thank you, just "with two exercises at the end", would expect other kind exercises, they are nice thoughts, maybe, just maybe, this will help me today.
nice question at the end. thanks for making the video. but to be honest, its far too long. this proof and the nice questions you posed could have been presented far more rapidly. remember viewers have the opportunity to replay anything they dont understand so all u need to do is make sure you explain the ideas concisely and accurately.
Very nice! I had the privilege of meeting him in Hungary. I have always been impressed by him and guys like Polya. His use of the "probabilistic method" for bounding Ramsey numbers was an incredible insight that has stayed with me since I saw it.
Wow, I'm jealous. I have read that he is a very eccentric character, he'd be a very interesting person to meet. I've also read that he had this idea that God would have a book of all the most simple, eloquent proofs, and that when he created what he thought was such a proof, he'd say 'one for the book'. While I obviously (I see you've subscribed) disagree with him, I think I'd have exactly the same idea if I were a theist.
This comment has received too many negative votesshow
This is the worst explanation and *pseudo*-proof of the theorem I've ever seen. Truly pathetic. Why continuously show yourself to be a pompous jackass?
proof is correct. It's in this form since Euclid. Remember this is a proof by contradiction. not necessary to say X can be a composite with prime divisors greater than all n primes, couse those prime divisors are not present in the list of n primes. X is the prime contradicting the ipothesis obtained by construction from the set of numbers 'generated' by the n primes. In such system, 59 and 509 would not exist. But the set of numbers is clearly not that with only n primes (till 13)
Ok, very good video BUT you cannot really assume that X is a prime. Only thing you can say is that the primes in the prime factorization of X is not in your finite list of prime numbers. So question 2 shouldn't really even be there...
That there is an infinite sequence numbers, by itself, is not a sufficient proof that there are an infinite number of primes. The purpose of the video was both to show that there were an infinite number of primes and to show the technique of proof by contradiction.
How do you show, that amongst the infinite number of numbers that all but a finite number are all composite? From any fixed finite set of prime numbers, for example, you can create an infinite list of multiples of them that are all composite. How do you know you can't find a large enough set of primes such that eventually the set of their multiples cover all numbers?
That doesn't even parse as coherent English to me. The definition of infinity has no relation to multiples of numbers. The standards of proof in mathematics requires that you being universally convincing to basically anyone versed in the practice of mathematics. Your responses here do not rise to that level.
Great video. I'd seen the proof before but still enjoy watching mathematics being shown on YouTube.
I think a video tutorial on the logical implication (truth table, negation, etc.) would help people understand why proof by contradiction works. I remember being confused as to why it worked when I first saw. This was only fully cleared up having had the logical implication explained to me.
hey websnarf... i absolutely love your mathematically related videos. this video was especially interesting and reminded me of something that i noticed a couple years ago... every fifth element in the fibonacci sequence is divisible by five, or so it seems. is it possible for you to prove this? or am i wrong?
Suppose t = f(5*n-1) and 5*s = f(5*n). Then f(5*n+1)=t+5*s, f(5*n+2)=t+10*s, f(5*n+3)=2*t+15*s, f(5*n+4)=3*t+25*s, f(5*n+5)=5*t+40*s=5*(t+8*s). So it is straight forward to complete the proof using the principle of mathematical induction (exercise to the reader.)
Congratulations! That's well done for someone of your age. Of course, I've long suspected that there was some sort of lurking genius behind that Welsh accent. :)
Aha! Paul must have sent you my way! I don't otherwise see how you would magically subscribe (appreciated, BTW) to me then suddenly decide to focus on my math proof video. My food challenge videos are way cooler. :)
Mister websnarf, I have a question (such question is connected, I guess, with "Re: Math Education: An Inconvenient Truth"): here, in Brasil (at least, in my poor opinion), the approach in regards to Math teaching is [basically] centered on "determine problems" (instead of "demonstrate problems"). What do you think about that? Could proof techniques be presented on basic school?
Proof is something you don't see until later in high school, and a lot of americans don't ever get to see proof by contradiction unless they continue with more advanced math in college. This kind of stuff can be hard for some people.
Yeah, i've used bad assumptions before too but used in projects I've known of them as a 'null hypothesis' which requires a test and I used a statistical test to help me out with that.
