Why isn't 1 a prime number?
blogs.scientificamerican.com
blogs.scientificamerican.com
Seems a rather cryptic way to go about things with your health on the line.
But it's interesting that a lot of people might credit iOS for adding this feature, when my (Android 2.2/2.3) MyTouch 4g had this functionality almost 10 years ago.
(Android user btw)
You can set a Lock Screen Message by searching for "Lock Screen Message" in the Android Settings.
You can also create an "ICE (In Case of Emergency)" contact.
I wonder if dialing 911 from their phone would give emergency services some helpful info. Do they automatically get caller ID (perhaps enhanced)?
A lot of areas have the ability to query a cell phone for its location, but it depends on how much that city/county/state(?) spends on 911. It's usually based on carrier location (so it's not really that helpful anyway), and calling from their phone probably won't help much since you're both in the same location. Caller ID might help if they have a recurring problem (like if they have epilepsy or something that the medics that show up would need to know), but for a concussion it's pretty unlikely they had called 911 before so it probably wouldn't have done much. I remember reading something about iOS sending medical cards somehow when you call 911, but I don't think that's a feature yet (and it relies on iOS, so if they didn't have an iphone it wouldn't help anyway).
Source: I help test emergency boxes (the things with the blue lights in parking lots), landline phones, and cell phones at my job every year. I have to confirm different info based on what I'm calling from. On cell phones, it's just the number the call is being made from. By the way, if you're wondering, 911 does not appreciate hundreds of test calls being made unannounced over the course of a day. We do it anyway :^) (we're required to)
I hope this person was okay. Concussions are truly hell when they don't clear up nicely.
Side note, I was told once by an occupational therapist specializing in brain injury that most concussions are from standing up from a crouch in the kitchen and hitting your head on a corner of a cupboard (MVA/sports are very common too). I didn't believe her at the time, but I don't write it off either.
Downside, if you're short you need a stool to close it.
Maybe there's a different solution though, I'm not sure
I ended up injuring my back and walking in the Early Asimo robot's "old man poopy pants gait" for an entire half a year, following a sneeze. You can google that. It's quite common to injure your back after a sneeze.
An odd royal and mathematical version of The Prisoner?
Or likewise, does the principle let you extend to other conclusions?
Perhaps the key insight of the 'too simple to be simple' principle is that we want the definition to impose both existence & uniqueness of something. In this case proper divisors (divisors strictly less than the number itself e.g. the proper divisors of 6 are 1,2, & 3).
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Let me give a concrete example of how it might be used.
- Uniqueness is captured algebraically here by the notion of a 'subsingleton' set: a subset where any pair of elements are equal. This can happen if (1) the subset has only a single element, or (2) the subset has NO elements (in which case the uniqueness requirement is vacuously true).
- Existence is captured by the notion of a 'singleton' set: a subset with exactly 1 element: any two elements belonging to the set are unique, AND such an element exists.
First lets apply this to the definition of the regular primes: the set of proper divisors of 1 is the empty set. The set of proper divisors of any other prime is the singleton set {1}. Thus the naive definition says 'proper divisors of primes form a subsingleton': it asserts they are unique, but not that they exist. The more sophisticated definition of the primes (which excludes 1) asserts the proper divisors of primes form a singleton: uniqueness and existence.
Now lets try to apply it (somewhat informally) to the fictional example of even numbers and 'even primes'. Examples:
2 is an even prime,
4 = 2*2 not an even prime
6 is an even prime,
8 = 4*2 is not,
10 is even prime,
30 is an even prime is as well, and so on.
Now what happens if we try and factor 60?
60 = 2*30
60 = 6*10
A unit is an operation you can apply multiple times and get back to the starting point, for example multiplication by 1 or -1. So these are units in the integers. Units are really important, but they don't get you anywhere on their own. We can think of non-units as building blocks.
So then the question comes up which building blocks can be broken down into smaller blocks, like "times 6" can be divided into "times 2, then times 3". Using a unit doesn't count, because you're still equally far from the starting point (so 6 = (-6)(-1) is not really breaking down 6).
Blocks that can't be broken down any farther are irreducible (like 7). But in common parlance, that's what we think of as the definition of a prime number. It doesn't include units like 1, because they're already broken down.
More generally, being prime isn't about being broken down, it's about building up. Suppose I tell you that n is divisible by 2, and n can be broken down into two pieces, n = ab. Then you know that at least one of a or b is divisible by two. So two is prime because anything that is built from two, when broken down into pieces, has a piece that can be built from two. (Example: 6 divides 12, but you can break 12 up into (3)(4), so 6 isn't prime.) This definition of prime also doesn't include units like 1 because they're not building blocks in the first place. Everything's already divisible by units.
