Funny true story about this:
One time a man came up to me and handed me his phone, and asked me to call his mom. He was about to pass out and he asked me not call an ambulance (God bless America) , he appeared to have a concussion.
By this time there was about 10-15 people around him and he could barely talk. I asked him what his phone code was and instead of just giving it to me he said "it's the first four prime numbers".
Immediately, about five people shout "1,2,3,5". I am no longer holding the phone, because I handed to someone else to make sure it was okay.
Sure enough, I was in a mathematical proofs class and we had just discussed this topic. So, I say "one is not a prime number".
Of course, we get the phone unlocked in the second try with "2,3,5,7" and the guys mom is on the way. Everyone thought I was a genius a hero.
As far as mild-moderate concussions go, there is unfortunately not much to do by way of treatment besides low stimulation rest.
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.
most concussions are from standing up from a crouch in the kitchen and hitting your head on a corner of a cupboard
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.
We got ahold of his mom, and one of the bystanders was a nurse. She didn't seem too worried, so I think everything was okay.
We were at a sports park and I think he hurt himself playing soccer.
Yea we considered this but then the rope hangs down into the counterspace beneath and gets in the way. Or in one case, hangs down and gets caught in the fridge when you close the fridge.
Make the rope half the length of the cabinet door, and instead of hanging it way out at the edge of the door, hang it halfway towards the hinge. The rope will hang slightly higher but it'll fit neatly inside the cabinet when the door is closed.
Unfortunately it's double hinged, so the point you'd need to pull down from to get proper leverage on it is at or near the highest point - hang a rope down there that's of useful length and it's by definition longer than half the length of the cabinet, and doesn't fold neatly in.
Maybe there's a different solution though, I'm not sure
Just to be clear, I'm not ragging on you or your comment at all, because I see this all over the place and it's not something specific to you.
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.
What a strange comment. GP wasn't claiming that iOS was the first to add this feature, and besides, does it really matter who added it first? They were just helpfully pointing out its existence for anyone who may not be aware.
The only reason I brought it up is because they called out iOS as "just adding it" when Android has had it for years, even though the original post said nothing about iphones.
Calling from a landline (or anything that you can't move without a lot of effort) will always give extra information such as where the phone is located (if you've ever signed up for landline service you entered this in somewhere, carriers are required to record it). Some even have specific information, like "3rd floor west stairwell at [address]", but that's only for public buildings. I'm sure you could call 911 and give them a heads up if you had a reason to believe you might call in the future, though. Office phones usually have a office number or at least a hallway (this is why calling 911 from a VoIP phone has a delay, it needs to route you through a real phone number with this information, and Cisco's solution for that is really slow).
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)
LOL the way to convince people the story real is certainly not to say "OK maybe I lied about the details, but trust me that the main part is still true" (though the story still seems plausible enough I'd still give it a ~30% chance of being true)
Very nice article! Also of interest, 1 not being prime can be viewed as a particular example of a more general phenomenon that occurs throughout mathematics, known as the 'Too simple to be simple' principle:
Do you mean the example of 1 not being a prime, or the example in the article of Z[root -5] not having unique factorization? You could certainly discover the former, but not the latter as far as I can see. The idea is that the principle should guide you on what the 'right' set of definitions are, so that your theory detects interesting theorems and isn't riddled with edge cases. You might make an analogy to programming, finding good primitive data types and interfaces so that you can implement elegant and efficient algorithms and well structured modules.
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.
Here I have uniqueness of proper divisors (they form a subsingleton), but only because existence of proper divisors fails for every even prime (not just 2)!
Now what happens if we try and factor 60?
60 = 2*30
60 = 6*10
It not great surprise that uniqueness of prime factorization fails as well (one of several problems with this informal example). The principle didn't help me find this example, it suggests that if I use the above notion of an 'even prime', i'm not going to get a good set of theorems.
I was glad that the article got into the definitions of irreducible, prime, and unit in more general algebraic structures! This question is all about these three concepts.
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.
> A unit is an operation you can apply multiple times and get back to the starting point, for example multiplication by 1 or -1.
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.
