It's called descending dungeons
Holy cow, what is that number? Would you like to guess?
You don't have a clue
If you want to describe a moderately large number, Brady
You describe it as writing a tower of numbers. For instance,
you might write 2 to the 3 to the 4 to the 5
And that's not a very big number but it's big enough. There are different ways to interpret it. Do you mean 2 to the 3?
to the 4
to the 5 or do you mean 2,
to the 3 to the 4 to the 5 like that
So there are two different ways; parenthesising it from the top down or from the bottom up,
and if you do it from the bottom up the tower essentially collapses. 2 cubed is 8 so we would have 8 to the 4th,
to the 5th; but 8 to the 4th is 2 to the 12 and to the 5th
This is 2 to the 60. So this is about 10 to the 18. It's big but it's not very big.
Now, whereas, if we do it from the top down, this is the official way to do it,
umm 4 to the 5 is 2 to the 10th, which is 1024.
So we have 3 to the 1024,
2 to that power. Now 3 to the 1024 is pretty big. So this is 2 to the 10 to the 489
And that's a pretty big number. So, for emphasising
from the top down, that's the default in most computer languages.
Okay, so this gives you some pretty big numbers, on the other hand my good friend Marc LeBrun had an idea
he said, um, instead of having towers, let's have dungeons.
Let's go down
He says we're going to use the following notation, which actually is in common use, I'll let a sub b mean
read a
in base b, and we do that all the time. For instance in
binary notation
7 is 1 1 1
So you would write 1 1 1 sub 2, which means interpret 1 1 1 in base 2, is equal to 7
in base 10.
And the way you
read this, you say, all right. That one is a 1 that one is a 2 and that one is a 4. So this
Is 4 + 2 + 1 which is indeed 7. so that's the notation.
Now, if
a had just one digit. It's kind of boring. If we took 3 sub 17
It's just 3 in 10. Single digit numbers don't change, under this operation.
So, the first interesting number is 10. And we're going to
rebase it in base 11, and we're going to rebase that in base 12, and rebase that in 13 and...
go for a while, down to n steps and stop.
What do we get? What are these numbers?
Well again, you have to decide how to parenthesise it. There are two ways, from the top or from the bottom. And in this case,
parenthesising it from the bottom up, is the best. 10 sub 11, for example, just to get started. What is 10 sub 11?
Well, that 1 means 11 and that 0 means you've got zero digits. So it's 11. Okay
What about 10 sub 11 sub 12? Well, we're going to parenthesise
from the bottom up, so that's 10 -
now 11's rebased to base 12. That 1 means 12 and that one means 1 so 12 plus 1 is 13.
So that's 10 sub 13
Now, 10 sub 13, that one means 13 and that zero means you don't have any more digits to add. Okay.
Now what about 10 sub 11 sub 12 sub 13? And again, we put in the parens.
Well, so it's 10 sub 11. 12 sub 13. What's that?
That 1 is 13 + 2
um dollars, is 15 so that's 11 sub 15
But now, what is 11 sub 15? Well, that one is a 15 is worth 15 dollars. That one is worth 1 dollar
So that's 16
And 10 sub 16 is 16
So the sequence that we're getting starts with 10, and then we get 11, 13, and 16
Let's do another one. So 10 sub 11, 12,
13 ,14. 13 sub 14
is 14 + 3. It's 17
Okay
12 sub 17
Is 17 + 2 is 19.
Now we have 11
Sub 19.
11 sub 19. That one is 19. That one is,
makes 20. So that's 20, and then we have 10 sub 20, which is equal to 20.
So the next term in the sequence is 20. And we keep going and the question is how big do these numbers grow?
Well, we could also have parenthesised it. Let me show you what it looks like if we do it the other way around. From
top down.
10, then we have 10 sub 11. Well, that's 11 again
Now we have 10 sub 11, sub 12, but i'm going to put the parentheses around the top two
So 10 sub 11, is 11. 11 sub 12
is um
13 again, same as it was here. 10 sub 11, sub 12, sub 13;
with parens like this.
So this is equal to 11 sub 12,
which is 13 sub 13. That one means 13 dollars, that 3 means three single dollars
So we get 16. Well, that's what we got before
But soon it's gonna change. If we parenthesise these dungeons, from the bottom up,
we get one sequence 10, 11, 13, 16, 20.
And then, the next one is 25. After 25
we get 31, 38 and 46, and so on. And then, it slowly increases.
as you can see, and after a while it starts to get bigger and bigger and bigger. And after
we've gone to a hundred or so steps, we're starting to get some pretty big numbers.