Prime integers have both the prime property and the irreducible property. Either way, I think it makes sense that 1 isn't included -- being prime and irreducible are properties of building blocks, and 1 is a unit.
Nope. That is an idempotent[1], or a root of unity, i.e. x^n = 1 for some n >= 0. A unit is something which has an inverse, though that inverse need not be a power of x itself. For example, 1/2 is inverse to 2 in the rational numbers.
Mathematicians have struggled with this for centuries before agreement was reached. See “The History of the Primality of One: A Selection of Sources” (https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.p...)
The “math looks nicer” argument isn’t unique to this example. It also is, for example, the ‘reason’ that 0⁰ equals 1, or that there are as many integers as rationals.
I'm curious what you mean here: The integers and rationals have quite different algebraic and geometric properties, and the fact that they have the same cardinality has always struck me as a deep result, rather than just a matter of definitions.
“two sets have the same size iff there’s
a 1:1 mapping between items in the set”
“two *finite* sets have the same size iff there’s
a 1:1 mapping between items in the set”
Why we don't consider units to be prime is more than just it "makes more math look nicer."
As discussed in the article, once we explored different types of number system (the example given is an adjoin of the square root of negative 5 to the integers) we were able to classify numbers that no-one would ever 'intuit' as prime: the units.
The defining characteristic of these non-primes is shared by '1' in the integers/rationals/reals so it is consistent to call 1 a non-prime as well.
Is it? That seems different. The math proof for that is pretty easy.
a^n * a^m = a^(n+m)
Is the argument just that we should say the above is true if n or m != 0?
One of the nice reasons to choose 1 over 0 for 0^0 is to make the statement "the cardinality of the set of functions from a size n set to a size m set is m^n". There's exactly one function from the empty set to itself.
"My mathematical training taught me that the good reason for 1 not being considered prime is the fundamental theorem of arithmetic, which states that every number can be written as a product of primes in exactly one way. If 1 were prime, we would lose that uniqueness. We could write 2 as 1×2, or 1×1×2, or 1594827×2. Excluding 1 from the primes smooths that out."
When the sequence has one element, then that element is the product. So, 2 is the product of (2).
When the sequence is empty, the product is 1.
p_S = \prod_{x \in S} x
And then later find out that the set S contains one element. Or it might even be the empty set in which case you'd interpret the above as equal to 1. This makes it work that
p_{S union S'} = p_S p_S'
in all cases.
I assume most people on this list are engineers. They have college level math but not necessarily number theory. Number theory requires a similar thinking to the problems we solve in all the time in engineering fields.
I just explained the unique product of primes argument to my mechanical engineering husband. He got it right away and also thought it was awesome!
Set_1 includes 1, Set_2 does not.
We have two possibilities for naming.
Option A: Set_1: Primes. Set_2: Primes except for 1.
Option B: Set_1: Primes and 1. Set_2: Primes.
You can then show the differences in the properties that make having Set_2 have the default name an easier to use convention than letting Set_1 have the default name. But at the core is the understanding that this is an arbitrary distinction done just for ease of use. Had we called Set_1 the Primes, or if ever someone became Galactic Emperor for a day and made the change, then Set_2 would get a new name for ease of use and we would just use it instead.
There are many other sets that are really close as well, which have their own properties.
Side note, SET_1 is listed on oeis https://oeis.org/A008578 and still includes 1. It is called "Prime numbers at the beginning of the 20th century".
I reached solace regarding this itchy issue (not even joking) by considering the following: Primes are the minimal set of numbers to generate another set of numbers, using multiplication (with repetition, yada yada) For the positive numbers, [2, 3, 5, 7, ...] is all we need, considering that, as many others have said, 1 = Product([]).
In that sense, to generate all integers, the integers you need are: [-1, 0, 2, 3, 5, 7..]
Similarly, to generate all gaussian integers (complex numbers with integer components,) you can follow the following link: https://en.wikipedia.org/wiki/Gaussian_integer#Gaussian_prim... however, am not sure why 0 is not considered a prime number in this context.
Gaussian primes are symmetric over the real and imaginary axes, and to me it's quite curious why 0 is not considered prime there, but I guess uniqueness of the multiplication matters too (0 can be repeated multiple times, which could make some theorems about uniqueness of factorization a bit cumbersome as well.)
So, if you have a set with multiplication and addition, that is called a ring. A very nice class of rings is called fields. This is where division is possible. Another way to state that 'division is possible' is to say that multiplication is a group operator on the entire set except zero.
Edit: for historical reasons prime elements are slightly different from the usual prime numbers, and 0 remains a bit of a special case even now.
