Is 1 a Prime Number?
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Check out my series on building numbers from the ground up:
• Mathematics from the G...
Okay, we all know 1 isn’t prime. But it wasn’t always that way. It also wasn’t always considered to be a number at all. Join me on a deep dive into mathematical history to see how our concepts of numbers and primes changed over time.
A huge thank you to my patrons for keeping the channel alive. Please consider supporting me by following the link above!
An enormous thank you to Chris Caldwell et al. for their compilation of sources regarding the history of 1 as a prime. Their work is much more thorough and extensive than my video and covers many, many more examples of 1 defined as prime or otherwise. My video is intended as a rough narrative through line regarding the broad(ish) consensus regarding the status of 1. Having such an extensive list of sources was really helpful when writing. You can find their article here:
Caldwell, Chris K. et al. “The History of the Primality of One: A Selection of Sources.” Journal of integer sequences 15.9 (2012).
All music by Danijel Zambo.
TIMESTAMPS:
00:00 - Intro
01:20 - Antiquity
09:17 - Medieval
13:50 - Modern
23:59 - Outro
REFERENCES
[1] - Euclid. The Thirteen Books of The Elements, Vol. 2 Books III-IX. Translated by Heiberg. Dover Books on Mathematics, 1956.
[2] - Martianus Capella. et al. Martianus Capella and the Seven Liberal Arts. Vol.2, The Marriage of Philology and Mercury. New York ;: Columbia University Press, 1977.
[3] - Isidore, and Stephen A. Barney. The Etymologies of Isidore of Seville. Cambridge: Cambridge University Press, 2006.
[4] - Khuwārizmĭ, Mu.hammad ibn Mūsá, Robert, and Barnabas. Hughes. Robert of Chester’s Latin translation of al-Khwārizmĭ’s al-Jabr : a new critical edition. Stuttgart: Franz Steiner, 1989.
[5] - Stevin, S. The Principal Works of Simon Stevin. Vol.2B. Mathematics. C.V. Swets & Zeitlinger, 1958.
[6] - Morland, Samuel. The Description and Use of Two Arithmetick Instruments : Together with a Short Treatise, Explaining and Demonstrating the Ordinary Operations of Arithmetick, as Likewise a Perpetual Almanack and Several Useful Tables : Presented to His Most Excellent Majesty Charles II ... London: Printed and are to be sold by Moses Pitt ..., 1673.
[7] - Moxon, Joseph, and Henry Coley. Mathematicks Made Easie, or, A Mathematical Dictionary ... The second edition, corrected and much enlarged by Hen. Coley ... London: Printed for J. Moxon ..., 1692.
[8] - Rahn, Johann Heinrich, Thomas Brancker, and John Pell. An Introduction to Algebra. London: Printed by W.G. for Moses Pitt ..., 1668.
[9] - C. Goldbach, Letter to Euler dated 7 June 1742. Correspondance math´ematique et physique de quelques c´elebres geometres du XVIIIeme, siecle by P.-H. Fuss, Vol. I, pp. 125-129, Academie Imperiale des Sciences, St.-Petersbourg, 1843.
[10] - J. G. Krüger, Gedancken von der Algebra, nebst den Primzahlen von 1 bis 1000000, Lüderwalds Buchhandlung, Halle, 1746.
[11] - Euler, Leonhard, and John Hewlett. Elements of Algebra. Trans. John Hewlett. Cambridge: Cambridge University Press, 2009.
[12] - Gauss, Carl Friedrich et al. Disquisitiones Arithmeticae. New York: Springer-Verlag, 1986. Print.
[13] - Gregory, Olinthus. Mathematics for Practical Men : Being a Common-Place Book of Pure and Mixed Mathematics, Designed Chiefly for the Use of Civil Engineers, Architects and Surveyors. 4th ed / rev. and enl., by Henry Law. London: J. Weale, 1862.
[14] - Hardy, G. H. (Godfrey Harold). A Course of Pure Mathematics. 6th ed. Cambridge Eng: The University Press, 1933.
[15] - Hardy, G. H. (Godfrey Harold). A Course of Pure Mathematics. Centenary ed. Cambridge: Cambridge University Press, 2008.
[16] - Sagan, C. Contact. Simon and Schuster, 1985.
[17] - J. B. Andreasen, L.-A. T. Spalding, and E. Ortiz, FTCE: Elementary Education K-6, Cliffs Notes, Wiley, Hoboken, NJ, 2010. - บันเทิง
The number one should really be considered a planet.
I disagree - it deserves to be a dwarf planet
@@steampxnk2037 I was going to reply to say that exact same thing hah! You beat me to it!
One is obviously a vegetable
Would that Mercury ='s 1
Is 1 a sandwich?
I've always rationalised it even further by thinking that 1 is a sort of "superprime", which is sort of true because it is the multiplicative identity. We have to separate it from the normal primes because it screws with a lot of theorems involving prime numbers.
0 could be considered "supercomposite" since LCM(0,a) is a.
you are gay😂
Or you alter the theorems involving prime numbers. All of this is just made up anyway.
Primes become more and more common as you come down from infinity, culminating in infinite prime density at 1, or as you say, super-primeness
@X22GJP I disagree that it's all made up. I think alien civilizations discover the same numbers, and have the same debates.
I'm glad you're not strictly talking about math problems but also about the history and philosophy behind how we define things. And a video on deductive reasoning? Everything I ever wished for from a math channel.
Plenty on TH-cam
Another great video! I liked the bit about the Totient function as I hadn't come across it before!
I have come across it when determine size of various groups, such as U(20) would be 20 - 8 = 12 elements U(n) groups are groups when you pull out the numbers that are not relatively prime to n. And this group is a group under multiplication.
In the Contact movie/book it makes some sense to include '1' since the aliens were replying to early radio and TV emissions from 1936 when possibly the 1 was listed as a prime by some sources as you just shown.
Also, it makes sense to start a sequence with the fundamental unity of measurement in the same way that the
messages on the Voyager included distances based on a multiple (in base 2) of the fundamental frequency of the hydrogen atom.
So there is an option that we do not send 1 as a prime to the aliens and they may think we're awful and destroy us. Or vice versa.
@@forbidden-cyrillic-handle No. There is no way in which it is mathematically sound to actually consider 1 to be a prime number.
