Why is the binary of a number irrational? If 1=.00110001 then it can just be written as 110001/100000000 and is therefore rational. What am I missing?
when students look for help on youtube about Bijections, they find Day[9], I wish day[9] was allways the number 1 result for everything I search on youtube.
I love the diagonalization proof, it's just so elegant.
Unfortunately it doesn't work that well with binary expansions <_<. If, after a certain point, you keep picking 1, you hose yourself, because 0.xxxx0111... is equivalent to 0.xxxx1000... So it's possible the new guy WAS covered, you were just too stupid to notice.
However, it works with any higher base (for instance, decimal expansions) where you can just go "also don't pick the highest number possible."
I really didn't like school, or education in general when I was younger. But now that I'm older, and my career is as a games programmer, I really wish my maths teacher had said, "You know, if you learn this stuff, you will know how to figure out what trajectory a cannon needs to fire and hit it's target." Then I would have aced maths. I'm not bad at math, but I would have been so much better otherwise!
@day9fan : Question: when was this video filmed? :)
@day9tv : Also, I hated math when I was in primary / high school, because all my teachers didn't know how to make me understand its logic, so I was awfully bad at math... but if I would have had a teacher like you Sean, I would have been very good! You make things fully understandable and that's awesome in many ways! ^^
@Taranuo That's not true! Of course, the same argument can't work for the rationals, because we know that the rationals can be put in one-to-one-correspondence with the naturals, but the reason for that has nothing to do with infinite sets of 'infinitely long numbers', whatever that means ;) The reason is that, although Cantor's diagonal argument (which Sean is presenting) goes through in exactly the same way producing a number that is not on the list, there is no reason for it to be rational!
@KnowledgeIsPowerKiP Because you need an infinite set of infinitely long numbers for the argument to work.
In rational numbers you only get finite representations and periodic representations. And a periodic representations isnt really infinite. ;)
Or a bit more mathematical: You can just map all periods to a set P1,...,Pn. Then you have n periods and m finite numbers, so you can create a list (bijection) of the finite numbers and the P's
Does any one know how to actually make a bijection between all the natural numbers and all the rational numbers? I mean, it is easy to make a bijection like ie 0 - 0, 1 - 1, 2 - 1/2, 3 - 1/3, ..., n -1/n, but that, of course, accounts only for a single number the top of the fraction, giving infinite rational numbers between 0 and 1. Any tricks?
@theWebWizrd Imagine the set of fractions lying on a grid, where the (x,y) position on the grid corresponds to the fraction x/y. Then, you incrementally take diagonal slices of that grid and start listing off numbers. The first slice only goes through 1 number, so 0->1/1. The next slice goes from (2,1) to (1,2), which passes through 2/1 and 1/2. The next goes from (3,1) to (1,3), so we get 3/1, 2/2, 1/3. (cont'd next post)
@theWebWizrd Then, you go through the list of fractions you made, and remove any duplicates (like 1/3 = 2/6) . Now, you've listed off all the numbers which you can write as x/y. In the end, your list starts off with 1, 2, 1/2, 3, 1/3, 4, 3/2, 2/3, 1/4, ... Another way to think about it is to incrementally write down all fractions x/y where x+y = 2, then 3, then 4... Since we looped over the entire (x,y) space without duplicates, we have exhaustively looked at all rationals exactly once.
Nice explanation! As someone who teaches math (usually to people who want nothing to do with the subject), showing my students that some infinities are more infinite than others is one of my favorite things to do. This blew my mind in high school. I think it goes over better if you compare rationals to reals instead of rationals to irrationals, though.
And high five for counting zero in N... I work in logic/computability, and 0 is most definitely there. Silly definitions...
If anyone is still curious about infinity or aleph's, read up on Georg Cantor who literally went insane thinking about infinity. The diagonal argument that Day9 used at the end of the video is the famous diagonal argument that Cantor made to prove there can be no bijection between natural and irrational numbers, leading to a larger infinity known as Aleph-1.
:D this is the first proof we had to learn in analysis class in the first year of university. It's basically the only proof that I remember from the 5 years I've studied this stuff lol.
the reason for this would be x*infinity=12*infinity=5*infinity=... Because fuck you mathematics ^^. If this would be valid (and it has to be if infinity is to be used in math.). My cellphon costs me 10^9999999999999999999999999999999999999 bucks a sacond to use but only 10^-4 per minuite. The only number that won't follow this is 0. All hail 0. Just some thaughts. Sorry for all eventual language flaws.
