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  • we're gonna do A ll the numbers on number five.

  • We've done a lot of the numbers, some would say, but we haven't done all the numbers.

  • Originally, we did the whole numbers and these are the classics.

  • My guys, we've done 11.

  • That wasn't nearly one.

  • We've done 3435 17 Right?

  • All the whole numbers sitting here.

  • But then there are other types of numbers.

  • If we go one, step out the Russian ones, the ones that are ratios, these air, you know, you get 17th you get I don't know things over 12.

  • You get all sorts of today angle.

  • The rational numbers we've gone beyond that day on the rational technically include the whole numbers.

  • They're subset.

  • But I'm doing this as what people call Venn diagram, which is wrong.

  • It's an oil.

  • A diagram showing every single possible combination, negative numbers, whole numbers.

  • You know, I put in the 12th because I thought I was being hilarious.

  • And then I immediately thought I wasn't gonna put negatives on this diagram.

  • So I regret that for two reasons opening the negative can and the expression on your face.

  • So I'm gonna make that a plus that we are.

  • In fact, this whole sheet of paper is just gonna be the reels.

  • Positive.

  • You know, it works for negative reels.

  • One my saying have the negatives.

  • It's fine.

  • But the sheets are all the reels.

  • And inside here, I've put whole numbers and then I put rational numbers.

  • If you can obviously get complex numbers coming up, we're not gonna do that.

  • We're going to stay down here and I'm gonna raise We work our way out until we get a greater distance out.

  • The number five has ever gone before, right?

  • We're going for the new number file record.

  • How far out into the weird reels?

  • Can we go?

  • But we're done rational next one up, the constructive ALS and these often aren't mentions.

  • You don't have to add this in as a category, but I quite like constructive ALS Maur.

  • Importantly, what?

  • People tend to go for the next one out of the algebraic.

  • Okay, so Simon did a fantastic video about algebraic numbers.

  • And when you go outside, transcendental numbers mean this number is really, really important.

  • No one you and so a lot of the number categories referred to in or out off these different sets.

  • So rational numbers or everything inside the blue line, irrational numbers or everything outside the blue line.

  • You've got constructively numbers or anything inside the Purple line.

  • Un constructive.

  • All numbers are outside this and this light blue line out here.

  • Algebraic numbers, everything inside there and transcendental numbers, everything outside of them.

  • So constructive.

  • All numbers are things that you can construct with a pencil and a compass and a ruler so fi you can do that.

  • The golden ratio because you can do Route five so you can get five you can do route to that lives in here is kind of fun.

  • Algebraic numbers are the solution to an algebraic equation.

  • If it's a square root or lower, you can put in constructively.

  • You can draw it if it's high in that you can't.

  • So the cube root of two isn't algebraic number, and then outside algebraic.

  • You've got beyond that, right?

  • So you got things like Pi Pi lives out here.

  • Pies.

  • Transcendental e is transcendental.

  • The natural log off, too.

  • That's out there, things that are a nice, neat solution to another break equation.

  • I let they're out there right This is how people tend to categorize all the numbers.

  • This is the fringe of kind of what we understand in mathematics and what we've done on number file.

  • So he was the first number that was proven to be transcendental.

  • And so that was proven in 18 73 so reasonably recent, given that some of these of thousands of years old we knew existed but didn't know where it went pie.

  • We didn't know where that went up.

  • I was proven to be out here in 18.

  • 80 to so hee to the power of pie was proven to be out here in only 1934.

  • That was a more recent woman.

  • Messed approved is out there and there are loads which we don't know pi to the A.

  • We don't know each of the we don't know pi to the pie.

  • We don't know, right?

  • These are all on the cusp.

  • We know that one off e times pi or e plus pi.

  • One or more of those a transcendental we don't know which or both, right, But, you know, at least one of them is most numbers that we know are algebraic.

  • Sit around here and we don't know if they're definitely transcendental or if they're still on your break.

  • For the most part, we haven't got a clue, right?