Oops, thanks! (been a while) Those are examples of units, but units are more generally "building blocks of 1" (if you apply a unit, there is some other unit that gets you back to 1). Thanks again for the correction.
The problem here is that “is not prime” doesn’t imply “is composite”. The number 1 is neither prime nor composite. Why? Because that is the choice that makes more math look nicer. If 1 were considered a prime, lots of theorems would have to say “let p be a prime >1”, rather than “let p be a prime”.
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.
> The “math looks nicer” argument [...] is, for example, the `reason' [...] 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.
As far as I can tell, the idea of “same size means establishing a bijection” appears (in a formal sense) at essentially the same time as it was discovered that there are pairs of infinite collections where you can’t do that (i.e. with Dedekind and Cantor in the late 1800s), so I don’t think that second definition ever actually existed in a meaningful sense, although I think your intuition seems reasonable (albeit “bending” holds the wrong connotation, imo, as this would seem to be a rather principled generalization of an existing concept rather than an arbitrary decision among a number of somewhat reasonable options). Regardless, I’d be very interested to read a survey of the history of ideas about cardinality/ordinality if you have a good one, since modern treatments of these ideas are radically different than how humans understood them even 150 years ago.
In what way is the last one a matter of making it 'look nicer'? Either there is a bijection between these or there isn't - that seems like a concrete thing that's not a choice of terminology or aesthetics.
By making the choice that that applies to infinite sets, too, you go against strong intuition such as that, for example, there are twice as many integers as even numbers (that intuition is very strong; even math students will start out thinking it is true, not “I wouldn’t know”)
The point of the article is that we don't consider a unit in a number system to be prime, and in the integers/rationals/reals 1 is a unit.
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.
I was going to say "because we said so", but this is a far^N more interesting comment. Elegance is a very useful criterion in many areas of mathematics, science and art. Stating the obvious, I know, but still.
Right, we want to define 0^0 in a way that fits in with the usual rules of exponentiation. The problem is that 0^n is always 0 for nonzero n, but a^0 is always 1 for nonzero a. So 0^0 can't continue both of these patterns by being both 0 and 1.
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.
Also, if one claims 0⁰ must equal zero “because multiplying by zero always gives you zero”: there are zero zeroes in the product 0⁰, or, said another way, there are no zeroes in 0⁰. So, why would it have to be equal to zero?
Bad example, if 1 isn't prime, 2 x 1 isn't a product of prime, so not all the number can be written as product of primes. My point is that you can't write an introductory article, supposed to be understandable to the masses, using a form able to confuse further the readers. A student, reading this, I don't think will understand this explanation that seems contradictory.
"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."
It's a good example just a poor rendition of the definition of the fundamental theorem of arithmetic. The proper definition includes "either prime or the product of primes". Many mathematical texts shorten this to "the product of one or more primes" which doesn't make much sense to me but if you take it as acceptable to use "product" with a single number then some shorten it even further to "the product of primes" (primes now representing "prime numbers" not "multiple primes") resulting in what you see in many places/this article.
It does make sense in a mathematical context though. You might write something like
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
All positive integers can be written as a product of primes, where a product is a sequence of numbers that are multiplied together. In the case of a composite number like 45, the primes are (3,3,5).
When the sequence has one element, then that element is the product. So, 2 is the product of (2).
Furthermore, the ordered sequence of primes that multiply to any given positive integer is unique. If 1 were considered prime, that wouldn't hold, as you could keep adding as many 1s as you like to the start of the sequence: (1,3,3,5), (1,1,3,3,5), etc. Thus 1 encodes no useful information here.
I think the best way it was explained to me was say that both exist as sets of numbers, which for now we will call Set_1 and Set_2.
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".
Sloppy wording, yes, bad example, not really. "just" requires you to consider "2" to be the product of all numbers in the multiset {2}. Let "product of primes" mean "product of one or more primes" and it works.
I thought the unique product of primes argument was great!
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!
Yeah, but it also says "in a unique way" and 1 breaks that since it sends any other number to itself under multiplication. So maybe a prime number also has the property that it sends any other number not to itself when multiplied by ? maybe that's already in it ?
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.)
