# Googol?

## Definition

A Googol is a $1$ followed by $100$ zeros. That's

## Alternately,

A Googol is $10^{100}$.

Seems more compact yet, somehow bigger!

# Googolplex

A Googolplex is a $1$ followed by a googol zeros!

## Alternately,

A Googolplex is $10^{10^{100}}$.

That's a lot bigger!

## Comprehending a Googol

One way to think about a Googol is as the number of zeros in a Googolplex. Then it make sense to ask questions like the following:

• How long would it take to write down a Googolplex?
• How many pages would that fill?

We might also compare that to a famous large document collection of knowledge, like the bible or the Library of Congress.

## Size of the King James Bible

According to Wolfram Alpha, the KJV has

• $789,650$ words with
• an average word length of $4.08$.
• That's a total of $3,221,772$ characters

If we account for punctuation and a space every fourth character or so, that's still fewer that $4.2$ million characters.

## Library of Congress

How many letters are there in all the books in the Library of Congress? This was a bit harder.

• According to Wikipedia, there are about 25 million books in the Library of Congress.
• According to this guy the average book has about half a million characters (including spaces) on about 250 pages.
• That makes about $12,500,000,000,000 = 12.5\times10^{12}$ or $12.5$ trillion characters.

## Fast typing

Suppose we could type one character every 4 microseconds. That's

• $1$ character per 4 thousands of a second, or
• $\displaystyle \frac{1 \text{ch}}{\frac{4}{1000} \text{sec}}$ $\displaystyle = \frac{1 \text{ch}}{\frac{1}{250} \text{sec}}$ ${\displaystyle \times \frac{250}{250}}$ ${\displaystyle = 250 \text{ch/sec}}$, or
• $2000$ characters every $8$ seconds

I wonder what that looks like?

## Typing the KJV Bible

If we could type one character every 4 microseconds, I wonder how long it would take to type out the KJV? That's

• $\displaystyle \frac{1\text{sec}}{250\text{ch}}$ $\displaystyle \times \, 4,200,000\text{ch}$ $\displaystyle = 16800\text{sec}$ or
• $\displaystyle 16800\text{sec}$ $\displaystyle \times \, \frac{1\text{hour}}{60^2 \text{sec}^2}$ $\displaystyle = 4\frac{2}{3}\text{hours}$

A pleasant afternoon

## Typing the Library of Congress

If we could type one character every 4 microseconds, I wonder how long it would take to type out the LoC? That's

• $\displaystyle \frac{1\text{sec}}{250\text{ch}}$ $\displaystyle \times \, 12\,500\,000\,000\,000\text{ch}$ $\displaystyle = 50\,000\,000\,000\text{sec}$ or
• $\displaystyle 50\,000\,000\,000\text{sec}$ $\displaystyle \times \, \frac{1\text{min}}{60 \text{sec}}\frac{1\text{hour}}{60 \text{min}}$ $\displaystyle = 13\,888\,888\text{hours}$ or
• $\displaystyle 13\,888\,888\text{hours}$ $\displaystyle \times \, \frac{1\text{day}}{24\text{hours}}\frac{1\text{year}}{365\text{days}}$ $\displaystyle = 1585.5\text{years}$

## Using exponents

We can write this a little more compactly

$$12.5\times10^{12} \text{ch} \cdot \frac{1\text{sec}}{250\text{ch}}\cdot \frac{1\text{min}}{60 \text{sec}}\frac{1\text{hour}}{60 \text{min}} \cdot \frac{1\text{day}}{24\text{hours}}\frac{1\text{year}}{365\text{days}}$$ $$=1585.5 \text{years}$$

## Typing a Googolplex

If we could type one character every 4 microseconds, I wonder how long it would take to type out a Googolplex? That's

$$10^{10^{100}} \text{ch} \cdot \frac{1\text{sec}}{250\text{ch}}\cdot \frac{1\text{min}}{60 \text{sec}}\frac{1\text{hour}}{60 \text{min}} \cdot \frac{1\text{day}}{24\text{hours}}\frac{1\text{year}}{365\text{days}}$$ $$=\frac{10^{10^{100}}}{250\cdot60^2\cdot24\cdot365} =\frac{10^{10^{100}}}{7884000000} <\frac{10^{10^{100}}}{10^{10}} \approx 10^{99} \text{years}.$$

This is still so unbelievably huge it's silly.

The number of particles in the universe is estimated to be $10^{80}$.