@@angelmendez-rivera351 I take it you didn't watch the video.
@@digitig Angel is a very well read mathematician, I'd say whatever Angel says overrides the video...
@@MagicGonads But this needs a historian of mathematics. If you read old mathematical papers, by the best mathematicians in history, you find that there was little agreement over whether to count 1 as a prone or not. There is agreement *now*, but as recently as my schooldays in the 60s, our textbooks still included 1 in the primes. And they weren't wrong, they were just on the verge of being rendered out of date.
next topic will be about number theory, i see..
I love that you've kept up with the humor. It's really fun to watch your videos and I look forward to more!
I seem to remember 1 becoming "un-prime" when I was about 10. As you're classifying that as "medieval", I now feel positively ancient!
Thank you for resolving a 50+ year annoyance! I learned some maths before formal schooling from some of my dad's 1930s textbooks: 1 was listed as a prime.
Then I went to secondary school and got (literally!) knuckle-rapped for answering 1 was prime, because the "new maths" had decided 2 was the first prime.
The discrepancy has always irritated me ever since. Your explanation from Gauss finally makes me happy!
Nice video: thank you.
I'm sorry the inconsistency got you into trouble but glad to have cleared it up!
I feel like the totient function excluding 1 isn’t because 1 doesn’t work as a prime, but because it isn’t being counted as a factor. If you count all the numbers less than 20 that share a factor and you consider 1 to be a factor, then there are none because 1 is a factor of everything.
It works either way, if 1 is prime then totient of anything should be zero as the formula says
There's another definition of the totient function that is a little more structural and clearly doesn't depend on this choice. The totient of n counts all the numbers less than n which are units mod n. That is, it counts the numbers m < n for which there exists an associated integer d such that dm = 1 mod n, meaning that dm - 1 is divisible by n. This definition is clearly independent of whether you decide 1 is prime or not, and then including 1 as a prime totally messes up the count as shown in the video.
The reason 1 is not being counted as a factor is because it is not a prime number.
@@angelmendez-rivera351
I’ve seen you typing on a lot of these comments and I just have this to say:
If I say all green animals should be considered plants because they have the same color, the current definition of “plant” is irrelevant because we are discussing the definition itself.
Definitions are non-universal, made up by humans and always changing. Therefore you can discuss hypothetical scenarios were the current definition does not exist.
@@Alex-02 *If I say all green animals should be considered plants because they have the same color, the current definition of “plant” is irrelevant because we are discussing the definition itself.*
If you say all green animals should be considered plants because they have the same color, then this does not address the current definition of "plant" at all, and as such, it does not replace said definition. Also, you chose a poor example for an analogy, because not all plants are green. In any case, if you want to define categories of objects by their color, then all you have defined is the color itself. This has nothing to do with biology. You can even define a subclass of the category of living beings based on color. Again, though, this has nothing to do with biology: this is simply about the color of the living beings. You are still including green fungi, green protozoa, green bacteria, etc., all in this category. There is no biological property that green animals share with plants, such that no other entities share that property, living or non-living. Therefore, it is ontologically inadequate to insist that they are in the same category.
*Definitions are non-universal, made up by humans and always changing.*
Well, this is just a false belief. Definitions are not mere labels you come up with arbitrarily on the basis of preference for labeling objects of your preference. Definitions have to be consistent with reality. To apply definitions to a set of properties, you first have to ensure that these properties actually are well-defined, and that there exists at least one object which has those properties. Then, when you apply the label to this set of properties, it follows that the label applies to _all_ objects satisfying these properties. If you want to single out certain such objects as being qualified, and excluding the rest, then you need to specify properties which are satisfied exactly only by those objects you are trying to single out. Also, for this to be justified, you cannot do this on an ad hoc whim. Otherwise, that renders all definitions as completely redundant and useless. What makes definitions useful is precisely their non ad hoc nature. Finally, there is the problem of decidability. A definition has to be decidable, because if there exists no algorithm by which anyone would ever be able to tell if a class of objects satisfies the given definition or not, then it is completely pointless, and as such, the 'definition' does not actually define anything.
How precise you want to get with all of this, it all depends on what academic discipline of study you are a part of, or whether the definition is just intended for colloquial nonsense. In mathematics, though, the highest level of achievable precision with these definitions is required.
On the note of universality, mathematics actually are universal. Mathematics are independent of culture, nation, religious belief, etc. Yes, the types of notation you use to talk about mathematical concepts vary from language to language and culture to culture. The mathematical concepts themselves, though, are universal. A quasigroup is defined in Norway in exactly the same way it is defined in South Africa. A topological space is defined in Malaysia in exactly the same as it is defined in the United States of America. The works that you see published by mathematicians from New Zealand are not in disagreement regarding the concepts and their properties with the works published by mathematicians in China. You seem to insinuate that mathematical definitions are on the same caliber as definitions of words you find in the Urban Dictionary. Yes, those words do vary in definition from location to location, and even just from person to person, and really, those words are not defined in any particular way at all, they are used arbitrarily, because they are not used to discuss concepts, they are just used to communicate basic bits of information about a particular ill-defined thing. This is not how mathematical definitions work at all, though.
*Therefore you can discuss hypothetical scenarios were the current definition does not exist.*
I can guarantee you that there is no hypothetical scenario in which the current definition would not exist and in which people have the knowledge of ring theory that we do today.
21:33 that was BY FAR the most convincing argument i have heard for demoting pluto from planethood.
I think what I find so appealing about your videos is that they tend to take ideas that a lot of mathsy people are probably familiar with (the representation of numbers as sets, the idea that one is not a prime, etc.) and shine a light on why we got there, rather than just the nuts and bolts of how those ideas work. I hope this isn't accidentally an insult, but I feel like the most enticing part of your videos has been that they straddle the border between mathematics and philosophy, albeit clearly on team mathematics. I think there has been some exploration of this space, but usually not with the level of sincerity and depth that you bring. Once again, an exceptional video!
Oh man, "until next prime" is a great closing stinger. If you used it just as a one-off for this episode, I urge you to consider making it your standard closing line.
the gap between primes increases very fast, we may never meet again
Primes: *literally means first
Also primes: *excludes the first number
According to Peano, 0 is the first number anyway. 😛
@@irrelevant_noob still excludes it.
