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The Prime Constant
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0:00
I'm going to draw a number line we're going to need a lot of paper for this which is why we're running this one uh on the continuous stream of brown paper cuz I'm going to do a number line starting from zero we'll put one maybe here and I'm going to continue this number line and I'm going to label all the Primes we'll do them in green as we go along if that's okay so every every single Prime I'm going to mark on this uh we may I feel like that might not be enough paper maybe I've got the scale
0:33
okay fine this quite it may be the longest number five video ever and uh there they all are there that is the 0.4 1 4 hang on let me get the rest of these digits 6 8 2 5 0 9 and maybe more digits 8511 and it keeps going there you go H all the
1:05
Primes so it's like you know Pi is the circle constant e is the exponential constant this is the prime constant and it's a number that represents all the Primes every single Prime I'm willing to hear more oh oh more more okay we'll do more so uh as St viewers you can try and predict where we're going going with this so I have declared accurately that this is all the Primes I'm now going to do something seemingly unrelated and then we uh we'll work out how we combine the
1:40
two so the unrelated thing I'm going to do is writing fractions in base two okay we'll do that a half now in base 10 we write a half as 0.5 but then in base two we write that as 0.1 then you think well hang on how you know how how did we extend the idea of base 10 because normally when we do binary it's almost always whole numbers at least in popular mathematics but in reality if you're doing things with numbers they're very rarely neat and tidy like an integer you got to have
2:15
um your decimal places we can have a think about how we represent them in base 10 how we do it in base two and the differences between them and a half is nice and neat in both cases now some numbers are not necessarily nice and neat so if we want wanted to do um a third in base 10 that can't be done it's really annoying so you end up with 0.333 and then you never stop doing threes it's threes forever so that three you could do it's like repeating or something like that whatever symbol you
2:48
like to use for that and in base I'm going actually going to look it up real quick oh it's also a mess here is it 0 0 1 0 0 1 1 0 0 1 1 and then that repeats it's like 0 0 one one over and over again right and that just carries on going going going going going we often don't think it really through in uh base 10 because it seems so obvious but what you're doing is you're taking many decimal places you've got and then dividing it by 10 the power of how how many there are so in this case
3:32
and I've not picked particularly great examples in this case here what you're actually looking at that's going to be 5 ID 10 that gives you a half and this case over here that's 1id 2 that's a half quite nice here we kind of if we draw a line here that would be 3 over 10 or 33 over 100 or 333 over 1,000 or 3333 over 10,000 and so on and so on so on so if we actually ended up with 333 over a th000 you can see that's close to a third but not quite and the more we have
4:02
the closer we get but we'll never quite get there this is the same idea here so what we could do is do 0 1 0 0 1 1 so we're going to stop here and we divide that by oh you know what I'm going to switch this to base 10 so that in the binary of that that's 19 and then we divide it by 2 to the power of um how many places have we done 1 2 3 4 5 6 2 the^ 6 which is 64 and that is close to a third no it's it's not as close as this because this is a bigger base so it converges faster but this is
4:44
reasonably close and the more we did of these the more we would get you can try and assemble a number that's you know a fraction in base two to see how it would look as a fraction so in a recent video and I didn't mention this in the video I looked at a bug in Minecraft where there was a weird behavior when Boats were falling from very specific Heights and and the code wasn't doing what it was meant to do and that was because in the code it had to represent uh 0.4 and so
5:14
0.4 in base 10 is nice and neat but 0.4 is not nice and neat it's you know 0.0110 011 and then that repeats 0 0 1 1 and so so so it's a mess and this is because five five is a factor of 10 and so you get a nice neat one whereas five is not a factor of two or any power of two and so terrible so actually We complain a lot about base 10 being bad for writing fractions because it's only factors are two and five and they're the only fractions where you get nice neat
5:46
um multiples of those you get nice neat decimal expansions base 2 is even worse and so because 0 4 never terminates in base two and when you put that into binary code you have to cut it off somewhere you have you only end up with an approximation of 04 so even though we can have an exact point4 in base 10 in base 2 you only get an approximation and so that was that was what was causing the bug in Minecraft which I found very very um interesting and we'll link to
