## Sunday, January 6, 2008

### Visualizing large numbers

This is a very hard thing to do. While it is easy to think about small numbers it is very difficult to imagine large numbers. It is largely because they seem to be totally unrealistic and not related to reality. For me it is sometimes difficult to visualize even not very large numbers - 100 for example. And visualizing say 10^50 is nearly impossible for everyone.

A bit of trivia: what is the largest named number?

Surprisingly it is a very well known one - googolplex. It is $10^{(10^{100})}$.
Why I said it is well known? Well the Google main office complex is named after it. So it is a very well known number (for such a huge and totally not connected to reality number) .
At this point you are probably wondering why I said that it is totally not connected to reality, that is, there is nothing in the physical world that it describes. After all the universe is so huge. So why wouldn't it contain a googolplex of something? However it turns out that there is only about $10^{86}$ atoms in the whole universe. This is almost zero compared to the googolplex!!

When I first learned that there is only $10^{86}$ atoms I actually said: "and this is all? Such a small number... " I said so because I couldn't visualize it. It had no meaning for me, and I new that there were numbers much bigger.

Lets see another example. Suppose we take a sheet of paper 0.1 mm thick and we will cut it in half, stack it and then cut in half again. We will repeat this 100 times.
In the end we will have a stack of $2^{100}$ paper sheets. Since the stack height is representing the number, it should help us visualize it, right? Well lets see how high will the stack be.

$2^{100}$=1,267,650,600,228,229,401,496,703,205,376

This is in 0.1 mm. In light years it is about 13.4 billion light years. This is close to the radius of the whole universe...

The problem with visualization of such numbers should be rather obvious now. Nothing we have in our world can help us visualize them. But what about smaller numbers - 10^6 for example?
Such numbers are possible to visualize using small objects. But it is still difficult.
For example you can visit this site to see how a 10^18 pennies looks like.

For most of us this is not an issue. We don't use such numbers were often, and we are not asked to visualize them. However, there is an interesting underlaying fact in all this. Those numbers can be written, we can use them in equations (I see no reason why but it is possible) but we have no use for them. They don't describe anything in our world. Then why do they exist? And do they exist at all, if they are useless? As I already said, I believe that math is discovered and that it has a "life of its own". While the existence of such numbers is not a proof to this view, it is still something to think about - if most of the numbers that we have are of no use to us then why we invented them, if they were invented by us?

#### 1 comment:

Michael Ayers said...

I wanted to try and visualise large numbers too. In the end I created a poster (which you can see here: http://stickinsect.wordpress.com/2010/05/04/1-million-dots/) that has one million dots on it.

When you really see how many dots this is, it make you realise how big some of the bigger numbers that you talked about here must be!