Did Hector Salamanca just call me a numerologist?
Only if you discard scientific rigour.
No, he called you a *ding* *ding* *ding*
Super interesting!
The different scripts for the 3 signs is a nice touch that I didn't notice until they were together at the end.
I've been so looking forward to this, and you did not disappoint! Thank you for amazing math content that tickles my brain in a very wonderful way.
Great video. I really liked how you used this topic to talk about definitions and properties more broadly. I like how even if you’re talking about a ‘simpler’ topic you try to find the conclusion that helps a student to learn about math more broadly
Absolutely brilliant series of videos. A thousand thanks for taking the time to create these. More as soon as possible, please!!! 😃
Another Roof, you have been selecting some awesome math topics! To create some connective tissue, I noticed the Euclidian definition of the unit and then the number as a collection of units sounds a lot like the foundation for the "empty set" definition of numbers you outlined in a previous video. I didn't realize that method of number definition had such old historical precedent...
Anyway, looking forward to future videos. Keep up the good work!
Until I started looking into it, I had no idea how long the "1 is its own thing and not a number" idea persisted!
My next video will be something very different.
@@AnotherRoof lol (x-1)! + 1 is divisible by x only if x is prime or 1
@@ValkyRiver Everything integer is divisible by 1.
I never knew Hector Salamance was into pure math
Expert on video: if p|ab then p|a or p|b
Me, an intellectual: if plab, then pla or plb
As soon as I read that wrong, someone came through the window and took my degree away.
It's a cool video!!! Im hoping to see other topics in your channel cuase you explain so well but I enjoyed this one as well!
Many more videos to come my friend!
the still ongoing disagreement over the inclusion of 0 in the "natural numbers" is extremely annoying, i hope we solve it soon enough that we don't have to constantly use the unambiguous "positive integers" and "nonnegative integers" all the time
Great video as always. I would really appreciate your aproach for the history of number 0.
Another perfect video! Congratulations!
like the number 6
If it were perfect, it would be 6, 28 or (God forbid) 248 minutes long.
I just love your style of explaining!
Thank you for making such great content!
AAAAA i'm so glad you mentioned the distinction of primes/units in general rings like the integers! That's what I was watching for the whole time lol, definitely wish you mentioned a bit more about it but fantastic video as always!
I didn't want to go into too much detail on rings because they deserve their own video!
@@AnotherRoof I just binged your channel (+subscribed!) and from the Q&A you 100% seem like my kind of person, lol, so i have no doubt whenever you get around to it it'll be absolutely brilliant!
@@lexinwonderland5741 I agree with you that more should have been mentioned on it, since the distinction between irreducible elements and units in a ring is ultimately at the root of why 1 cannot be considered a prime number.
Look, do not get me wrong. I think that understanding how mathematical concepts from antiquity involved into mathematical concepts today as our understanding of mathematics improved and became more refined is very fascinating, and certainly an important kind of knowledge to have in general. However, as far as answering the question "is 1 a prime number?," the history is not enlightening at all: it ultimately does not answer the question. Yes, I know that mathematicians in the 1700s thought of 1 as a prime number, this is all well and fine, but that tells us nothing as to whether 1 actually is or should be considered a prime number or not. These questions are questions regarding the relationships between various mathematical concepts at a foundational level, not questions about names and conventions that mathematicians vote on. If you want to get at the question of whether 1 is a prime number or not, then you ought to compare the prime numbers with 1, analyze their properties and their roles within the integers, then compare how these things extend or fail to extend when you move on to other mathematical structures, like polynomials and Gaussian integers. *This* is how you answer the question.
Appealing to the history of mathematics actually reinforces most people's misconception that 1 should be considered a prime number, and reading the comments to this video has resoundingly confirmed this suspicion. I think that discussing the history is perfectly fine when addressing the question "why did we ever consider 1 a prime number?" or "how has our understanding of prime numbers changed?" But neither of those questions is the question the video claims to address.
@@angelmendez-rivera351 Do you know a good video or resource explaining this ring idea? (I don't know anything about rings).
This channel is AWESOME. Thank you
1 may or may not be a number, but 2 is the oddest prime.
Is it due to the fact it's the evenest one?
I enjoyed the history-centred approach :)
I think it made more sense when I learnt about the ring theory way of thinking where you have units other than 1 (eg -1 for those wondering). Otherwise classifying 1 as it's own thing seems slightly hacky
Yeah, I'm surprised he didn't mention that, it's the most powerful argument against primality of 1 in my opinion
P.S. Fun fact: in my first language "units" are called "one divisors" by analogy with "zero divisors" and it's such a nice name to explain the concept quickly IMO
@@vladislavanikin3398 wow that actually makes a lot of sense re one-divisors
@@vladislavanikin3398 It is the most powerful argument, and I would say, the only actual argument that really succeeds. All other arguments rely on logic that is fallacious, uncompelling. For example, this whole "it makes the fundamental theorem of arithmetic easier to state" nonsense is highly specious outside of the context of unique factorization domains. In most other areas of mathematics, mathematicians are completely unbothered when theorems have exceptions built into them. Heck, even for other theorems about prime numbers, there are many, *many* theorems which have 2, 3, and sometimes even 5 as the exception, and yet no one bats eye in calling these prime numbers anyway. The double standard makes no sense. Besides, the validity of a theorem should not depend so much on the precise details of how its phrased. Otherwise, you could prove literally anything by choosing "the adequate phrasing." This makes it obvious that the actual answer to the question "why is 1 not a prime number?" has nothing to do with the fundamental theorem. The question is answered, as you said, by the fact that 1 is a unit, and that units are conceptually and fundamentally different from irreducible elements.
every natural number is a uniquely defined product of primes, unique up to ordering. the number one is the product of no primes, also known as the empty product.
It's a bit like defining 0! = 1 - We just do it because the alternative is to have to keep track of the special case of 0 (or 1 for primes) at every stage of a proof.
If you're writing a proof you get to decide whether 1 is a prime number for the purpose of that proof. So many things were defined for just one proof and then the definition stuck
I’m surprised more of the comments aren’t about that “water is wet” premise
Excelent video mate! Very useful for teaching purposes
I really enjoyed the historical walkthrough. Thanks for recommending this video during your charity marathon.