Whereas, on the other hand, if we parenthesise from the top down,
we get much, much bigger numbers, very quickly. From the top down, it grows really quickly. Yeah
Okay, we get the sequence 10, 11, 13, 20. So far it agrees, but then we get 30,
48, 76,
132,
420, and so on. And after a mere 30 steps - we're getting gigantic numbers, as you can see here
So that's the second sequence. - (Brady: Where are you taking us next?)
Well, I want to give you two more. Which is the other way, natural way, to
make a dungeon. If you give me a bit of paper I'll show you. In this sequence, we're going to start with 10
And then we'll do 11 sub 10, 12 sub 11 sub 10
13 sub 12 sub 11 sub 10 and so on and again we have to decide which way to parenthesise it. And the preferred way
is from the bottom up. So 11 sub 10 is that one means 10 dollars. And that one means one more dollar
So this is 11. So now, we're going to do 12 sub 11 sub 10.
Now 11 sub 10, as we just saw, is 11. So what we have is 12 sub 11.
12 sub 11 is 11 + 2, which is 13.
So the, so the sequences goes 10, 11, 13. Now, we have to do
13 sub 12 sub 11 sub 10. Okay, 13 sub 13,
by what we just did, and that's 13 + 3, is 16.
And that is, obviously, by what we just did, 14 sub 16, which is 20. Now,
we will do one more. 15, 14, 13, 12, 11, 10.
By what we know from the previous little calculation is
15 sub 20, but that one means 20 dollars and add five dollars. So that's 25 dollars.
Okay, so that's,
that's from the bottom up. If we do it from the top down, we'll get a different sequence.
This was pretty easy, because we didn't have to recalculate everything each time,
we could just use what we knew from the previous step. From - the difficulty with top down is you have to redo
the computation at each step. That's why um, the other one is preferred.
All right. So 11 sub 10 is certainly 11. No we have a 12 sub 11 sub 10.
Now, 12 sub 11
is 13, so this is 13 sub 10, which is 13.
And, you can see that, it's the same, so far. We have 10, 11, 13. All right
13, 12, 11, 10 is equal to, parenthesised
from
the top down. 13 sub 12 is
12 + 3 is 15. 15 sub 11 is 16
So. And then 16 sub 10 is 16. Still the same as we had before 14, 13, 12, 11,
10
Okay, 14 , 13 is 17. 17, 12 is 19
19 sub 11
Is equal to 20.
20 sub 10 is 20
Holy cow! All right, we'll go one more step.
(Brady: Breaking from the pack now.) - Yeah, we've got a different sequence
(Brady: and this again presumably the same thing will happen here?)
(Brady: But that one - ) - Yes, exactly the same. I have the same pictures, as I showed you. Okay, so now we've got four sequences,
parenthesised from the top down, from the bottom up. Yes. The question is: how big are these? Now,
what happens if you go down a dungeon n steps down towards the basement? What - how big is the number?
If you have a number, you take its logarithm,
It's much smaller than; if you take the log of the log of the number, it gets really tiny
and if you take the log of the log of the log
It has to be a pretty big number before you get anything, the - anything that's left after you take logs.
So with a tower, a tower of height n, you have to take logs
n -1 times to bring it down to earth, but these dungeons, they don't grow that fast, which is really surprising
What you get is that you only have to take logs twice before you get a number you can see
(Brady: But some of those numbers you showed me, like, after you've done hundreds of steps were massive)
(Brady: Surely logging them twice wouldn't bring you down to an - )
( a reasonable number?) - Yes, it does. Um, this is asymptotically
Yeah
Um, a hundred steps - we got a number that was that big,10 to the 80
If we were taking logs to the base 10, of a number like 10 to the 80. 10 to the power 80, take logs
becomes 80 times the log of 10, call it 80. Take the log again. Log of 80 is
just under 2.
So it drops down very quickly - (Brady: But i'm not surprised. It has a different behaviour because you're doing a completely different operation!)
(With the towers you were, like, multiplying numbers)
(but this operation you were doing with the dungeons, involved like)
(converting numbers)
(into bases and things so it was a whole different game you were playing.) - Yeah. You're absolutely right Brady. I
maybe
Was exaggerating a bit. Yes. I mean they do grow very quickly, and it is surprising when you work them out.
You saw those, those how rapidly they grow, those numbers. See,
initially, we thought that they would grow just like towers. So, it was really a surprise to find out that they don't.
One really nice example:
we've been talking about the notation a sub b which means take the number a and
read it as if it was written in base b what if you took a,
and read it in base a,
and read that in base a,
and you went all the way to infinity?
And in particular I want to take a to be the number 1.1
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[preview of part 2]: ...means 5 over 10, the 6 means 6 over 10 squared. If we take 1.1, sub itself...
forever