6:11
that video below if you want to check it out our point is you can flip between base 10 and base two with these these decimal numbers and now you think well hang where's our Prime constant coming from well I'm going to put the prime constant into base two and we'll see what it looks like so I will label it actually I'll put it underneath so the prime constant can be written as 0.01 1 0 1 0 1 0 0 0 1 0 1 and so on if you think well what's going on here well the first position is a zero positions
6:50
two and three are a one four isn't five is six isn't S is 8 9 and 10 are not 11 is 12 isn't 13 is and I'm both describing if it is a one and I'm describing if it's Prime so what mathematicians did was make a binary number so there's just a one wherever there's a prime and a zero wherever there's not and that's completely determined that like there are infinitely many primes but they're in a nice neat order and we know well we've found them to a point and we can fill
7:23
them in so that defines a very specific real number and it defines inbase 2 the Bas is arbitrary I put it in base 10 because that's easy to understand I going to assume that this number was manufactured with the knowledge of the prime numbers and it wasn't found correct and gave us the prime 100% it is completely useless at predicting prime numbers because we only know its value by finding the prime numbers and then putting them in there you might think it's just a novelty it's just like oh
7:54
well done of course we can do that just another wave it's just another wave encoding the Primes but you got to find them first to encode it what I think is kind of fun is you can do this for any sequence of integers where they all always increase which in mathematics we call the monotonic sequence it just means that next one always gets bigger which means if like they've already been sorted in numerical order and if you go into the online encyclopedia of integer sequences
8:23
you can see there's loads of sequences on there not all of them are monotonic but if they happen to be monotonic if they always increase and the primes do cuz they always go up so there is a number there's a Fibonacci constant between 0 and one which will encode every single Fibonacci number I say We'll encode every single Fibonacci number we don't know what it is we have to find the Fibonacci numbers first and then reverse it but for me that doesn't ruin the beauty of I mean
8:51
there are infinitely many and uncountably infinitely many real numbers between Zer and one like there's so there's so much going on in this tiny little range and I think it's incredible I think we often forget the power of having infinitely many digits after the decimal point like it's a long way down it never ends and that's why there are just so many numbers packed in there and it's just it's the same as like every possible conceivable monotonic series of numbers is is in
9:21
that that is cool that is cool I think it's amazing and so any sequence of numbers you can think of that always increases you can turn it into a real number somewhere between 0 and one and the most famous of all of those is the prime constant because the primes are just the the mascots of mathematics and so of course it's it's very funny to say we can encode the infinitely many prime numbers which are famously difficult to find into a single a single number and I think that's I think that's very
9:49
interesting I feel like because this encodes the primes the the the fundamental numbers and it's in base 2 which seems to be the nerd's favorite base this strikes me as a really good candidate for the number that would be sent to us by an alien civilization absolutely like if you beamed out 011 like on and off in this sequence there's no plausible natural uh event that would generate the primes like this now people might complain CU you like you can make a game
10:21
of life simulation that generates the primes but that's not going to happen randomly in the universe this like if you came across this sequence been beamed out you'd be like oh my goodness it's like beaming out the digits of pi but then again you could put Pi into base two and beam that but what if somewhere else in the universe the curvature SpaceTime is different and they got a different pie that'd be pretty terrifying whereas primes are always primes and so this this is it if
10:48
you want to if you want to if you want to yell hey we're pretty clever that's your number
10:59
thank you very much for watching this video If you enjoyed that and you want more Matt Parker in your life and let's be honest who wouldn't why don't you check out his latest book love triangle you can also check out his own YouTube channel standup maths and a playlist of all his previous appearances on number file I'm going to link to all of that down below in the description the comments the usual things love triangle give it a go it's pretty good
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