How can a number be "only" divisible by one and itself? 2, for instance, is divisible by 1, 2, 3, 4, 5, etc. You may end up with a fraction, but it is still divisible. What is it about prime numbers that disqualifies factional answers?
You clearly do not understand the term divisible .
First that term refers to integers.
So the statement that $\displaystyle a$ is divisible by $\displaystyle b$ means that each of $\displaystyle a.~b,~\&~k$ are integers such that $\displaystyle a=kb.$
That means that $\displaystyle 2$ cannot be divisible by $\displaystyle 3$. Do you see why?
That definition also is used in defining prime numbers: an integer is prime if and only if it has exactly two divisors.
Often even well respected textbooks and commentators (i.e. Keith Devlin the math guy) say that an integer is prime if "it is only divisible by itself and one". But that definition makes one a prime. Do you see why?
BTW: (Thanks to R L Moore for that distinction,)
for certain kinds of "structures", namely rings, one says that for a,b in a ring R: a|b if b = ma for some element m of R.
it turns out that the set of integers form a ring, with ordinary addition and multiplication. in *the integers* 2|4, but 3 does not divide 4. this is because there is no INTEGER m such that:
3m = 4 (which is because 4/3 is not an integer).
if one is speaking of the ring of rational numbers (which *is* a perfectly good ring), then every number is divisible by any other number except 0. this holds in any field (which is a special KIND of ring). because divisibility relations in fields are trivial, there's not anything particularly revealing about investigating them: they don't tell you much.
in rings that are NOT fields, however, divisibility tells you useful information. for example, in the polynomial ring Z[x], the fact that (x - 2) dvides x^{2} - 3x + 2, tells you that (x - 2) is a FACTOR of x^{2} - 3x + 2, and this means that x = 2 is a SOLUTION (root) to x^{2} - 3x + 2 = 0. this is useful.
in other words, in the context of integers, "divisibile by m" does NOT mean, "able to be divided by m", but rather "if divided by m", the answer is a "whole number" (where in this case "whole number" = "integer", not "positive non-negative integer"). that is: no fractions.
this is not arbitrary at all. number systems come in different flavors, to model different kinds of situations.
Another thing to add to that, the divisors are assumed to be positive integers, and only positive integers greater than 1 are assumed to be prime/composite. For example, -1 is divisible by 1 and itself, but -1 is not defined to be a prime number.
Also, another reason why 1 is defined to be not prime is, if you assume 1 is a prime number, then 2,3,5,... would be the product of two prime numbers, and therefore couldn't be prime.
There is another, deeper, reason not to consider 1 a prime: in the ring of integers, 1 is a unit (by this i mean it has a multiplicative inverse).
The reason why 3|1 in the rational numbers, is because there *is* a number m such that 1 = 3m, namely, m = 1/3. similarly, if in a ring R, the element u is a unit, that is, there exists v in R with:
uv = vu = 1
then in a sense EVERY element r of R is "divisible" by u: for, in order to find an m with um = r, we can chose m = vr:
um = u(vr) = (uv)r = 1r = r.
for example, we can certainly factor in Q[x], the linear polynomial 2x - 3 as: (1/2)(4x - 6), but one gets the sense that this is not really what we mean by "factor". we haven't really broken down 2x - 3 into "simpler" things, in some sense this is "trivial". in much the same way, we can factor 2x - 3 over Z as:
(-1)(-1)(-1)(-1)(2x - 3)
but again, one gets the sense that we are not really "doing anything" (because...surprise! -1 is also a unit).
when we factor a number into primes, we want a "unique decomposition", we are not going to get that if we allows units to be prime, because we can ALWAYS put pairs of units times their inverses "out in front".
for another example, there is a theorem that says that Z_{n} is a finite field if and only if n is prime. now suppose n = 1; this is tantamount to saying: 0 = 1, that is:
Z_{1} = Z/Z = {0}. this "almost" is a field. it satisfies every axiom except one:
0 ≠ 1
the definition of a field requires that the non-zero elements form an abelian group. groups are required to have (at least) an identity. but there ARE no non-zero elements of {0}. so calling 1 a prime, would mean that we would substantially have to revise a great deal of useful mathematics of finite fields (which are used every day in such areas as cryptography, logical circuit design, error-correcting codes, UPC codes, library cataloging and data control systems).
historically, the distinction of 1 as "something different" goes back even further. to the ancient greeks, 1 wasn't "a number", but a variable kind of unity, you "picked your unit" and then other quantities were "compared" to it (a process called commensuration). so it was something like an atom, thought of as being perfect and indivisible. this concept actually hindered their mathematics, as they didn't have "rational numbers" per se, but "ratios"; what we would call 2/3, they thought of as 2 of this, 3 of that (all reconciled by the basic unit that went evenly into both). fractions as we know them now took the "long way around" reaching europe by way of indian and arab mathematicians (who had much more sophisticated ways of expressing numerical concepts), and it is likely that the first wide-spread use of them in europe ultimately is due to the popularity of fibonacci's book, Liber Abaci.