Yes, that is proof of non-existence, but only for an omnipotent God. A deists God might not be described as omnipotent. Russells paradox pertains to the set of all sets, and asks is this set part of itself.
I loved the math. I am 35, My education is only high school level. But have always been ok with math. I actually answered the 2nd question though it made me think. Some geometry would be cool. This math was explained just a little above my level which is great, I dont mind being challenged with that.
I am really glad you answered Q2 on your own. (In some ways Q1 is actually harder.) Its really hard to make in a single math lecture something which can be understood, and also challenge everyone in the same way.
I absolutely loved this video. I thought it was clear and quick, considering your time constraints. I love this stuff. I liked the level and could probably go up a level or two, but I'm in my confort zone. Great stuff.
I keep rereading this comment, but I can't figure out the innuendo, or twisted sarcastic alternate meaning. So I'll just provisionally accept it as a complement for now, knowing full well what I am setting myself up for. ;)
That makes sense, but I always thought 1 was a prime. So if the prime cannot be factored into two smaller primes it is not a prime? I wrote this java program to test it, but I didn't include the factoring.
A 'long' would have been more appropriate, but since '// where iLimit=100', and I 'in error' assumed that 1 was a prime I felt that I didn't need a data-type greater 32-bits
Just use Ruby, it elegantly facilitates Bigint, Complex, Fraction etc. This and the Python underscore madness (as well as numbers not being objects) made me switch to Ruby (knew it way before Rails).
websnarf, I really enjoyed the proof. Being familiar with the "exists"- and "for-any"-quantor etc. I don't know the =><= symbol, very intuitive. But what are the thee dots? Abbrev. for "therefore"?
Thanks for clearing this up. There must be a genealogy or ethymology of some sort for math symbols. Anyways, watchmanzero should use an "unsigned" qualifier/modifier. On a 64bit architecture, "register unsigned long" certainly is fun _and_ fast. :D
Well, it could very easily be my fault. I did assume certain knowledge, familliarity with number theory properties of integers and so on. The problem is I tried to jam it into one video, and didn't want some of my more math expert subscribers to completely fall asleep. This is the problem with trying to do math lectures for such a wide audience.
I enjoyed this brainteaser. It's long time for me since I had to deal with math even if I turned out to become PHP developer. I'll enjoy any further vids on this topic from you.
sorry if you explained that.. I have a habbit of jumping the gun when it comes to questions.. I am just tired and should watch this video when i am not tired... seems like some things that require my full atention.
I did not explain this, but 1 is not considered a prime number. Its just a definitional thing, but there are also a bunch of math theorems that wouldn't work if you considered 1 to be a prime.
One step at a time. We will get to Lesbegue measures and all that sort of thing in good time, if people have the patience for it. I am trying to bring *everyone* aboard very slowly.
I love seeing this. This is the sort of thing I was never good at. The secondary big lesson for me is that when we prove something by contradition, we make a proof of our hypothesis and_nothing_else. I would have gone away thinking that the product of the sequence plus 1 is prime. Thanks!
Yeah, the added bits on the end are kind of important. I just want to demonstrate that proofs are rigorous and yet limited in scope. Indeed it saddens me that this is not explained at least to this level of clarity in real classrooms in our schools.
thank you, just "with two exercises at the end", would expect other kind exercises, they are nice thoughts, maybe, just maybe, this will help me today.
KissingTheBeeHive 2 years ago
nice question at the end. thanks for making the video. but to be honest, its far too long. this proof and the nice questions you posed could have been presented far more rapidly. remember viewers have the opportunity to replay anything they dont understand so all u need to do is make sure you explain the ideas concisely and accurately.
spaghettionetime 3 years ago
Uhh... yeah..
EcuaRiquenoCuban 3 years ago
Every number can be expressed as a square times a prime factorisation with each prime used no more than once.
For the first N numbers, which contain n primes, this yields a maximum of sqrt(N) * 2^n expressions.
The number of expressions cannot be less than N, as each number must equal one of these expressions.
sqrt(N) * 2^n >= N
2^n >= sqrt(N)
n >= .5*log_2 (N)
As N->inf, n->inf
- Paul Erdos's proof of infinite primes.