James Grime did a nice video on primes and the fundamental theorem of arithmetic over on numberphile, good companion to this video.
The saddest thing about Pluto is that the definition you cite actually *was* crafted to fit our _modern_ usage of 'planet'. For most of history planets were any large circular object - moons were called planets, for example. This usage was lost, briefly, and replaced by the planets of astrology. The planetary society than ham-fistedly shoved through the vote in the dying hours of a conference, and the rest is history.
This was an amazingly well made video!
This is so satisfying. Thank you so much.
2:23 all hail to the monad in the category of endofunctors.
I’m glad you managed to squeeze at least one brick into the video at the end. I was getting worried.
This is a really interesting video, and I say this as someone who didnt study maths beyond GCSE
What convinced me that 1 isn't a prime was the logarithmic integral curve matching pi(n), that was like Nature telling us that 1 isn't a prime.
My favourite definition is "a number that has 2 unique factors" (no further clarification needed, provided that 1 and itself count as factors). 1 only has 1 unique factor, which means it has even less factors than a prime. Also the least number of factors of any integer (0 and infinity have infinity factors).
Thank you for another one. Can't wait for me.
Interesting historical bent. Wasn't expecting that, but I enjoyed it all the same.
An approach I heard in grade school is this: A prime number is a natural number with exactly two unique integer factors.
1 doesn't have two factors, only one.
Not true. 1=(-1)(-1). You have to say 2 positive factors
@@victoramezcua4713 -1 is there twice. Two of them are not unique factors. Anyway he could have used the natural numbers again as factors instead of integers.
@@forbidden-cyrillic-handle Or, you take the more reasonable approach of noting that all prime numbers have 4 divisors (-p, -1, 1, p), and 1 only has 2 divisors (-1, 1).
@@angelmendez-rivera351 if I f we go that route then let's add also i and -i. Let's also add rational numbers. And why we study numerology at all? Obviously the initial definition of what is a prime number is numerology. Therefore let it burn.
@@forbidden-cyrillic-handle Uh... I should probably remind you that -i and i *are not integers.* However, if you want to talk about *Gaussian integers,* then that is perfectly fine, then we can talk about -i and i as units alongside -1 and 1. In fact, the concept of a prime number is well-defined for all GCD domains. The integers, the Gaussian integers, and the ring of polynomials over the integers, are just examples of that.
Using "Planet" here is kind of funny because of the original meaning of it. The ancient Greeks didn't have the same idea of what a planet was. The word means wonderer. They believed they were moving stars. People had no concept of the modern definition of "planet" until the 1600's when the telescope was invented. Interesting to see this come up in a video about how the classifications of 1 as a prime has also changed over time.
14:56 "Every integer can be written uniquely as a product of primes"
This was a simplification of what was highlighted in the text, but I'll address it anyways. That statement is false if one is not a prime, because one is an integer that would therefore not be able to be written as a product of primes.
To address this, I would say a better elaboration of an integer would be "Every integer can be written uniquely as a minimal number of prime numbers". This way, it retains the fact that one is not *necessarily* included in the factorization of a number. This elaboration would define 1 as a prime number, which I'll get to in a bit.
This would not have issue with the quotient formula, as you simply use the minimal number of prime numbers in the formula, which doesn't include one because its not needed and therefore not included in the minimum number of prime factors. It also works better with q(1) than the currently held definition, as the current trick to solving it would be to say q(1) = 1 * .... we ran into a problem didn't we? there is no prime factorization of one if one isn't a prime, so q(1) is not solvable using the trick. If instead we use my reasoning for integers and say one is a prime, we'd use the minimal number of prime factors of one, therefore we'd just use 1. q(1) = 1 * (1-1) = 1 * 0 = 0, there are no integers less than 1 that share a factor with it. This trick to solving the equation now still holds true, even if it was a trivial matter to do.
Also, the p|ab argument I heavily disagree with. Just because something is trivially true, doesn't make it problematic, it just makes it not informative. But it is no more informative than say 3|6*9 since all that's saying is 3 divides at least one, but we know it divides both, just like 1 divides both a and b. So just because it isn't useful if it was defined as a prime, doesn't mean it shouldn't be a prime.
By changing the definition, I made your erroneous simplification valid, I made the method for solving q(n) still hold true, and also solvable for q(1), and I've stated why p|ab is an inherently subjective reason for arguing against one being prime and therefore not a valid argument for a system that should be objectively true as much as possible. The only issue I can see is that there is a disconnect with the relationship between a number and its possible factors, as there is only one-way uniqueness now, meaning any set of factors points to exactly one integer, but one integer can be written as multiple factorizations. This is trivial in my opinion, as the wording we've used is "all prime factors other than one", when we could reinforce the definition by saying "the minimal number of prime factors". The only issue is saying it, because its not as succinct as "all prime factors", which again, subjective.
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Great video. I wonder if the primality of 1 was the "Is Pluto a planet" pub discussion argument basis of its day.
Of course we still have to contend with zero as an exclusion qualifier on so many functions. It could be a nice follow on to this video to do one on zero, especially if you get into varied number bases. Working title: [When did nothing become something?]
As a number theorist, I can tell you there are _countless_ reasons to exclude 1; so many I wouldn't even know where to begin. Excluding 1, if I tried to list all the theorems that need to be phrased "let p be a prime or 1," then I would be hard pressed to find even a single theorem. Including 1, if I tried to list all the theorems that would need to be phrased "let p be a prime other than 1," then I might have to list every theorem which refers to primes at all. Almost every pattern regarding prime numbers breaks at 1, suggesting it's more useful to exclude it.
I think that's pretty much the conclusion I reached in the video. I just wanted to streamline the examples I gave so as not to overwhelm viewers who weren't as familiar with the subject
@@AnotherRoof Yeah it’s definitely not the most intuitive fact, as evidenced by all the historical examples you gave. It only becomes obvious after seeing what we can do with primes.