How do you pair for example 7/11 as well as 3/7 in the same bijection? Mind that they have infinity digits thus the only exact way to write them is as the are in the question and the bijection has to follow the same rule for all numbers. And saying that this or that divided by 0 or infinity equals the same won't cut it since that leaves no uniquness to each bond. Imo infinity can't be used to count with in any ways, you may only make certain approximations. If not, x=12=5=7 and so on
Basically he supposed that he could write the rationals down in a list (thus making them countable/bijective with IN). Then he created a rational that is NOT in that list which leads to a contradiction.
Since "rationals are countable" leads to a contradiction, the opposite HAS to be true (basic logical deduction) which means "rationals are not countable" is true.
N.B. countable means "is in a bijection with the naturals" here.
Wait.....so...he makes this number.....says that no one's holding it's hand.....and that's supposed to mean something? What does that number mean? Why isn't anyone holding it's hand? So confused
@pepkin88 Yeah, that makes sense. Technically neither set is "standard" (at least according to my number theory prof) so I was more commenting on which he used as opposed to which was right.
With how he explained a mathematical concept that should go completely over my head, it is no wonder why it is so easy to learn from him when he teaches Starcraft! Way to go Day[9]!
@dennypwnsall It comes from Mathematics, so either abbreviation is fine. Also, it's English and Math; they should be capitalised due to them both being proper nouns.
This is pretty cool. When does he this stuff? Does he regularly do it? Ive never seen anything like this live around his daily. except for maybe the rope problem.
@SpiritOfSneedling It sure does get you far. Calculus wouldn't work without infinit(esimals), and the field of engineering would no longer exist without it.
@SpiritOfSneedling "playing with the speed of light is not gonna get you very far" is what you could have told Einstein a 100 years ago. But then you we wouldn't be able to produce enough electricity nowadays to run your computer.
Science is abstract. Science doesn't look for results!! And that's what makes it so efficient because 1 in a thousand scientific papers will change the world in a very significant and new way!!!
Science doesn't aim for results, but it generates the best results!
The BBC has done some brilliant shows on infinity that really mess with your head, but in a good way. Including proving that some infinities are bigger than other infinities :)
Why is the binary of a number irrational? If 1=.00110001 then it can just be written as 110001/100000000 and is therefore rational. What am I missing?
xNajda 30 minutes ago
when students look for help on youtube about Bijections, they find Day[9], I wish day[9] was allways the number 1 result for everything I search on youtube.
FuzzyCheeseStarcraft 1 month ago
This guy talks weird.
This guy talks weird.
This guy talks weird.
gardrek 1 month ago
forever alone circle in group 2 =((
sl4k3y 1 month ago
I understood everything except why that irrational number at the end has no friends.
JTProud 1 month ago
cant you just use limits? >.<
qtrWhileOne 2 months ago
when he repeats himself, he reminds me of Howard Hughes in the Aviator. lol
cking636 2 months ago
oh day9... someone watched to much jacob two two
RockinRolloXD 2 months ago
OMG more maths from day9 plz o_o so entertaining lol
therealjordiano 2 months ago
I love the diagonalization proof, it's just so elegant.
Unfortunately it doesn't work that well with binary expansions <_<. If, after a certain point, you keep picking 1, you hose yourself, because 0.xxxx0111... is equivalent to 0.xxxx1000... So it's possible the new guy WAS covered, you were just too stupid to notice.
However, it works with any higher base (for instance, decimal expansions) where you can just go "also don't pick the highest number possible."
AgaresOaks 2 months ago
Comment removed
lucifertu 2 months ago
he is so hot ... :(
LauVi92 2 months ago
I love the video URL.... AAAAAAAAAA
IhateMissouriALot 2 months ago
man whys he repeating himself so much :D when was this?
tubeaccmhaaa 2 months ago
...how'd he get on this from starcraft...?
Ceefaproductions 2 months ago 2
These were the days that Day[9] repeated himself a lot, gotta love em.
Uzee13 3 months ago 2
א = Aleph (Hebrew)
pila406 3 months ago
This has been flagged as spam show
א !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
pila406 3 months ago
alith/alith =1!
cweida111 4 months ago
infinity is pretty cool
cweida111 4 months ago
Natural numbers are cool, but you know what's cooler?
Integers!! Booyaaah! <3
Texus8 4 months ago
@Texus8 Pretty much the same thing...
cweida111 4 months ago
@cweida111 Yeah pretty much lol, I like the negatives too tho :P
Texus8 3 months ago
@Texus8 I once tried to make a bijection between all the negatives and all the double negatives, but my prof wasn't having none of that.