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people often grow up thinking that numbers are one kind of thing, and then they learn about "new kinds of numbers":
"counting numbers": 1,2,3,4.....
0 is usually learned "later", and is a bit odd at first (is it a number that means "no number", does that "count"?)
negative numbers are often next, with the usual existential dilemma: "what does -3 apples MEAN?"
then fractions, percentages, decimals, and irrational things like the square root of 2, and pi.
perhaps even later, one might learn of complex numbers, and their real and imaginary parts.
this entire process creates larger and larger number systems, and one is tempted to conjecture that there is some "grandfather number system" that contains "everything". but the truth is more complicated than that: there is actually a dizzying array of LOTS of different kinds of things that might legitimately be called "numbers", and some of them are incompatible. in other words, when we say: a is divisible by b...that doesn't make sense, really. because we have to know "what kinds of things ARE a and b?", first. in short: the meaning of some mathematical terms is not "absolute" but rather, context-sensitive. what is true for THESE kinds of things might NOT be true for THOSE.
I'm afraid I'm only more and more confused. I hope that by posting in this forum, I didn't give you the impression that I am more intelligent than I actually am. In my heart I'm more of a philosopher, but philosophy has lead me to mathematics and physics, realms of thought that require more structured thought. I try hard to understand certain concepts, but in order to do that, I need first to ask the obvious questions. So when I find out that prime numbers are so important to number theory, I try to find out why. Before I can even do that, I need to understand why prime numbers are the way they are and not some other way.
For awhile, it seemed that since every number is divisible by one, then it should obviously be important and not arbitrary that there is something special about numbers that cannot be divided by any number except one and themselves, because that is the very minimum.
But, it's not true that a number can only be divided by one and itself. Numbers could be divided by any number and still get a result. It may be a non-integer, but how does that make it any less of a number? Is it because "one" is the basic definition of all numbers? 2=2*1=1+1 / 3=3*1=1+1+1. Every possible number is simply a population of 1's. But then, since populations are defined by numbers and numbers are defined by 1, what does it mean to have a population of "two" 1's? It would mean that I have 1+1 1's.
Then we come to rational numbers. 1/2 is 1 broken into 2 pieces. But what should I call the value of those pieces? 1/2 of 1? That only begs the question "What is the value of 1/2?" We could say "The answer is .50." However, isn't that only putting a new mask on an already unknown person? It's as if I am asking "Who is Dave?" and you tell me "We also call him Micheal." This doesn't help if I never met Micheal, who is Dave, who I haven't met, so it should have to be the case that I haven't met Micheal...
Hmm... I feel that the more I think about it, the more lost I am. I feel that I just don't understand the concept of numbers. How to manipulate those "things" called numbers, I can understand, but what it is I am manipulating and what I am doing to them, I do not. I am only a monkey that learn to move symbols around in a certain game.
It's not a comfortable thing to ask "Why are prime numbers the way they are?" only to later reveal that I have trouble even grasping what it is I am doing when I add or divide, and allude that I don't understand much more.
Maybe there is some way you can help me? If so, then let's start by getting back to the first question: What makes a non-integer any less of a number? Maybe this has something to do with my incomprehension of division? I hate having to ask you to further dumb down your answer, taking up your time, further humbling myself, but it is what I must do to understand that which I don't understand already...
I feel for your. Philosophy was my undergraduate major. But a course in symbolic logic convinced me to go to graduate school in mathematics. If you really are interested in this, here is a book that I enjoyed. What is Mathematics Really by Reuben-Hersh.
I'm not going to get into rings, fields, or algebraic structures...that's a little more abstract than we need here.
Suppose $\displaystyle a$ and $\displaystyle b$ are integers and $\displaystyle b \neq 0$. By definition, $\displaystyle a$ is divisible by $\displaystyle b$ if $\displaystyle \frac{a}{b}$ is an integer. No need to show why it's true, it's just a definition. For example, 6 is obviously divisible by 3, but 5 is not.