One way to justify the non-primality of 1 is to categorize natural numbers by their order in terms of divisibility. We say a | b ("a divides b") iff there is a natural number n so that an = b. Then we say a < b iff a | b and not b | a. Then you can verify that < is a strict partial order over N, and moreover, that 1 is the least element and 0 is the greatest element. That is to say, for any natural number n, 1 | n and n | 0. For any prime p, we have 1 < p, but there is no other number n for which n < p. So if you draw a graph of
You are the math hero we always wanted but never deserved ❤
One is prime. I don't care if it messes up someone's arbitrary definition of some other thing. One is _the_ prime.
13:06 I do think that if we consider one to be a number, and define primality as "the numbers that can only be measured by 1", it would still make sense to exclude 1 from the primes, because *1 can't even be measured by 1* (a number can not be measured by itself: 3:23)... Interesting math history video anyway!
The 'convenience' argument is always funny to me because of how many algebraic theorems state "For any nonzero number x..."
It's a balance. The convenience of zero existing far outweighs the inconvenience of excluding it from many theorems.
@@MuffinsAPlenty I disagree. The truth is that 1 not being a prime number has absolutely nothing to do with the convenience of how we state the fundamental theorem of arithmetic. I dislike it when people who are educated in mathematics try to present everything as if it is "a matter of convenience," because that is just not true at all. 99.99999% of things in mathematics are never about convenience. Sure, _some_ things in mathematics are purely a matter of convention. The fact that we still use what I consider to be bad notation for derivatives, such as dy/dx, instead of using functional notation for them, such as D(y), really is entirely a matter of convention, even if there if one of the conventions is objectively superior to the other for three dozen reasons. It really is just notation. Using base 10 for our positional representation system to denote integers, instead of base, say, 60, like the Babylonians used to do, is s convention. In fact, using positional representations systems at all, rather than non-positional ones, is itself a convention. So, I am not saying there are no conventions in mathematics. My problem is that everyone on TH-cam presents so many things (like 0! = 1, or 1 is not prime, or the ideas about the radical symbol being used as a function, etc.) as conventions that actually are not conventions. 1 not being a prime number is not a matter of notation, and it is not a choice we get to make. It is an irrefutable mathematical fact, that when it comes to commutative rings, and how we classify objects, there are exactly four families into which these objects can and do fall into, and these four families are exhaustive, distinct, and mutually exclusive. We can characterize these families as (a) those objects x such that there exists some y such that x•y = y•x = 0; (b) those objects x such that there exists some y such that x•y = y•x = 1; (c) those objects which are neither of the above, and whose proper divisors are exactly the objects in (b); (d) those objects which are not in (a) and have proper divisors in (c). How we choose to label these four families with four distinct labels is completely arbitrary, yes. We can even choose to have multiple distinct labels for the same individual family, yes. However, we can never choose to insist that elements from distinct families actually belong to one single family, and should be labeled as such. This is not a choice we get to make, because it is conceptually inconsistent with the mathematics above, and it leads to ill-defined terminology. Calling 1 a prime number is entirely analogous to insisting that my Toyota is a plant.
Anyway, what this comes down to is, there is an actual conceptual reason behind why 1 is not and cannot be a prime number, no matter how much we would like it to be. It has nothing to do with how easy it is to formulate the language the factorization theorem in the English language when we reject 1 as a prime number. People should be taught the actual conceptual reason behind why 1 is a prime number, not this "it's more convenient" nonsense. And no, I am not saying we need to be formal about it. Simple intuitive explanations will do.
I know that, as far as colloquial language is concerned, you can arbitrarily coin words and make them mean absolutely nothing and use them in self-contradicting fashion, or make them have useless meanings for the sake of trolling, and you can arbitrarily change how you use those labels any time you want to. But, this is not a colloquial language we are dealing with, now, or is it? We are dealing with abstract mathematics and number theory. It is a serious discipline of study. Going around telling biologists that your Toyota is a plant, because you chose to change the definition of the word "plant" in some unspecified, ad hoc way to include your Toyota in the definition, is not how science works. Similarly, simply changing the definition of terminology ad hoc so that 1 is a prime number, that is not mathematics. That is just pseudomathematical crankery. Now, I am not actually accusing anyone of having done this. That is not the point I am making. The point I am making is that educators need to stop encouraging this idea that all definitions exist only according to convenience, and that we change them willy-nilly how we want to. This is true of colloquial language, but not of language in academic disciplines of research. Educators also need to stop presenting fundamental mathematical facts that we do not get to do anything about as if they are something we choose. Perhaps you think otherwise, but that would beyond incomprehensible to me. Maybe I am out of my element. Maybe my strong advocation for the idea that people should not be taught false things makes me unreasonable, although if true, that gives me very little faith in the human species. But I remain unconvinced that this is the case.
@@angelmendez-rivera351 Thank you for your reply! Sorry it took so long to get back to you. I would say that, above, I went with a more "expedient" response than I usually would. Instead of going into my usual spiel about how one can notice that 1 (and other units) do not behave like primes exactly, and with careful enough definitions, we can pinpoint why, I chose to merely respond to the op granting the position that 1 was de-primed for convenience. I think my response was less of a defense of de-priming 1 for convenience sake and more of "whether or not I agree with your conclusion, I don't find your reasoning convincing" and was sloppy in my own execution of that.
Now, I will say that I probably don't find saying 1 was de-primed out of convenience to be as bad of a thing as you probably find it to be. But I agree with your point that it isn't really actually the case.
Awesome vid!
Now I understand !
1 is a prime number IF we work in the additive/Goldbach domain.
It is not in the multiplicative domain because 1 is the neutral element of multiplication (which is most used because Goldbach's conjecture is not yet proved)
It's too bad we don't actually teach the fundamental theorem of arithmetic in a fully qualified way... the most accurate way to say it is that each number, up to an invertible factor (1 and -1 for integers) has a decomposition into a unique product of primes, and some invertible factor. Said that way, it extends straightforwardly to more complex sets, such as the gaussian integers.
If you include -1 in your notion of "number," then you should include the negative primes -2, -3, etc as well. But then the product is no longer unique, only unique "up to associates" (ie if you have two factorizations, the primes of each can be paired up so that each prime in one product is an invertible factor times the corresponding prime in the other). And do you include 0 as a number? If so then what is the unique way to write 0 as a product of primes and an invertible element?