Dasyati 1 month ago
Day9 the teacher
smallpiper2 4 months ago
I really didn't like school, or education in general when I was younger. But now that I'm older, and my career is as a games programmer, I really wish my maths teacher had said, "You know, if you learn this stuff, you will know how to figure out what trajectory a cannon needs to fire and hit it's target." Then I would have aced maths. I'm not bad at math, but I would have been so much better otherwise!
DigitalDecadence 4 months ago
I'm trying to study political science but instead i'm watching this. Using both sides of my brain at once hurts...
ApprenticeJim 4 months ago
@day9fan : Question: when was this video filmed? :)
@day9tv : Also, I hated math when I was in primary / high school, because all my teachers didn't know how to make me understand its logic, so I was awfully bad at math... but if I would have had a teacher like you Sean, I would have been very good! You make things fully understandable and that's awesome in many ways! ^^
Kitty4Catz 4 months ago
11:46 he say's that "getaremix" said i like a long thick infinity, then begins to like his lips LOL
corayis1 4 months ago
I want more Math from Professor J... anybody know where to find it?
DrunkZergling 5 months ago 42
@Taranuo That's not true! Of course, the same argument can't work for the rationals, because we know that the rationals can be put in one-to-one-correspondence with the naturals, but the reason for that has nothing to do with infinite sets of 'infinitely long numbers', whatever that means ;) The reason is that, although Cantor's diagonal argument (which Sean is presenting) goes through in exactly the same way producing a number that is not on the list, there is no reason for it to be rational!
Jouwmoedertje 5 months ago
Comment removed
BIueFlame 5 months ago
Comment removed
Nerianae 5 months ago
"Fuck you"
harm133 5 months ago
What is this from?
TerraceArchimedes 5 months ago
I don't really understand why you can't apply the proof at the end to rational numbers as well.
KnowledgeIsPowerKiP 5 months ago
@KnowledgeIsPowerKiP Because you need an infinite set of infinitely long numbers for the argument to work.
In rational numbers you only get finite representations and periodic representations. And a periodic representations isnt really infinite. ;)
Or a bit more mathematical: You can just map all periods to a set P1,...,Pn. Then you have n periods and m finite numbers, so you can create a list (bijection) of the finite numbers and the P's
Taranuo 5 months ago
you have a great teaching style. thanks for this
briansabbeth 6 months ago
Can you come teach math for this school? I actually understand shit when you explain it!
boredalex1 6 months ago 2
Does any one know how to actually make a bijection between all the natural numbers and all the rational numbers? I mean, it is easy to make a bijection like ie 0 - 0, 1 - 1, 2 - 1/2, 3 - 1/3, ..., n -1/n, but that, of course, accounts only for a single number the top of the fraction, giving infinite rational numbers between 0 and 1. Any tricks?
theWebWizrd 7 months ago
@theWebWizrd Imagine the set of fractions lying on a grid, where the (x,y) position on the grid corresponds to the fraction x/y. Then, you incrementally take diagonal slices of that grid and start listing off numbers. The first slice only goes through 1 number, so 0->1/1. The next slice goes from (2,1) to (1,2), which passes through 2/1 and 1/2. The next goes from (3,1) to (1,3), so we get 3/1, 2/2, 1/3. (cont'd next post)
abramsaustin 6 months ago
@theWebWizrd Then, you go through the list of fractions you made, and remove any duplicates (like 1/3 = 2/6) . Now, you've listed off all the numbers which you can write as x/y. In the end, your list starts off with 1, 2, 1/2, 3, 1/3, 4, 3/2, 2/3, 1/4, ... Another way to think about it is to incrementally write down all fractions x/y where x+y = 2, then 3, then 4... Since we looped over the entire (x,y) space without duplicates, we have exhaustively looked at all rationals exactly once.
abramsaustin 6 months ago
He says the circle will never be paired with anyone, but circle is Artosis' favorite color, so that is only half true.
AuberellaDorkella 7 months ago
2 people couldn't find each other at the ball
TheHuskyTier 7 months ago
I wish day[9] would teach my math lectures in university
flixt 7 months ago
Why can't the last number make any friends?
DjNajda 8 months ago
I love the way the outcome is totally anticlimatic xD I wish you would do more of these "math is awesome for dummies" <3
Elitios 8 months ago
Fuck that, I'll be a stripper.
iamme625 8 months ago 3
I love this explanation of the uncountable nature of the set of Irrational Numbers.
