Mathematicians can describe a very strange theoretical object, which has two very paradoxic properties.
It looks like a horn, or maybe a trumpet. Hmmm, I guess it resembles one of those dreaded vuvuzelas most. It becomes thinner and thinner along its length — which, by the way, is infinite. You know what? Here's a picture:
On the one hand this vuvuzela has an infinite surface area. Because as the vuvuzela becomes longer, its exterior and interior become bigger, thus this vuvuzela of infinite length has an infinite surface.
On the other hand, its volume approaches π. That's right, a finite value. Surprisingly, this vuvuzela of infinite length has an exact finite volume.
What is the length of the coastline of Great Britain?
If you'd ask the Ordnance Survey (the mapping authority for the United Kingdom), they might give you a number of 11,073 miles (17,820 km). That's all well and good, but what does this number actually mean?
If I said the coastline had an infinite length, would I be wrong?
This story is a follow-up to The Infinite Hotel.
David Hilbert had been getting quite some attention with his Infinite Hotel. Georg Cantor, a fellow mathematician who was always in for an impossible challenge, wondered if he could design an even bigger hotel. So that when it was completely full, there would be no way all those guests could ever fit in Hilbert's hotel. No matter what clever trick Hilbert would come up with (and as we know, he had quite a few of those).
So Cantor started to think. Hilbert's hotel had an infinite amount of rooms, each denoted by its own room number. The first was number 1, the next number 2, and so on — all the way to infinity. Basically, there was a room for every possible positive (whole) number. How could he design more rooms than that? Hmmm... what if, maybe, his hotel was to include rooms for all negative numbers as well? A list of all rooms would then go on indefinitely in both the positive and the negative direction! Surely he would then have twice as many rooms as Hilbert did.
Cantor soon realized it was not going to be that easy. Infinity times two? That's still infinity. Just arrange the rooms so that you can match them, and you'll see:
| Cantor's Room Number | 1 | -1 | 2 | -2 | 3 | -3 | 4 | -4 | ... | ∞ |
|---|
| Hilbert's Room Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ... | ∞ |
|---|
The idea sounded good at first, but the above shows that Hilbert's hotel would still be able to house every potential guest in Cantor's hotel. It doesn't matter if you change the labels on the doors; the number of rooms is still infinite.
In Greek mythology, Achilles was a hero of the Trojan War. Some 2500 years ago, philosopher Zeno of Elea included him into one of his paradoxes. Achilles was quite the runner... or maybe the paradox shows otherwise.
Suppose Achilles was in a 100 meter foot race with a tortoise. The tortoise is given a 25 meter head start. It would seem obvious that when Achilles eventually starts running, he will easily overtake the tortoise and then comfortably reach the finish line (with time to spare). Looking at it logically though, one could wonder how Achilles can ever beat the tortoise.
This story is a follow-up to The Highest Number.
David Hilbert had worked hard for it, but after what seemed like an eternity, his Grand Hotel was finally completed. So many people had been involved in building it, it was impossible to keep count. But there it was, up and running, with more guests coming in every day. And Hilbert's Grand Hotel was grand alright. You see, this hypothetical hotel had an infinite amount of rooms. A feature Hilbert was keen to advertise: “The hotel that always has a room available!”
Business was good. Hilbert enjoyed his tasks as manager, as quirky situations tend to occur a lot in a hotel with infinitely many rooms. Such a situation often required a mind-bending solution, just the kind of challenge Hilbert liked. And with the hotel quickly filling up, things were about to get crazy.
- 1 × 102 = 1 × 10 × 10 = 100
- 1 × 103 = 1 × 10 × 10 × 10 = 1000
- 1 × 104 = 1 × 10 × 10 × 10 × 10 = 10000
Basically, powers of 10 can be used to define the number of zeros that follow the ‘1’. So here's some large numbers you might know: a million, a billion, a trillion, a quadrillion... These can be written respectively as 106, 109, 1012 and 1015. You might knew these first few, but that list actually goes on for a while. Take for example a vigintillion, which is 1063. Are there any higher?
Of course! How about the googol? It is notated as 10100. That's a ‘1’ with a hundred zeros! Is this the largest number? Nah, we can put at least the centillion (10303), septuagintacentillion (10513) and the millinillion (103003) on the table.
We need to do better than this, so let's push it into crazy territory. A googolplex is 10googol — yes, that's 10 to the power of a googol. A ‘1’ with a googol zeros. 1010100. Fine! It is a 1 with 10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 zeros. So as you can see I just wrote out a googol. I cannot write out a googolplex: there is not enough room in our universe to do so. See, the number of atoms in the observable universe is estimated to be around 1080. This means that writing down a googolplex requires far more zeros than there are atoms in the universe!