Mozza314 3 years ago 3
Very nice! I had the privilege of meeting him in Hungary. I have always been impressed by him and guys like Polya. His use of the "probabilistic method" for bounding Ramsey numbers was an incredible insight that has stayed with me since I saw it.
websnarf 3 years ago
Wow, I'm jealous. I have read that he is a very eccentric character, he'd be a very interesting person to meet. I've also read that he had this idea that God would have a book of all the most simple, eloquent proofs, and that when he created what he thought was such a proof, he'd say 'one for the book'. While I obviously (I see you've subscribed) disagree with him, I think I'd have exactly the same idea if I were a theist.
Mozza314 3 years ago
This comment has received too many negative votes show
This is the worst explanation and *pseudo*-proof of the theorem I've ever seen. Truly pathetic. Why continuously show yourself to be a pompous jackass?
nortexoid 4 years ago
@nortexoid its not his fault you have a learning disability...
happinin 10 months ago
Find me please a general formula for representing all primes otherwise you suck balls!
EVEN RECURSIVE FORMULA!!
sotatorfak 4 years ago
This proof is totally wrong!!!!! The conclusion that X is not prime leaves the proof incomplete. He should have ended the proof by saying that
1: X is a prime greater that all the primes less or equal to Pn OR
2: X is a composite with prime divisors greater than all the primes he listed.
IN BOTH WAYS THERE WILL STILL BE A PRIME GREATER THEN Pn.
Furthermore, his method will mean that we can generate bigger primes using known primes.
bonzionet 4 years ago
CORRECTION: The conclusion that X is prime leaves the proof incomplete. I GUESS HE ADDED THAT LATER. DIDN'T GET LAST PART.
bonzionet 4 years ago
proof is correct. It's in this form since Euclid. Remember this is a proof by contradiction. not necessary to say X can be a composite with prime divisors greater than all n primes, couse those prime divisors are not present in the list of n primes. X is the prime contradicting the ipothesis obtained by construction from the set of numbers 'generated' by the n primes. In such system, 59 and 509 would not exist. But the set of numbers is clearly not that with only n primes (till 13)
corlhat 4 years ago 2
Ok, very good video BUT you cannot really assume that X is a prime. Only thing you can say is that the primes in the prime factorization of X is not in your finite list of prime numbers. So question 2 shouldn't really even be there...
LovicProductions 4 years ago
I might put up a similar video demonstrating in just a few lines with real analysis that the real numbers are not countable.
adb4 4 years ago
Why isn't -1 a prime number?
crambo0349 4 years ago
Because that's not the definition of a prime number.
websnarf 4 years ago
This has been flagged as spam show
@crambo0349
Why are you so fucking retarded? For the love of god!
hamsterpoop 7 months ago
This may be a silly question, but anyways:
Can't we already deduce that there are an infinite amount of prime numbers simply because there theoretically is an infinite sequence of numbers?
smaakjeks 4 years ago
Or was the point simply to show the principle of proof by contradiction?
smaakjeks 4 years ago
That there is an infinite sequence numbers, by itself, is not a sufficient proof that there are an infinite number of primes. The purpose of the video was both to show that there were an infinite number of primes and to show the technique of proof by contradiction.
websnarf 4 years ago
Why is that not sufficient proof? Just curious is all :)
smaakjeks 4 years ago
How do you show, that amongst the infinite number of numbers that all but a finite number are all composite? From any fixed finite set of prime numbers, for example, you can create an infinite list of multiples of them that are all composite. How do you know you can't find a large enough set of primes such that eventually the set of their multiples cover all numbers?
websnarf 4 years ago
Because by the very definition of infinity you can never have the multiples (exponentiated) for all numbers?
smaakjeks 4 years ago
That doesn't even parse as coherent English to me. The definition of infinity has no relation to multiples of numbers. The standards of proof in mathematics requires that you being universally convincing to basically anyone versed in the practice of mathematics. Your responses here do not rise to that level.
websnarf 4 years ago
My apologies then. I was asking as best as I can from pure curiocity.
smaakjeks 4 years ago
Nice video, except that the fundamental theorem of arithmetic needs the extra statement that the natural numbers are greater than 1.
jasondh22 5 years ago
Great video. I'd seen the proof before but still enjoy watching mathematics being shown on YouTube.
