I've got one for you Brady,
I know you love a new type of number. Today we're doing practical numbers.
You like the sound of that? So apparently this is meant to be a good number to use when designing
weights and measures. So things like um weights or coins or you know lengths and things like that.
20 is a practical number and I'll tell you what that means. So the idea is we're going
to design a set of weights, it's based on the number 20 and the divisors of 20;
divisors or factors if you prefer of 20. So imagine I've got a weight of 20 units - whatever
we're going to call them. The next weight is based on the divisor so it's going to be 10
because it divides 20. All the other numbers that divide 20: 5 will be the next weight,
and we're going to have - what am I going to have next - 2?
No 4. Then I'm gonna have 2 and then I have 1; because you always have 1 don't you. There we go,
so that is 20 and the divisors of 20. Now the idea is I'm going to try and make every value up
to 20. So imagine I'm a merchant, I've got a set of scales. I've got this set of weights here, this
is a complete set of weights, and I might want to measure whatever number up to and including 20. So
let's check I can do that, okay. The idea is we're going to use each weight just once only, so we
won't need to have two sets of weights. The idea is I'm not- I don't need to use my weight 4 twice,
I should just be able to use one of each - so let's do that.
- (Brady: On your scales or?) - On my scales yeah yeah So
for 1 I can just use the 1 weight, so that's done; for 2 I use the 2 weight, that's done.
For 3 I would use the 1 and the 1. For 4 I've got a 4 weight so that's done. For 5
I've got a 5 weight so that's done as well; and maybe it gets more interesting with 6.
A 5 and a 1 would make 6, 5 and a 2 for 7. For 8 I can use my 5, a 2, and a 1. Again,
just to say, I don't have to use 4 twice I can do it this way. What do I need for 9? I haven't got
a 9 weight so- yeah I've got 5 plus 4 - great. Uh 10 I've got a 10 weight, good. Basically I'm
just going to do the same thing again now but I've got a 10 weight. So it's 10 plus 1, and then it's
10 plus 2, and then it's 10 plus 2 plus 1 and it's 10 plus 4. 10 plus 5. 16: 10 plus 5 plus 1; 10 plus 5
plus 2; 10 plus 5 plus 2 plus 1. 19: 10 plus 5 plus 4 and then that gets me up to 20. And
I can do every number up to the number itself and I can actually keep going until I get to like the
sum of all the weights uh which would add up to 42 in this example; and then I stop because I
can't go any further because that's the complete set of weights. So that's kind of nice. So that's
called a practical number because the idea is that you could design a set of weights that do that.
The idea was that the old weights and measures are meant to be kind of superior,
this is what they say, because they were based on these kinds of numbers. They were based on 16,
like 16 ounces in a pound. In the old days it was 20 shillings to a pound, as in pound sterling - money.
12 is another practical number - inches in a foot. 28 is a practical number. So apparently the old weights
and measures were based on these kind of numbers because you could design sets of coins or sets
of weights and then you could use them like this in this combinations to make any number that you
wanted until you got as big as you could get. So this is what they say, that's why they're called
practical numbers. Oh because yeah the old weights and measures are far superior - much better than 10
and what we use now, our decimals and our metrics. Thing is, I tried to find a set of weights that
actually do this; or a set of coins, don't think it exists. I don't think this was actually done.
I just think this is mathematicians going "these numbers are far superior". I don't think this was
actually ever used; with one exception which is 16 ounces in a in a pound, which is these weights.
This is the 16, this is the 8, this is the 4 this is the 2, and this is the 1. Although I think this
is more based on powers of 2 rather than being 16 is a practical number. This is basically binary. So
if I'm using different combinations of powers of 2, that is what a binary number is. That I
found but I didn't find any like 4 gram weights; I didn't find any 4 pence coins or 4 cent coins;
and that's where it falls down. So I've seen sets are nearly like this but but they don't do
it because they don't have the full thing. So let me show you the sequence of practical numbers, let
me show you what they are. 1 is a practical number, 2 is a practical number, 4, 6, 8, 12, 16,
18, 20, 24, 28 - some in between - 100 it's worth mentioning is a practical number; and so on.
So these are the practical numbers here. - (They're not rare are they?) - They're not completely rare. We
do know how many there are, I will tell you about that, but I think it's worth looking at a set that
fails; or a number that fails uh to be practical to see why it isn't practical. So we've got all
these practical numbers, so what's wrong with the ones in between? Okay let's take a look at what
doesn't work. So if I had a set of weights with 14. 14, right divisors of 14: 7, 2 and 1 - what can
I make? I can make 1, good. I can make 2. I can make 3 as 2 plus 1; and then I'm stuck, I can't make 4.