Mercer321s 8 months ago
0 has generally been included in N since the 1600's
Gowerly 8 months ago
When did he do this?
terakahn 9 months ago
so when does the number J come into the equation?
uberleethatguy 9 months ago
Carlsberg don't do Maths teachers, but if they did, they'd probably be a little like this.
tomtom11go 9 months ago 4
Nice explanation! As someone who teaches math (usually to people who want nothing to do with the subject), showing my students that some infinities are more infinite than others is one of my favorite things to do. This blew my mind in high school. I think it goes over better if you compare rationals to reals instead of rationals to irrationals, though.
And high five for counting zero in N... I work in logic/computability, and 0 is most definitely there. Silly definitions...
shadowrunner103 9 months ago 4
My discrete mathematics final is tomorrow. Thanks for the review.
protolink24 9 months ago
i think my brain melted
Vikzmanizzle 11 months ago
0 is not a natural number!
Then again, neither is J...
simonlikepie 11 months ago 56
@simonlikepie It is if you're a computer scientist.
BibisMcBryde 3 months ago 5
@simonlikepie 0 is a natural number.
Tzeapaedi 3 months ago 2
@simonlikepie There's plenty of debate over that, and I agree with Day[9]; it's more useful in virtually every area to include 0 in the set you use.
Also, did you miss the variable declaration? Because J=9.
v3l0c1r4p70r 2 months ago
@simonlikepie ah, but it is an integer ;)
manfriendlyness 1 month ago
Total mindfuck.
MrDerekftw 1 year ago
This comment has received too many negative votes show
Holy shit I suck at math and this isn't interesting.
tayteeklw 1 year ago
On our next class I want you to explain SC2 using Game Theory :P
bstriant 1 year ago 3
which daily was this from? was this from a preshow?
tristancow 1 year ago
A wild ALEPH has appeared!
FragRaptor 1 year ago 2
Day9, I wish I could have had you as a math teacher
Animostas 1 year ago
If anyone is still curious about infinity or aleph's, read up on Georg Cantor who literally went insane thinking about infinity. The diagonal argument that Day9 used at the end of the video is the famous diagonal argument that Cantor made to prove there can be no bijection between natural and irrational numbers, leading to a larger infinity known as Aleph-1.
piemaster89 1 year ago 3
As someone with a math degree, I appreciate the clarity of his explanation :-)
(Despite the fact that he included zero in the set of natural numbers.)
TheJollyJoker414 1 year ago 4
:D this is the first proof we had to learn in analysis class in the first year of university. It's basically the only proof that I remember from the 5 years I've studied this stuff lol.
Day J should do that more often
khyran 1 year ago
part 2
the reason for this would be x*infinity=12*infinity=5*infinity=... Because fuck you mathematics ^^. If this would be valid (and it has to be if infinity is to be used in math.). My cellphon costs me 10^9999999999999999999999999999999999999 bucks a sacond to use but only 10^-4 per minuite. The only number that won't follow this is 0. All hail 0. Just some thaughts. Sorry for all eventual language flaws.
riotgage 1 year ago
How do you pair for example 7/11 as well as 3/7 in the same bijection? Mind that they have infinity digits thus the only exact way to write them is as the are in the question and the bijection has to follow the same rule for all numbers. And saying that this or that divided by 0 or infinity equals the same won't cut it since that leaves no uniquness to each bond. Imo infinity can't be used to count with in any ways, you may only make certain approximations. If not, x=12=5=7 and so on
riotgage 1 year ago
What daily is this from?
lookoutitscaleb 1 year ago
Didn't get the last part....
SchickeMutti 1 year ago
@SchickeMutti
Basically he supposed that he could write the rationals down in a list (thus making them countable/bijective with IN). Then he created a rational that is NOT in that list which leads to a contradiction.
Since "rationals are countable" leads to a contradiction, the opposite HAS to be true (basic logical deduction) which means "rationals are not countable" is true.
N.B. countable means "is in a bijection with the naturals" here.
khyran 1 year ago
I actually got it. Confusing, but well explained. Sean should seriously consider starting a math show ;)
pit173 1 year ago
Please become a teacher. You obviously like it, looking at all the dailies.
tehhhhhd 1 year ago 2
In all my life I've never had an interesting math tutor. Ever. That all changed today.
nickrowan 1 year ago 2
that last part kinda hurt my brain >.< (then again, I just woke up)
But D9 is pretty good at explaining things in a simple way. He should become a teacher :) (coz he's fun too)
stoikr 1 year ago 5
Ain't nothing to it, Day[9] made math do it.