I think a video tutorial on the logical implication (truth table, negation, etc.) would help people understand why proof by contradiction works. I remember being confused as to why it worked when I first saw. This was only fully cleared up having had the logical implication explained to me.
tomfitzyuk 5 years ago
They also have some videos on Advanced math: Polynomial Approximation, Convergence, and the
Concept of infinity vs. Many.
Newton1692 4 years ago
That was neat. I enjoy your math lessons =)
JDRedstone 5 years ago
hey websnarf... i absolutely love your mathematically related videos. this video was especially interesting and reminded me of something that i noticed a couple years ago... every fifth element in the fibonacci sequence is divisible by five, or so it seems. is it possible for you to prove this? or am i wrong?
josda1000 5 years ago
Suppose t = f(5*n-1) and 5*s = f(5*n). Then f(5*n+1)=t+5*s, f(5*n+2)=t+10*s, f(5*n+3)=2*t+15*s, f(5*n+4)=3*t+25*s, f(5*n+5)=5*t+40*s=5*(t+8*s). So it is straight forward to complete the proof using the principle of mathematical induction (exercise to the reader.)
websnarf 5 years ago
ahhahaha im well proud of myself, i sort of understood this haha:)
X
loopylorny 5 years ago
Congratulations! That's well done for someone of your age. Of course, I've long suspected that there was some sort of lurking genius behind that Welsh accent. :)
websnarf 5 years ago
hahahah i told pdoeman that i liked math yesterday and he was like "wow ive been talking to a genius" lmao
hahah not quite but its nice for ppl to think it hahahah
xx
loopylorny 5 years ago
Aha! Paul must have sent you my way! I don't otherwise see how you would magically subscribe (appreciated, BTW) to me then suddenly decide to focus on my math proof video. My food challenge videos are way cooler. :)
websnarf 5 years ago
hahah i knew of you anyway haha but yeh he sent me the link to this sooo..
ill check out ur other vids too
loopylorny 5 years ago
In my opinion, this explanation wasn't long. Good video (exposition), man. O tempo é "uma bobagem".
12ytjljlcrf 5 years ago
Mister websnarf, I have a question (such question is connected, I guess, with "Re: Math Education: An Inconvenient Truth"): here, in Brasil (at least, in my poor opinion), the approach in regards to Math teaching is [basically] centered on "determine problems" (instead of "demonstrate problems"). What do you think about that? Could proof techniques be presented on basic school?
12ytjljlcrf 5 years ago
Proof is something you don't see until later in high school, and a lot of americans don't ever get to see proof by contradiction unless they continue with more advanced math in college. This kind of stuff can be hard for some people.
websnarf 5 years ago
long winded explanation of a simple idea
mwilliams0 5 years ago
Yeah, i've used bad assumptions before too but used in projects I've known of them as a 'null hypothesis' which requires a test and I used a statistical test to help me out with that.
WorldinMotion 5 years ago
Can I instead simply map Russel's paradox to it, and say I'm done? (Can god make a stone he cannot lift?)
websnarf 5 years ago
Yes, that is proof of non-existence, but only for an omnipotent God. A deists God might not be described as omnipotent. Russells paradox pertains to the set of all sets, and asks is this set part of itself.
gklr 5 years ago
why isn't infinity factorial plus some other prime number not the greatest prime #
yigit2681 5 years ago
Infinity is not a number. So neither is infinity factorial.
websnarf 5 years ago
Aw you ruined it...I was so enjoying that "cant prove a negative" nonsense
mranenome 5 years ago
I loved the math. I am 35, My education is only high school level. But have always been ok with math. I actually answered the 2nd question though it made me think. Some geometry would be cool. This math was explained just a little above my level which is great, I dont mind being challenged with that.
It was great you covered proof by contradiction.
cjunk351 5 years ago
I am really glad you answered Q2 on your own. (In some ways Q1 is actually harder.) Its really hard to make in a single math lecture something which can be understood, and also challenge everyone in the same way.
websnarf 5 years ago
I absolutely loved this video. I thought it was clear and quick, considering your time constraints. I love this stuff. I liked the level and could probably go up a level or two, but I'm in my confort zone. Great stuff.
howtofoldsoup 5 years ago
I keep rereading this comment, but I can't figure out the innuendo, or twisted sarcastic alternate meaning. So I'll just provisionally accept it as a complement for now, knowing full well what I am setting myself up for. ;)
websnarf 5 years ago
Damn, I posted with my other youtube account. Oh well, I just use this one for now.