Because I haven't got a 4 weight to use and I can't add up to 4 either. The problem is you
need to be able to reach the next divisor, that's that's real- that's the crux of it. You have to be
able to get to the next divisor like stepping stones. You need to reach the next one and then
that should get you to the next one, that's the idea. So yeah 4 I can't make, because 7-
you know 5 I can't make, 6 I can't make, because I can't get to 7, I've got this big gap. Then 7
I can make because I've got a 7. Then I can do 7 and 1 for 8, 7 and 2 and then I'm stuck again. 11
and 12 I'm stuck, 13 I'm stuck, and I've got a 14 like that. So yes you get these gaps. The divisors
have to be fairly close together, that's it. - (Okay. Are they impractical numbers those ones or?)
We can call them impractical numbers; they're not called that but they are now, thanks Brady.
Uh so yeah the divisors have to be fairly close together, so practical numbers tend to be
um powers of 2; they tend to be perfect numbers which have nice divisors that add up to make the
number itself. They tend to be factorials because that's, you know, divisible by 1, 2, 3,
4, 5, 6.. so you can get away with that. Uh primorials, that's when you multiply the primes to
get - the consecutive primes. So if I do 2 times 3, or 2 times 3 times 5, or 3- 2 times 3 times 5
times 7, those are primorials. So again those are have the divisors fairly close together.
And um highly composite numbers which Brady knows as? - (Uh uh anti-primes) - Anti-primes. Which if you go to
Wikipedia now it says highly composite numbers, also known as anti-primes - no that's just Brady.
That doesn't count, you can't put that on Wikipedia! - (What do you mean 'just Brady') - That's just Brady!
-(All right) - Again, these are numbers though where the
divisors are close together. A nice um test for a practical number though is you can just test
the prime divisors. So if you've got a practical number and it's divisible by a prime P, if you can
make P - 1, that's for all your possible prime divisors, if you can make P - 1 you can do all
the numbers - it's a practical number. So there's a nice little test for it, so you don't have to test
every number. That will get you your stepping stones, that will get you from one divisor to
the other, that's what you need. - (How many practical numbers are there?) - How many practical numbers are there?
(I would have thought they were infinite) - So yeah there are there are infinite practical numbers.
In fact a practical number multiplied by a practical number is a practical number.
Also a practical number multiplied by a divisor is a practical number - I'll show
you an example of what I'm talking about. I've got say 6 is a practical number, and I've got 2
as a practical number. So yeah 2 times 6 is 12 and 12 is a practical number. But I could also
do 6 times 3 where 3 is not a practical number, but it is a divisor of 6. And that gets me 18 and
that is a practical number - it is on my list. So you can make practical numbers from practical
numbers; so hopefully you can quickly see that you're going to have infinitely many practical
numbers, including the powers of 2. So yes you do have infinitely many of them. But we also know
how they're related to the primes as well. So there's a whole bunch of things that you can
do for primes that haven't been solved that you can solve for practical numbers. There's a
um a twin practical number result; so twin primes means you've got two prime numbers that are 2
apart, the conjecture is there's infinitely many of them, that hasn't been proven. Well for practical
numbers it has been proven. Practical numbers that are 2 apart; there are infinitely many pairs
of those, so tick yes that's been proven. Goldbach conjecture which says every even integer greater
than 2 is the sum of two primes - not proven. Yeah that is proven with uh practical numbers, so every
even number is the sum of two practical numbers, that's nice. Umm prime number theorem which is
a proven thing, so we know how many primes there are. Number of primes less than x is approximately
x divided by the natural log of x - it's a famous result, it's called the Prime Number Theorem.
There's a result like that for practical numbers as well; I'll show you what that looks like. Let's
call it P of X, number of practical numbers less than x. And that is actually quite similar, it's
x over log x multiplied by a constant, I'm going to call it C for constant. That constant- well we
knew that was the case, we knew it was going to be something like that. That was actually calculated
finally by Andreas Weingartner in 2020 and this is the value: so C, this constant, is equal to 1.33607 and stuff.
What that means is there's about 34% more practical numbers than there are prime numbers.
(So they're kind of rare, all right) - Yeah I guess they are. - (And they get more sparse)
(the higher you go?) - Yeah it's just- yeah, you can see yeah it's like the um like the prime number theorem.
At this point I was telling you that two practical numbers multiply together
makes a practical number; or a practical number multiplied by divisor makes a practical number. That
means we can actually minimise this sequence, we can get like the atoms of practical numbers. Like
the primes of practical numbers, a minimal sequence. Which is this: 1, 2, 6, 20, 28, 30 and so on. So all the
other practical numbers can be made out of this sequence of minimal practical numbers.
So here's a longer list of these atoms of practical numbers; they're also known as primitive practical numbers.
It's sequence A267124 on the online encyclopedia of integer sequences if you want to dig deeper. And
also on screen you see those names? They're not like the discoverers of the numbers or anything,
they're actually Numberphile Patreon supporters. And if you'd like to support Numberphile on Patreon
there'll be a link on the screen and in the video description. We appreciate anyone who can
help us out; but we also appreciate everyone who watches the videos, so thanks for being one of them.
And we'll be back again soon - practically before you know it - with another video.