A prime number is a positive integer that is divisible by exactly two positive integers (this definition is better than saying "divisible by 1 and itself" because what happens with "1?"). We use the definition of "divisible" mentioned previously -- 5 is divisible by 1 and 5, and no other integers. Hence 5 is prime. 1 is not defined to be a prime number.
What you don't seem to understand is that mathematics is in many ways a game. We play by rules.
Divisibility is defined on the positive integers.
That in no way excludes the fact that $\displaystyle \frac{5}{3}$ is not a perfectly good number. But $\displaystyle \frac{5}{3}$ is not an integer. That means that $\displaystyle 5$ is not divisible by $\displaystyle 3$. It does not play by the rules of divisibility.
yes 0.6 is a number. but it isn't an integer. the very word "fraction" means "part of a whole". the integers (well, natural numbers, really, but that's another story...) are the "wholes" being referred to. in mathematics, "divisible" has a different *meaning* than "can be divided".
in more basic terms, an integer is anything that can be made by adding 1's and -1's together, in any combination. now suppose 3/5 was an integer, which we'll call k because we don't know WHICH integer it is.
3/5 = k.
this means that 3 = 5k.
suppose k < 0.
then 5k < 5*0 = 0, so we have:
3 = 5k < 0, which isn't right. apparently, k can't be less than zero.
suppose k > 1.
then 5k > 5, so 3 = 5k > 5, which also isn't right.
so the only integers which might possibly qualify are k = 0, and k = 1.
but 5*0 = 0 ≠ 3, and
5*1 = 5 ≠ 3.
so whatever k is, it's *not* an integer. now, in fact, since we know k = 0.6, we've shown 0.6 is not an integer. not too terribly surprising, you're never going to get 0.6 no matter how many 1's and -1's you add together (if you only pay with, and receive dollar bills, and never make change, you're not going to wind up with 60 cents at the end of the day, if you started with nothing).
"divisibility" has to do with integers, it is a function of a *particular* number system (there are weird number systems in which 2 isn't prime. these questions are dependent on which system we're in). it's not kosher to say: i'm asking about a property of 2 as an integer, and then "switch" to the rational numbers to get an answer. i'll make another analogy:
we say a number is even, if n = 2k, for an integer k. for example, 26 is even, because 26 = 2*13.
well, if we allowed k to be "anything", then we could say "3 is even", because 3 = 2(1.5). but that just destroys the utility of our definition of "even". surely if we are using a word to describe something, we must have a reason for doing so, right? if "even" just meant "any number" why use the term at all?
with integers, divisibility is used to distinguish which integers are multiples of other integers. when we say 6 is divisible by 3, what we mean is: 6 is a MULTIPLE of 3. in other words, when you learned your "3's times table", 6 was one of the numbers you learned (in particular: "3 times 2 is 6"). this is useful, in reducing fractions, for example:
3/12 = 1/4 (because.....3 divides 12, or: 12 is divisible by 3).
the chief benefit we gain by this, is we can "factor" a number into primes:
240 = 2*2*2*2*3*5
and this factorization is *unique*....in some sense primes are the "atomic elements" that we use to build the "chemical compounds" of numbers with.
it's not *obvious* that this should be true. after all, we can write:
12 = 2*6
12 = 3*4
and this factorizations are definitely *not* the same.
it's not that 3/5 is not a number. it's a perfectly good number...but it's not the right KIND of number. numbers come in different "flavors", they aren't just all living in the same "jar".
If I may, we aren't excluding rational numbers at all, but rather we are naming numbers with different qualities. When you begin looking at abstract algebra you start to see some very interesting qualities of prime numbers (irreducibles too, but those end up being the same in the integers). The real point is that it gives us a way to identify some numbers, we aren't excluding rational numbers, they have their own name, and composite numbers also have some nice qualities.
This sort of question is very difficult to answer without looking in to the deeper mathematics, since without the mathematics we easily start to grapple with philosophy
That is the exact reason I am asking for! Brilliant! I understand a lot of what you're saying, you speak so clearly, I was really happy to read this post in particular.well, if we allowed k to be "anything", then we could say "3 is even", because 3 = 2(1.5). but that just destroys the utility of our definition of "even". surely if we are using a word to describe something, we must have a reason for doing so, right? if "even" just meant "any number" why use the term at all?
However, it seems that you misunderstood my question;
you asked my very question yourself, the answer to that -and why we say 3 is not an even now- is the answer I am looking for!