The greatest common divisor of any two primes is one. To say that two numbers are coprime, is to say their greatest common divisor is 1. If we considered 0 a prime, it would be a bit weird (and messy) that two prime (0 and any other prime number) numbers aren't coprime.
[1]: https://en.wikipedia.org/wiki/Prime_number
[2]:https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.h...
For some "minus five" is an operation, not a number, and for others "negative five" is meaningless, the correct expression would have to be "the negative of five". There are entire PhD theses written on the topic of how to avoid confusing young children when introducing them to numbers less than 0, and the words we should, and should not, use in that context.
Yes, there are.
Those PhD theses seem a little misguided (if well meaning) to me. Language is riddled with ambiguity - "panda eats shoots and leaves" for example. Despite that ambiguity we generally manage to get along. Teaching critical thinking in general might be a better idea than fixating on specific issues.
That's a strong statement to make as a throw-away. You haven't read them, you haven't done the research, so I'd ask - what experience do you have of tracking the effect of specific language on the acquisition and development of mathematical skills in a significant sized population of very young children?
I suspect the answer is not a lot, and dismissing someone's research on the basis of effectively no information is poor form.
Then, the number 1 is the single element of the first row and all the prime numbers are the atoms immediately above 1.
I guess you can say that 1 is just an absolute unit.
Shorter answer: Convenience.
[1] http://mathworld.wolfram.com/PrimeNumber.html (3rd paragraph)
There are infinite natural numbers and infinite prime numbers, but you need each and every prime number to generate any other natural number.
So I'm puzzled to learn that 1 is not a prime number because I do not know how to generate 1 if we exclude 1 from the prime set.
Some people really care about 1/n being possible for every natural number N.
1 != Sn
Sn = Sm implies n=m
n + 1 = Sn
n + Sm = S(n + m)
n × 1 = n
n × Sm = (n × m) + n
Induction
One way to state unique prime factorisation including 0 is to treat the set of primes as a pointed set with 0 as the point. The smash product makes the category of pointed sets into a symmetric monoidal category. Then unique prime factorisation can be stated by saying that the multiplicative monoid of natural numbers (with 0) is isomorphic to the free commutative monoid object on the set of primes.
I.e. X * 1/X = X/X = 1, always.
I guess, technically, it's a division needed to get the inverse, so it's perhaps it doesn't count :-)
Edit: Ah, yeah, now with help from the reply below I see my mistake - you're saying the inverse itself cannot exist since we're only dealing with natural numbers, that makes sense.
I guess that's pretty much the reason that only multiplication in allowed in the OP's original question, allowing division lets us get all kinds of fractional numbers which wouldn't be allowed.
In each context, it's the "same" 2, but the structure we put around it is different. The structure is what makes it interesting.
2-the-rational is not prime. There are no primes when every non-zero element has a multiplicative inverse. An element with a multiplicative inverse is called a unit and for the same reason we exclude 1 from being prime we exclude units from being primes in other structures.
Every non-zero real number is a unit. Every non-zero rational is a unit. Only -1 and 1 are units in the integers. Only 1 is a unit in the naturals.
From a programmer's point of view, a similar thing happens with Lisp. Lisp is a very elegant language due to its minimal structure, but not a very useful one with respect to how often it is used.
The first is: "numbers that can be multiplied together to get all other numbers". This mental model by necessity excludes 1.
The second is: "numbers that cannot be 'made' by multiplying other, smaller numbers together". This mental model is "looser," and ends up including 1, and maybe 0 and -1.
The first mental model tends to be more mathematically useful, so becomes the official definition of prime.
\* Note that I explicitly use "mental model" here instead of "definition," because I am discussing different ways that different humans try and _understand_ different sets of numbers.
(Or to put it differently, the set of primes forms a generating set for the monoid of natural numbers under multiplication, not the semigroup. Semigroups are kind of dumb. :P )
Engineers: Eh, maybe?
The guy doing the engraving on the gold plaque on the side of the space probe: What is that, a potato?
Aliens: They treat root vegetables as numbers.
Alien anthropologist: They worship plants.
Alien TV presenter: In showing us agriculture, they introduced us to mathematics!
(Me: I think I need to cut down on the cold medicine.)
In other words, because semantics.
a number with no factors beside 1 and itself, excluding zero
1 does not meet that definition, since 1 is already counted, the other factor needs to be different.
Actually it's the other way around:
'Whenever m or n are divisible by p, then their product m×n must be divisible by p.'
The opposite is not true, i.e. the statement is not reversible and still true. For instance:
6x4 = 24; 24 is divisible by 8, i.e. 24 mod 8 = 0, yet 6 mod 8 = 6, and 4 mod 8 = 4.
[ed] I've stepped into a cognition discontinuity, move along, nothing to see.