Xjaonn 1 year ago 3
Wait.....so...he makes this number.....says that no one's holding it's hand.....and that's supposed to mean something? What does that number mean? Why isn't anyone holding it's hand? So confused
wcr4 1 year ago
Day[9] writes his "7's" like i do :) :)
I cant believe how happy its made me...
TheShadowsaken 1 year ago
@TheShadowsaken dude I know right. Me too!
lookoutitscaleb 1 year ago
I wish day9 was my teacher
mslow117 1 year ago
when was this recorded?
StMichaelOfTheWing 1 year ago
He included zero in the naturals :O
Amzipotatosheep 1 year ago
@Amzipotatosheep You can say: On the tree sit 5 birds. 5 is a natural number.
You can say: On the tree sit 0 birds. 0 is a natural number.
You cannot say: On the tree sit -3 birds. -3 is not a natural number.
You cannot say: On the tree sit 1.4 birds. 1.4 is not a natural number.
You cannot say: On the tree sit π birds. π is not a natural number.
You cannot say: On the tree sit 4+7i birds. 4+7i is not a natural number.
pepkin88 1 year ago 4
@pepkin88 Yeah, that makes sense. Technically neither set is "standard" (at least according to my number theory prof) so I was more commenting on which he used as opposed to which was right.
Amzipotatosheep 1 year ago
day9 likes to repeat things.
he likes to repeat things.
he likes to repeat things.
thermos26 1 year ago 5
This is Galileo's paradox
rdubwiley 1 year ago
This is silver-level math. Gimme some masters league shit!
kamikrazi123 1 year ago 2
Seans been very cryptically enveloped by heart break lately, it kind of bum's me out....
cascoll2000 1 year ago
11:46-11:56 was so awkward rofl
hcferago 1 year ago
this blew my mind.
CradLeRcker 1 year ago
holy sausage fest
2theAJ 1 year ago
alif! alif!
martnik 1 year ago
Damn I was watching this at 6 a.m! Do I love Day[9] or what?:P
Dagaen 1 year ago
@Dagaen you do...
dsvw 1 year ago
@Dagaen If you really loved him, you'd move to California to be in the same time zone :'[
TheeAntiFOB 1 year ago
Zero:Natural numbers::Pluto:planet
GoingViral 1 year ago 3
With how he explained a mathematical concept that should go completely over my head, it is no wonder why it is so easy to learn from him when he teaches Starcraft! Way to go Day[9]!
Ziggy488 1 year ago
If I was studying Maths, I would totally want you to be my profesor Day J!
justgodlik3 1 year ago 70
@justgodlik3 i think you should study english, as you cannot spell Math
dennypwnsall 5 months ago
@dennypwnsall In the UK Maths is short for mathematics, it's not very uncommon.
astronautg117 5 months ago 4
@justgodlik3 i think you should study english, as math is not plural
dennypwnsall 5 months ago
@dennypwnsall It comes from Mathematics, so either abbreviation is fine. Also, it's English and Math; they should be capitalised due to them both being proper nouns.
88Arbs 5 months ago
This was before or after the show?
0rland0walkthrough 1 year ago
i hope he presents his thesis in this manner
italiagymnast91 1 year ago
That was fun
CyberWolf008 1 year ago
This is pretty cool. When does he this stuff? Does he regularly do it? Ive never seen anything like this live around his daily. except for maybe the rope problem.
TidalMaster 1 year ago
Playing with infinity is fun, but it doesn't get you very far.
SpiritOfSneedling 1 year ago
@SpiritOfSneedling It sure does get you far. Calculus wouldn't work without infinit(esimals), and the field of engineering would no longer exist without it.
xcusemeprincess 1 year ago 2
@SpiritOfSneedling "playing with the speed of light is not gonna get you very far" is what you could have told Einstein a 100 years ago. But then you we wouldn't be able to produce enough electricity nowadays to run your computer.
Science is abstract. Science doesn't look for results!! And that's what makes it so efficient because 1 in a thousand scientific papers will change the world in a very significant and new way!!!
Science doesn't aim for results, but it generates the best results!
khyran 1 year ago
The BBC has done some brilliant shows on infinity that really mess with your head, but in a good way. Including proving that some infinities are bigger than other infinities :)
kimded 1 year ago
Being able to pair things in math, doesnt make you able to pair yourself with someone in life...
at least I can do the math-part :/
Stonie2007 1 year ago 60