Stating the full version precisely is actually very difficult to get right; there's way too much complexity for someone encountering the fundamental theorem of arithmetic for the first time to handle. I think it's much more reasonable to teach the version that restricts to positive integers first, and then at some point - after they've gotten comfortable with the idea but before seeing factorization in other rings - to ask them to think through and discuss how they would generalize the statement to work with the integers, without being able to refer to a notion of "positive." This would give them an opportunity to contend with the nuances of units, associates, zero divisors, etc before moving on.
@@japanada11 Yeah, I forgot to add that unique factorization only applies to multiplicatively cancellable numbers, which discounts zero. Thanks for pointing that out.
I would take the view that -3 is a prime number, but it's "the same" prime as 3. Viewed that way, 6 = 2*3, 6 = (-2)*(-3), -6 = (-2)*3, and -6 = 2*(-3) are all "the same" factorization of "the same" number. (This is slightly different from what I said before, I know.) Explained by example like that, I don't think it's too hard to follow, though the precise abstract statement is indeed too complex to use as an introduction to the subject. I just think it's a shame that when we teach it, we don't even address the issue of negative numbers at all.
Fair point! Even if students are presented with the standard easier definition, I think it's important to get them to explore what happens with 1, with 0, with negative integers, with what happens if you allow fractions, etc. It really is a shame that unique factorization (as with so much in precollege math) is often presented as a fact to be learned and internalized, rather than a launchpad into many deeper questions!
Loved it!!
Until Next prime could become your outro text, it was amazing!
The one thing I don't like about 1 not being a prime is that then it doesn't have a "prime signature"
It's weird that 1 is impossible to construct using only prime numbers
1 is a special number that is neither prime nor composite. It is similar to zero who is neither positive nor negative.
Doesn't it kind of have a prime signature, that being a completely empty set of primes?
Yeah this is an interesting point but because of 1's nature as the multiplicative identity there's really no escaping the fact that 1 is just weird and special and doesn't behave like any other number.
If you allow for empty products, then 1 is possible to construct using only prime numbers. 1 is the product of no prime numbers :P
@@trolledwoods377 Yes.
This is a great video! For some reason, I had a really strong sense of déjà vu while watching it
maybe because my recording setup hasn't changed since my first video and won't be changing any time soon? :P
If I was an archaeologist I'd point out that talking about "hunter gatherers" as the pre antiquity form of society is... Antiquated
I get where you're coming from, but I find the planet analogy a bit iffy, as it was decided without asking any planetary geologists.
That said, I loved the video and I especially love your taste in movies.
Very Modern:
Consider Gaussian primes. There are four "one"s and it's not negligible e.g. 2=i(1-i)^2. In such cases, one is considered to be in a special class separated from primarity.
Great presentation! Humour and maths (yes, maths) - great combo :D
Very interesting, thanks.
Another great vid
But ... if 1 isn't a prime then how can you write 1 as a sum of primes? In order for every integer to be written as a sum of primes 1 in and of itself must be prime
I thought Eratosthenes, and his eponymous Sieve, were sufficient to show that 1 is not prime: “if a number is prime, eliminate all of its multiples and continue on until you find the next prime.”
Since if you eliminate all multiples of 1, there are no numbers left, 1 cannot be prime.
I think that's putting the method before the definition.
One could also use that line of reasoning to prove that 1 is the only prime, for example.
Neat idea though and thanks for watching!
1 is my favourite number and this is just 1 cool reason to love it, pretty neat that it was once counted as it's own thing in the past, simpler times, also finally got how someone might see 1 as prime if seen in a additive sense rather than multiplication sense, this topic has interested me forever and this video did it justice, nice work!
The concept of a 'prime' number makes no sense in an additive sense. It is inherently and necessarily a multiplicative concept.
@@angelmendez-rivera351 I meant additive sense as repeated addition
@@1ToTheInfinity That does not address anything I have pointed out.
@@angelmendez-rivera351 seeing 42 as 7+7+7+7+7+7 or as a rectangle prioritizes the *pairs* of factors that multiply to 42, and since 1, 5, and 79 both can only be divided by 1 and itself, it makes perfect sense 1 is among the primes, but in a modern factorization sense of 2*3*7, 1 acts differently than 2,3 or 7 now that it is part of composing numbers, in which 1 doesn’t help with the prime factorisation
@@1ToTheInfinity *...since 1, 5, and 79, are all examples of positive integers which can only be divided by 1 and itself, it makes perfect sense 1 is among the primes...*
No, it does not make sense. The problem is, "only divisible by 1 and itself" is not the definition of a prime number. It never really has been the definition of a prime number. The video talked about how, in the past, different definitions of prime number were used, and the one which was used the most was this notion of being "measured" by another number. For example, "the prime numbers are numbers that are measured only by the number 1 and nothing else." This is the definition that you encounter Another Roof mentioning throughout the video, because that is the definition that was used in antiquity and in medieval times. But saying that a number is divisible by x is _not_ the same as saying the number is measured by x. This was explicitly stated in the video too. Why are they not the same thing? Well, because (a) the mathematicians of those times did not consider numbers to be measured by themselves. In other words, 1 measures 7, but 7 does not measure 7. Again, this was explicitly stated in the video. The other reason is that (b), well, it simply is not true that 5 is divisible by only 5 and 1. 5 is also divisibly by -5 and -1. But for a long time, negative integers were never considered in number theory. This is an oversight that needs repair.
Let me go back to part (a). As mentioned, 1 measures 7, but 7 does not measure 7. That is how mathematicians used to think of it, but why? What is the difference between saying that 7 measures 7, and that 7 divides 7? The difference is that 1 is a proper divisor of 7, but 7, although it is a divisor of 7, is not a _proper_ divisor of 7. The distinction between a divisor of x and a proper divisor of x is actually the exact distinction as the distinction between subset and proper subset. It is also completely analogous to the distinction between "equal or less than" and "less than." y is called a divisor of x if and only if y divides x. But, a proper divisor is more special. y is called a proper divisor of x if and only if y divides x AND x does not divide y. Now we get it: 1 divides 7, but 7 does not divide 1, so 1 is a proper divisor of 7. The whole point of proper divisors is that we only care about the divisors of x that are "simpler" than x: we do not at all care about the fact that x divides itself, because that is just completely useless, and trivial. Like, all numbers divide themselves anyway, so do we care about x being a divisor of x? This is why the concept of proper divisors exists. And the concept of proper divisors is the concept that the mathematicians of old were alluding to when they said "x measures y." The old language of "x measures y" translates into the modern language as "x is a proper positive divisor of y." Positive, because again, negative numbers were never considered seriously by European mathematicians prior to like the 1500s.