  • If you look at the entry for transcendental numbers on Wikipedia or Math World, this is a list.

  • Here's the only ones we know And that's it, right?

  • Some of our favorite big old numbers.

  • Graham's number in here.

  • Googol plex in here, right?

  • Whole numbers.

  • Doesn't matter how big it is.

  • It's in there.

  • They're infinitely many whole numbers.

  • But they're also infinitely many rational numbers.

  • Same infinity, their infancy.

  • Many construct herbal numbers, imaginary numbers, but a ll countable infinity is the smallest infinity possible.

  • Lee.

  • Many off these numbers in here and there is one more circuit out, you know, what should we check on the last one?

  • I can add another loop and it cleans up all of these and it puts the ball in a neat bow.

  • This is the collection off computer poll numbers, computer.

  • One of them was.

  • That means we can compute them so we can compute.

  • We can compute pie, right?

  • It's not the solution to a nice equation, but I can write down a system by which you will get the decimal place right?

  • And so we do this.

  • We print them out on a very long bit of paper rolling out on runway.

  • It's hilarious, right?

  • We're here on a runway because for some reason, Brady has printed out the 1st 1 million digits off pie.

  • This came down to Alan cheering in 1936 and even remembers that sharing invented the computer with the Turing machine.

  • And that was in his paper on computer ble numbers he was looking at If you can compute all numbers and he showed their number that exist out here, but we just don't know what that I called the dark numbers right?

  • All these numbers that we know they exist.

  • Cheering showed us, but they're they're so hard to grasp and occasionally we see them.

  • But no, often we're here.

  • Why isn't pie out here?

  • We can compute pie.

  • Well, it takes forever like I could write down a infinite Siri's, which gives you pie, and I can give you the rules for writing at the infinite Siri's and I could write them on a postcard or some finite amount of space and go here are the rules for getting pie, you're gonna have to do them forever, but the rules are finite.

  • So for everything else in here, I can write a description of how to get all the digits out here.

  • They're only define herbal by writing out all the digits.

  • And and that's actually most numbers in here.

  • This is the countable infinity land, right?

  • There's there's there's infinitely many of these, but the smallest, infinitely many are here is a bigger infinitely many.

  • So the vast majority for the strongest definition of vast majority you can come up with.

  • All the numbers are not compute herbal.

  • So we live in this nice little island of numbers that makes sense.

  • And then outside is this vast, vast world of a ll.

  • The reels, which are only define herbal by writing at their digits on, were spotted a few.

  • So there's one court.

  • Are the child in constant?

  • I'll have to double check.

  • I got that right.

  • Tch Item.

  • Vaguely speaking is the probability for a certain way of writing a computer program.

  • If you generate a computer program at random, it will run and come to a stop, right, And that probability is a naive way of describing it, and it depends on how you write a program.

  • So, in fact, there are lots of these constants, but we know they're allowed here.

  • The only way to get them is to work out every single digit.

  • Individually, there's no equation of algorithm that spits it out.

  • It's an un computer ball number, and they're so mysterious and hard to understand at the fringes of our comprehension of numbers.

  • But they're out there, and the scary thing is, most numbers are out there.

  • How can we even know one of them?

  • It seems it feels like an unknowable unknown.

  • It is insane that we even know a couple of them, because when I describe the numbers out here, I'm super hand way because I don't understand them.

  • I've read.

  • I have tried to read the paper.

  • I'm like, Man, this is beyond me like the mass to try and grapple with.

  • The numbers out here is insane.

  • That what's incredible as we have done it, right?

  • So if you look up un computer ble numbers, there are a few examples of one's out here, although interestingly, when you get these weird numbers, you can you can just you can kind of make artificial ones.

  • Okay, So I'm now gonna kind of ruined my lovely need diagram by putting on a whole new category.

  • This is the category off.

  • What accord?

  • Normal numbers.

  • Which is a bit of a silly name.

  • It just means that every possible sub grouping of digits is equally likely to be in there.