You might be happy to know that 0 is considered a prime element in typical cases (barring those cases where there are 0 divisors, i.e. where two elements can be multiplied to create 0, obviously this is not an issue with regular numbers).
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.
> I am not sure why 0 is not considered a prime number in this context.
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.
> I am not sure why 0 is not considered a prime number in this context.
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.
Wikipedia [1] also mentions that: "Most early Greeks did not even consider 1 to be a number, so they could not consider its primality". Both articles (scientific american & wikipedia) refer to the same paper [2].
I always raise an eyebrow at statements like ‘most early greeks’, because who knows how many of them actually read Euclid or had any exposure to math at all.
That depends - this divide is not just "USA English Speaking People" versus the rest. Within the UK, Australia, New Zealand, and Canada, there is a split between people who say "minus 5" and people who say "negative 5".
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, and who on earth says "cheerio, pip pip"? "Toodle pip" or "cheerio" or "pip pip" are acceptable.
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.
> Those PhD theses seem a little misguided (if well meaning) to me.
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.
I was thinking more of UK (the only place where "pip pip" makes sense as a stereotype) versus the rest! Are you saying there are those in the UK who say "negative five"? That would surprise me.
As the article says, one could consider this a matter of definition. The definition of choice for me (and some other category theorists?) is that the prime numbers are the second row in the devisibility lattice. (That is, n < m iff n | m.)
Then, the number 1 is the single element of the first row and all the prime numbers are the atoms immediately above 1.
Short answer: 1 could be counted as a prime number, and in fact it used to be counted as a prime number by some mathematicians [1]. These days, most people just choose not to do that, since this makes most proofs and definitions more convenient.
1 is the empty product. You generate it by multiplying nothing together.
(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 )
The problem is that the usefulness of primes comes from the ability to break down a number into a unique set of primes, and including 1 as a prime defeats that usefulness. While your understanding of primes is an elegant idea, in sciences, usefulness trumps elegance.
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.
It almost seems to me that there are two very similar, overlapping mental models* of "primeness".
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.
There are actually many circumstances in which we do want to treat 0 as a prime. For example it generates a prime ideal in the integers.
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.
Prime numbers are defined as elements of rings, so we need two operations, the other being addition (understood as a group operation). So you generate 0 by 1-1.
I would argue that Zero is generally considered to be a natural. The naturals are often constructed via an initial object, and you can call that initial object whatever you want (some people write Z or Zed), but it's metaphorically a zero.
And other people care about natural numbers being the set of cardinal numbers for finite sets. Others want the set of natural numbers with addition to be a monoid. And actually most-known series in analysis are usually indexed starting from zero, because that's when the formulas are usually simpler.
I'm just saying to get to 1 through multiplication you can always multiply any number by its own inverse (which is another way of saying divide any number by itself)
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.
It's important to separate the structure from the objects. There's 2-the-natural-number, 2-the-integer, 2-the-rational, and 2-the-real.
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.
I've always thought of 1 as the definition of what a prime number is. Therefore, because it's generally bad practice to define a thing in terms of itself, 1 shouldn't be categorized as a prime number. Instead, it's like the Aristotilian Ideal of prime made linear.
tl;dr: "...whether or not a number (especially unity) is a prime is a matter of definition, so a matter of choice, context and tradition, not a matter of proof."
That shows the author doesn't understand the issue well, because every definition is semantics. That's what the word means. The question is why certain semantics are prefered.
The article mentions some of the historical context around defining 1 as a unit and also suggests its usefulness as a number due to base 10 arithmetic.
I also think it is interesting to call the article pedantic and then make a similarly pedantic argument just beneath that claim.
It's against the guidelines to say it I think. But I'm pretty sure at this point he didn't read the article. He has just read the title and thought he had enough knowledge on the matter to answer it without reading the author's writing.
honestly this is where math stops being logical and follows emotions. The logical reason one isn’t prime is because we don’t want to ruin theorems that were based on faulty logic so we continue with the faulty logic... yay?
If p is a prime number, then the product m×n being divisible by p means that either m or n is divisible by p. In fact, this is how prime numbers are defined when generalized to generic integral domains.