So, now that you know what the distinction is between divisors and proper divisors, and now that you have watched the video and you understand that prime numbers were always ultimately defined in terms of proper divisors, albeit in a different language, it should be clear why 1 is not a prime number. You see, the definition of a prime number always has been "a positive number which is only measured (positively) by the number 1." Translating this to the modern language, this means "p is prime if its only positive proper divisor is the integer 1." _THIS_ is the definition of a prime number. This is what it has been since basically forever, in concept, even if the language used was different. But look: the number 1 has no positive proper divisors, since its only positive divisor 1 itself. So, it does not actually satisfy the definition of a prime number. The problem here is that most teachers, and most textbook authors, believe that the actual definition is too complicated for grade schoolers (i.e, children) to learn. So they simplify it down, they get rid of all the "technical details," and so they just tell the children that 'a prime number is divisible only by itself and by 1.' But what this is a mistake, because this is completely misleading: you have changed the definition itself altogether by getting rid of the technical details. You *cannot* get rid of the technical details, because they are the _most_ important part of the definition, and not the _least_ important part. Prime numbers always have exactly 4 divisors: -p, -1, 1, p. The proper divisors are -1 and 1. But -1 and 1 have no proper divisors at all! They are fundamentally different from the prime numbers, and do not satisfy the definition of a prime number, so they belong to an entirely different classification system. -1 and 1 are called "units," or unitary numbers. One of the defining properties of units is that they divide _all_ numbers. Also, the product of two units is always a unit. Notice how this can never be true with prime numbers: the product of two prime numbers necessarily is a composite number, by definition. Also, units cannot be divided by any prime numbers at all. They can only be divided by units. 2 cannot divide 1 or -1. -7 cannot divide 1 or -1. Units are fundamentally different from prime numbers. Saying 1 is a prime number is like saying "a car is an animal." Like, no, that is just way off. Also, 0 is not a prime number either. It is also not a unit, because you cannot divide by 0. 0 is what is called a zero divisor. In the integers, 0 is the only zero divisor, but this is not true for all systems of arithmetic, actually. For example, when you work with matrices, there are some matrices not equal to the zero matrix, but are zero divisors anyway. A zero divisor is a quantity x such that x•y = 0 for some nonzero y.
By the way, this distinction between primes, units, and zero divisors, is not exclusive to the integers. It holds universally. It holds for matrices, polynomials, associative vector algebras, the rational numbers, the complex numbers, the Gaussian integers, the dual-integers, the split-complex integers, etc. It holds for all commutative systems with associative multiplication and addition. A unit is some quantity x such that for some y, x•y = 1. The integers -1 and 1 are units, and they are the only units in the integers. In the Gaussian integers, the units are 1, -1, i, -i. In the rational numbers, everything is a unit, except 0, which is a (trivial) zero divisor. The analogue for prime numbers in more general settings is called an irreducible element. An irreducible element is an element that (a) is not a zero divisor, (b) its only _proper_ divisors are units. Remember: units have no proper divisors, so they are not irreducible elements. A composite element is just a product of two or more irreducible elements, just like in the integers. Note: how you factorize composites into irreducibles need not be unique in general. But, in the integers, it is unique.
Now that you know all this, you are probably going to object that this all just "the multiplicative perspective," and that when you look at it from "the additive perspective," it is very different. But that is not the case at all. Why? Because the definition of a prime number has *absolutely nothing* to do with how you write numbers as repeated sums of integers. In fact, there is nothing interesting to point out: all integers can be written as sums of two integers in infinitely many ways, and all nonzero integers can be written as sums of 1 or -1 alone. So, ultimately, the additive structure does not matter at all. My point is, there is no such a thing as "the additive perspective." Insisting that there is one is borne from a severe misunderstanding of how number theory actually works.
I'm currently doing research with prime numbers. Interestingly, I sometimes find one fits as being prime, and sometimes it doesn't. I'm not convinced one way or another but it is easier to not think of one as prime.
It's not really accurate to say that Gauss was the first to prove the prime number theorem. The unique representation of numbers as products of prime factors was known even to Euclid. To understand why Euclid couldn't present a proof, look at his proof that there are infinitely many primes. It actually just proves that there are at least 4 primes. The point of the proof is that the same reasoning can be applied to any multitude of primes, and thus anyone who understands the proof can see that there are infinitely many primes.
Similarly, Euclid's lemma (Elements VII Props. 30-32) can be applied repeatedly to eventually divide any number into prime factors. Euclid IX Prop 14 proves that this decomposition is unique. Many ancient authors clearly assumed this fact, so it was evidently known. Perhaps nobody ever thought to prove it, or had the notation to prove it in full generality, or perhaps those proofs are lost. But Gauss was merely writing down what people like Euler were already using in practice and had been for thousands of years.
The problem is one is the first "number" so it should be prime. That is what prime means, like prime minister.
All numbers are derived from 1, that is why they are divisible by one. Why not go back to the original idea and call the "prime numbers" incommensurate.
Exciting!
11:06
Water is makes things wet, not wet itself.
Great video. I would like to see some similar subjects. For example, for modern French system 0 is a natural number and 0 is positive but not strictly positive. For modern american system, 0 is not a natural number, it is non-negative but it is not positive.
(Personally I perfer without negation, namely to say positive/strictly positive rather than positive/non-negative, but just my subjective thought)
Would like to know how historically these things are taken into considerations.
In my opinion, I don't consider 1 as a prime bc of the unique prime factorization of numbers, if 1 is prime, we have an infinite ways to express any number: 2 = 2*1 = 2*1² = 2*1³ and so on
It really just sounds like the answer is "It's more complicated than a binary answer." And let it be used where it should, and not used where it shouldn't.
I watched the video, and it was cool!
7:11 is a fun book to read!
06:21 Live long and prosper.
Not sure why 9 would be considered perfect...