watchmanzero 5 years ago
That makes sense, but I always thought 1 was a prime. So if the prime cannot be factored into two smaller primes it is not a prime? I wrote this java program to test it, but I didn't include the factoring.
watchmanzero 5 years ago
public static void FindPrimeNumbers(int iLimit)
{ for(int iCnt=1, iPrimeChk = 0;iCnt<=iLimit;iCnt++) { for(int iCnt2=1;iCnt2<=iCnt;iCnt2++) if (iCnt%iCnt2==0) iPrimeChk++; if (iPrimeChk == 2 || iCnt==1) { DisplayPrimeNumbers(iCnt); iPrimeChk = 0; } else iPrimeChk = 0; }
} // where iLimit=100
watchmanzero 5 years ago
int? Why not a long? In fact aren't there bigInt classes that are appropriate here?
websnarf 5 years ago
A 'long' would have been more appropriate, but since '// where iLimit=100', and I 'in error' assumed that 1 was a prime I felt that I didn't need a data-type greater 32-bits
watchmanzero 5 years ago
Just use Ruby, it elegantly facilitates Bigint, Complex, Fraction etc. This and the Python underscore madness (as well as numbers not being objects) made me switch to Ruby (knew it way before Rails).
websnarf, I really enjoyed the proof. Being familiar with the "exists"- and "for-any"-quantor etc. I don't know the =><= symbol, very intuitive. But what are the thee dots? Abbrev. for "therefore"?
leporidus 5 years ago
Yes, the three dots means therefore.
websnarf 5 years ago
Thanks for clearing this up. There must be a genealogy or ethymology of some sort for math symbols. Anyways, watchmanzero should use an "unsigned" qualifier/modifier. On a 64bit architecture, "register unsigned long" certainly is fun _and_ fast. :D
leporidus 5 years ago
Well, it could very easily be my fault. I did assume certain knowledge, familliarity with number theory properties of integers and so on. The problem is I tried to jam it into one video, and didn't want some of my more math expert subscribers to completely fall asleep. This is the problem with trying to do math lectures for such a wide audience.
websnarf 5 years ago
I enjoyed this brainteaser. It's long time for me since I had to deal with math even if I turned out to become PHP developer. I'll enjoy any further vids on this topic from you.
kricert 5 years ago
I'm glad you enjoyed it; I enjoyed making it. I was thinking about doing either games or a little geometry next. But we'll see.
websnarf 5 years ago
Okay teacher one question..
Is 1 a prime number.. it is only divided by 1.. and itself.. which is one..
thefakeyeti 5 years ago
sorry if you explained that.. I have a habbit of jumping the gun when it comes to questions.. I am just tired and should watch this video when i am not tired... seems like some things that require my full atention.
thefakeyeti 5 years ago
I did not explain this, but 1 is not considered a prime number. Its just a definitional thing, but there are also a bunch of math theorems that wouldn't work if you considered 1 to be a prime.
websnarf 5 years ago
I liked the reminder that any conclusions leading to the contradiction were also false. The audio was too soft, though.
OparinsBulldog 5 years ago
I moved the microphone to as good a position as I could manage it, but I am "wire length limited".
websnarf 5 years ago
Did you crank up the volume? Just asking the obvious.
OparinsBulldog 5 years ago
and there are as many primes as natural number, and as many as integer and as many as fractions???? (the set Q)
exactly aleph0. and love to see math induction ,and you did a 30 second proof in 20 minutes, nice work (not sarcasm)
blmutantx 5 years ago
One step at a time. We will get to Lesbegue measures and all that sort of thing in good time, if people have the patience for it. I am trying to bring *everyone* aboard very slowly.
websnarf 5 years ago
I love seeing this. This is the sort of thing I was never good at. The secondary big lesson for me is that when we prove something by contradition, we make a proof of our hypothesis and_nothing_else. I would have gone away thinking that the product of the sequence plus 1 is prime. Thanks!
DarwinsHamster 5 years ago
Yeah, the added bits on the end are kind of important. I just want to demonstrate that proofs are rigorous and yet limited in scope. Indeed it saddens me that this is not explained at least to this level of clarity in real classrooms in our schools.
websnarf 5 years ago