter does not break anything, and in fact leads to some useful results.

I don't agree that this contradicts the original definition of a product. When defining something, mathematicians are usually very careful to state in in something like the following form:

"DEFINITION. If a and b are two real numbers, then ab is said to be the 'product' of ab."

This definition says that, if there are two real numbers a and b, then we can say that ab is their product. Emphatically, it does *not* say that no other things at all can be called products. This definition leaves open the possibility that other uses of the term "product" might exist. I would wager that any textbook on arithmetic or algebra that uses a definition of product at all uses one like the above, that leaves open the possibility of extending the definition.

(A good definition in mathematics should always be extensible.) Here is how I might define a product:

-The product of one number is just that number.
-The product of n numbers is the last number times the product of the first n - 1 numbers.

This gives a recursive definition of the product of n numbers, where n is any integer. It leaves open the possibility of a product consisting of zero numbers, or a product of things that are not numbers, and so on, so that if a useful, consistent definition for such things is found, then it can be added.

My definition doesn't contradict the other definition I gave, because my more general definition agrees with the earlier one whenever the earlier one is applicable. Instead, my definition generalizes the earlier one: it includes the earlier definition as a special case (when n = 2).

Answer 2

There is no hard rule on this. Some sources list 1 as the first prime number, others list 2 as the first prime number. Your argument is perfectly reasonable, but so is the alternative view. There are conventions when doing certain mathematics. For example does 0! = 1 ? It is only 1 because it fits the scheme of things when discussing factorials. Whatever math you do, there are always 'boundary cases', i.e. cases where you have to make certain exceptions or certain assumptions. You may have heard of perfect numbers. You can prove these always end either in either '6' or '28'. But wait a minute! Isn't 1 also a perfect number? It is the exception to the rule. Or is it? Take your pick .

Answer 3

Mathemagician's answer is good. It's not naughty to define new concepts just because they are useful. Just on the contrary, this is very wise and makes Math simpler. In Math, we have several useful definitions that apparently don't make sense but are useful. For example:


x^0 = 1 for every x
x^1 = x for every x
0! = 1
C(n, 0) = 1 (n choose 0)

Defining that the sum or of the product of one element is this element itself makes things simpler when we deal with summations or products. For example, Sigma (i =1, n) a_i or Product (i =1, n) a_i still make sense for i =1, the result is a_1. Of course this is just a convention established to make things easier. If you want to avoid this, you can state the fundamental theorem of arithmetic as "Every positive integer greater than 1 is either a prime or equal to a product of prime numbers".

Answer 4

To answer a few comments I have read, 1 is NOT a prime number, nor is it a composite number.

A prime number MUST have EXACTLY two factors: 1 and itself. The number 1 fails this definition because it only has one factor: 1.

A composite number MUST be a product of TWO or MORE primes. Since 1 only has one factor, and there is no prime factors of the number 1, it fails this definition as well.

There is a website I ran into that explains primes through the fundamental theorem of arithmetic that you might want to check out:

http://primes.utm.edu/glossary/page.php?sort=FundamentalTheorem

Answer 5

Firstly 1 is not a prime number.
I think you may be confusing this with the fundamental theorem of arithmetic.This states that every positive integer larger than 1 can be written as a product of one or more primes in a unique way.

Answer 6

a prime is a product of one prime, e.g. 13 is 13 x 1.

the only prime number there is 13.

so it is really that a prime is the product of one prime times 1.

in case you are wondering, 1 is not a prime number.

Answer 7

it is the "fundamental theorem of aritmetic" you will learn the proof if you studied mathematics at university. All natural numbers can be written as a product of primes the theorem states this and its also is proved.I could write out the whole steps for you but its too long.If your intrested in maths you will definately learns this in your first year at uni unless u can prove it now.

Answer 8

Your question makes no sense, but I can't believe there are two idiots on here who think that 1 is a prime number!

Answer 9

That seems to be true unless one of the elements is 1, which can always be the case...
ie 13 = 13.1 p = p.1
and remember 1 is a prime number as well.

Answer 10

You might want to remember that ONE is a prime number, too.