6, on the other hand, does meet the definition of a perfect number: its non-self factors add to itself. A property shared by very few numbers. 28 and 496 are the next two smallest such numbers.
0:06 ok, here's my argument and we'll see how correct (or more likely how wrong) I am after the video:
In my opinion 1 is not a prime number, because prime numbers need to have 2 factors with one of them being 1 and the other one being the prime number itself. 1 can't be prime, because it only has 1 factor. The factors aren't 1 and 1, that makes no sense, because you can't have multiple identical factors. And numbers need to have exactly 2 factors to be prime
You, sir, are a prime example of a clever man.
I watched Contact a few weeks ago and I'm sure the prime number broadcast started at 2 in the film. I wonder why it was changed from the book.
No, not at all. There are many environments and interpretations that magnify just how much of a non-prime 1 is. The mu function value at 1 is +1, decidedly not -1 as would be required. This isn't just arbitrary assignment of value, either. Play around with the Legendre formula for the prime counting function to convince yourself that the [x/1] = [x] term in the sum needs to be positive, and could not at all be negative.
In this context of a generalized Sieve of Eratosthenes, allow me to give my interpretation of 1. The primes are primes because they cast out all composites from your list of numbers from 1 to n. The number 1 writes down the list from 1 to n. This is essentially recognizing the pattern 1+1+1+...+1, 2+2+...+2, 3+3+...+3, 5+5+...+5, ... . The number 1 being added again and again is the table being made, and the primes being added is the actual Sieve in action upon that table. Since 1 can do nothing other than write the table down again (or erase the entire table as it goes, leaving neither primes nor composites, if that is your view), it is therefore decidedly and definitively not prime.
Since the definition of prime can vary, the only requirement is that any result referring to primes has to be consistent. So just because the same word is used two different ways doesn’t mean they are the same thing. A great example is the word “horse”. One use is a horse is a mammal with four legs, elongated head, hoofs, etc. But does a sea horse fit the definition. No. But it does share the property of elongated head. What about saw horse? It does share a property with the animal horse because it is used to carry a load.
So if we used a different word it would clear up the problem. So let’s use the word “aaa” to refer to prime like numbers that are the set 3,5,7,11,… “eee” are the prime like numbers in the set 1,2,3,5,7,11,… and “iii” are the prime like numbers in the set 2,3,5,7,11,…
Then the fundamental theorem of arithmetic could be stated using “iii” with no qualifications. To state it with “eee” you would have to add “except 1”. And so on.
So the question is not so much, what is a prime? It is what is the easiest way to define a set of numbers that have certain properties, like divisibility, factors, and so on. The easiest way is to use one word.
Again the main goal is consistency and your definition determines how other properties are stated.
Now, is 0 a natural number????
When I was at school - and I don't quite remember which degree of school it was - we did consider 1 to be one of primes, but many of our statements looked like "for all prime numbers greater than 1 ...". So yeah, 1 in my mind satisfies all logical requirements on primality but considering it a prime is impractical because then you have to exclude it way too often.
If you have to "exclude it too often," then it literally *does not* satisfy the conditions for primality. You would not have to exclude it so often if it actually did. Besides, if you take a look at the actual definition of a prime number, you will see that 1 is not one.
@@angelmendez-rivera351 why are you spamming the comment section?
@@TykoBrian7 I am not. Please refrain from making baseless accusations. Such behavior does make a good look for you.
5 seconds into the video "But 1 is not a number..."
[here we go again.jpg]
I missed a trick there -- an infinite loop of videos would really rack up the views!
11:10 Stevin doesn't describe the attributes of the number components, but what constitutes them. One molecule of water is comprised of a water molecule, just as is a bucket of water made of water molecules. Their wetness is not relevant. It's not a fallacy.
Well now that I think about it totient being always 0 if 1 is a prime totally makes sense, because every single number is divisible by 1 so all numbers share factor so there is exactly 0 of numbers that do not share factors
The song by Three Dog Night says, "one is the loneliest number". Now we know why.
14:45 So if 1 is not prime, then 1 is not an integer? Maybe 1 is not a number? Maybe every number is 1 or part of 1?
1 is the Normal. It is not any particular length, but what you take it to be, and then all other numbers are normalised by 1.
Is there a 0? Is it an integer? What primes make up 0?
Maybe there is 0 (nothing) and 1 (all) - and everything else is between these limits?
Maybe 1 is like the Unit Circle and 0 is its centre?
Can all the infinities greater than 1 be mapped into the interval less than 1 and greater than 0?
It seems like 1 is the Unity of all numbers.
1 is the Great Normal. A set or category is normalised by 1, and then can be counted 1, 2, 3 etc.
Without a set or category, numbers have no meaning.
There is nothing that is 2 or any other number compared to 1, if they are not part of a common set or category.
Every individual thing is just 1, until there is a set or category.
Amazing video!
There was one part that that seemed a little off to me though. At 19:34 when you talk about how "1's behavior doesn't mirror that of the other primes" you give the example that since 3 | 4 x 6, claiming that because 3 is prime it only divides 6 and not 4. But 2 is also prime, and it divides both 4 and 6. And in general, if the two numbers a and be are not coprime, you can generate infinitely many examples where a prime divides both numbers in the product.
The way you phrased the theorem made it seem like an exclusive or ( p | ab -> p | a XOR p | b ), in which case 1 should not be considered a prime only if the theorem requires a and b to be coprime. On the other hand, if it's an inclusive or, and a and b are free to be anything, then 1 should be considered a prime.
Could you clarify which kind of OR you were going for in the video?
he never said only 6 and not 4, he just said either 6 or 4. both would also count under the definition of 'or'.
@@nathanisbored That's what I'm confused about, if it's an inclusive or, where it can divide both 4 and 6, then how does 1 not mirror the behavior of other primes?
It's inclusive or. And I do say the result is true for 1 but that it's *trivially* true (because 1 divides everything) so the result where p=1 is a result that doesn't really say anything about underlying structure of integers.
It's obviously inclusive OR as you just showed with the 2 dividing both. The use of the word "either" when talking about this was the part that confused you
18:58 well this isn’t wrong, it’s internally consistent. If one is prime, then every number shares a factor with twenty- one. So phi of n would be zero.
isn't this the same reason we don't include the zero ring as a field?