  • And a lot of people say, like, for example, pint every goes your phone number somewhere and pie.

  • Your name is somewhere in pi.

  • The complete works of William Shakespeare.

  • If 10 digits somewhere in pie, we don't know that we could move pie into here, which we may we yet do.

  • We may yet do It may be in here.

  • Everyone seems to think it is.

  • All the digits we've checked imply it's in here.

  • But we had not yet managed to prove that.

  • We don't know if pies in here We don't know if he's in here.

  • We don't know if route tours in here.

  • Even though for all of these if you look from the digits you confined any string of digits you want.

  • I found my name in all of them, right?

  • Because They've all got a lot of digits that suitably random and you confined sub strings in there.

  • But we've not made to prove any of those in normal numbers.

  • Would you like to say one number?

  • We have nice to prove.

  • So this I love this number.

  • It's called 10 pronouns.

  • Constant is normal.

  • It's one of the few numbers we know is normal Constant, Const.

  • Essen.

  • And because it is just climbing under the S o champion ounce, Constant is one of the few numbers we know is normal.

  • And it goes 0.1 23 I memorized it.

  • 456789 Well, 10 10 11 12.

  • And it's just a ll.

  • The whole numbers.

  • 14 15 16 set.

  • And so on.

  • That's that's 1819 to 0.

  • I've got them all, and so on.

  • That's like brute.

  • Forcing the problem really is it really is.

  • I call this an artificial number because champion ound just went Can I find a number which is normal?

  • And he came up with this procedure.

  • He just went right.

  • You just put all the numbers in order.

  • All the interviews in base 10 and you know it's true, right?

  • Cause whatever you want to find, it's in there eventually because it's just all the whole numbers listed out and then turned into digits.

  • But that's all numbers out, right?

  • About your digits.

  • Perfectly valid number, right?

  • And it is computer ble, right?

  • So actually, it's like the least efficient way of doing it.

  • It is.

  • There's a slightly more efficient one.

  • The Kirtland air Dish number is the same idea, but only the prime.

  • So it's 23571113 So on 17 Except right.

  • So exactly the same idea, slightly more efficient is also normal.

  • So we didn't start with these numbers and go.

  • I wonder if they're normal.

  • These people sat down and went.

  • I'm gonna make a normal number.

  • How can I do that?

  • And they generated these Now Champion Allen's number is transcendental.

  • So this is it in base.

  • 10 off the different basis and it is computer ble.

  • It is normal.

  • It is transcendental.

  • So transcendental Mrs.

  • Outside algebraic kids there and not the speed of light champ announced.

  • Constant sits in this part of the diagram, So the only normal numbers we know for certain are artificial ones that have been constructed for that purpose.

  • We have never started with a number and then discovered it is normal.

  • We have no test.

  • There is no process for taking a number, improving its normal like mathematics.

  • Hopefully, one day we'll have a test.

  • Currently we don't.

  • Which means all the normal numbers we have are artificially generated.

  • We're yet to take anything from here and show that it's allowed to move over this line.

  • There's an elephant on your diagram.

  • There is over here.

  • Well, very things.

  • You said that already.

  • This section is empty and this is the only properly empty section off the diagram.

  • Now, up here we had that one number whose name I couldn't remember properly.

  • Tryto all this set of numbers do with with a rock program will halt.

  • We've got numbers in this category, right?

  • So this section we've got numbers, this is completely empty.

  • And that's because the only normal numbers we know off the one that we made for that purpose and the fact that we made them for that purpose means we have a rule for generating them, which means they must be computer ble.

  • I have an UN computer ble normal number would be incredible, but this is currently empty.

  • But we have managed to prove one thing Mr Proof that most numbers are normal and most numbers are on computer ball.

  • So actually, this is the biggest section.

  • This is numbers, right?

  • This is a trivial blip right in the world that this is where numbers are right and we have none.

  • So in the main category of numbers, where a ll the numbers are apart from a few trivial side effects, right?

  • We know zero them, you know, as mathematicians, we think we're getting somewhere.