Back to the Math and Logic Page
Magnitude in relation to:
- #General Prefixes
- #Information
- #Size of Information
- #Money
- #Time
- #Population
- #Speed
- #Popularity
- #More
On the topic of really really big numbers
2200 years ago, a Greek mathematician named Archimedes was wrestling with the idea that there could be numbers too large to count, too enormous to even conceive of. The number of sands on all the beaches of the world, the number of stars in the sky—for all practical purposes for someone living in 212 BC, these quanities were essentially infinite.
Archimedes sat down one day and decided to prove otherwise. His goal was to calculate the number of grains of sand it would take to fill the entire universe, stuffing it full of sand across its entire expanse. He started by working out for himself an upper bound on the number of grains of sand there could be on a beach, the entire earth, and the universe itself, assuming a similar ratio between the earth and the sun as to the sun and the rest of the universe.
Archimedes came up with a final figure of 10^64, as the most grains of sand it should take to fill the universe. As it turns out, based on what we know today he overestimated on his figures by a factor of a thousand or so. But still, for someone figuring this all out on pencil and papyrus with only a view of the stars and his own wits, and considering he had to invent the idea of exponents to pull it off, it's an amazing feat.
This page is written in a similar spirit. Today, computers are able to deal with numbers on a scale beyond what any of us can keep in mind. The concepts of millions, billions, and beyond are going to become more and more immediate as we start dealing with things on that scale. This page is here to put it all into perspective.
General Orders of Magnitude
|
Numerical Value |
Magnitude |
Computer Equivalent |
|
10^3 |
A Thousand |
2^10 = A kilobyte |
|
10^6 |
A Million |
2^20 = A megabyte |
|
10^9 |
A Billion |
2^30 = A gigabyte |
|
10^12 |
A Trillion |
2^40 = A terabyte |
|
10^15 |
A Quadrillion |
2^50 = A petabyte |
|
10^18 |
A Quintillion |
2^60 = An exabyte |
|
10^21 |
A Sextillion |
2^70 = A zettabyte |
|
10^24 |
A Septillion |
2^80 = A yottabyte |
The "Computer Equivalent" entries are in the table partially to show how the SI prefixes work, since they follow the same form as Kilometers, Megameters, etc. Computers work in a base 2 rather than base 10 system however, so a kilobyte is actually 1024 bytes(2^10), rather than 1000 bytes.(10^3) Similarly a megabyte is 1024 kilobytes, and so on up the scale. Your hard drive is likely measured in gigabytes, while the web consists of terabytes of information.
As a side note, the fact that we need to have names for numbers beyond the thousands is a relatively recent phenomenon. The largest named number used by in the ancient greeks or in the Bible was a Myriad Myriad, or one hundred million, where a Myriad itself represented ten thousand. When the world's population can be measured in the ten of millions rather than in the billions, you just don't need to think on that scale.
Information
The following are estimates from Kevin Kelly, Editor of Wired magazine, on how much information has been "published" by the human race since the days of the Sumerian Clay Tablets until today.
|
Numerical Value |
Magnitude (rough estimates) |
Average size(bytes) |
|
500,000 |
The number of movies produced |
700 megabytes |
|
3,000,000 |
The number of videos, TV shows and short films created |
1 gigabyte? |
|
25,000,000 |
The number of pieces of music ever composed |
5 megabytes |
|
32,000,000 |
The number of books published |
5 megabytes |
|
500,000,000 |
The number of images created |
2 megabytes |
|
750,000,000 |
The number of articles and essays written |
50 kilobytes |
|
100,000,000,000 |
The number of public web pages created |
17 kilobytes |
Thus, the total "published" information content of the human race to date: Roughly 6.37 Petabytes.
That may seem huge, but even all that information is very filtered compared to all that's out there. According to a study at Berkely, 5 exobytes of new information are generated every year (most of that being recorded on hard drives). Also according to that study:
Size of Information
|
Size in Bytes |
Magnitude Represented (loose estimates) |
|
750 megabytes |
The amount of information stored in the human genome |
|
10 terabytes |
The size of the 19 million books in the library of Congress |
|
170 terabytes |
The size of all web pages on the surface web in 2002 |
|
274 terabytes |
The size of all the instant messages sent in one year |
|
400 petabytes |
The size of all the emails sent in one year |
|
5 exabytes |
The size of all the words ever spoken by human beings |
|
17.3 exabytes |
The size of all telephone calls made worldwide in one year |
The size of all telephone calls made in a year refers to amount of space it would take to record the audio for each one, which is why it's so much bigger than the size of all the words ever spoken.
The amount of information stored in the human genome could probably be compressed a good bit since you have a lot of repeated sequences with amino acids. But uncompressed, you could store the DNA of everyone alive on the planet in 4.875 exabytes. (if you factor in twins and the fact a lot of that information is going to be extremely similar, you could compress that down to a miniscule fraction, but compression is another topic)
Money
|
Dollar Amount |
Value Represented |
|
$8,500 |
The average salary throughout the world |
|
$40,000 |
The average United States salary |
|
$4,000,000 |
The statistical value of a human life |
|
$78,576,000,000 |
The Gross Domestic Product of Peru |
|
$100,000,000,000 |
The estimated wealth of William Henry Gates III |
|
$8,230,000,000,000 |
The Gross Domestic Product of the United States |
|
$28,225,000,000,000 |
The Gross Domestic Product of the world |
The Gross Domestic Product of a country is the value of all goods and services produced within one year.
The statistical value of a human life is calculated by estimating the amount you're likely to contribute to the economy over your lifetime, and how much you're worth to society. If the government has to decide whether to spend 10 billion dollars on a program that would only save the lives of 100 people, they're probably not going to do it, since the program would only produce a value of 400 million dollars from all the lives saved.
Time
|
Numerical Value |
Magnitude Represented (loose estimates) |
|
29,200 |
The number of days you will probably live |
|
2,190,000 |
The number of days in recorded human history |
|
5.0 * 10^12 |
The current estimated age of the universe in days |
|
3.587 * 10^15 |
The number of days before every star in the universe dies |
|
3.665 * 10^39 |
The number of days before every proton in the universe decays |
Only 3.587*10^15 shopping days until the practical end of the universe.
Size
coming soon
Computation
coming soon
|
Number |
Amount Represented |
|
19,683 |
The number of possible Tic-Tac Toe positions |
|
2,000,000,000 |
The number of operations your computer can perform in a second |
|
10^11 |
The number of possible seven digit lowercase passwords |
|
10^18 |
The number of possible positions in a game of Checkers |
|
10^50 |
The number of possible positions in a game of Chess |
|
10^170 |
The number of possible positions in a game of Go |
|
|
|
See Games and Complexity for more on the implications of how hard some of those problems are
Population
|
Number of People |
Population Represented (est) |
|
2,933,462 |
The population of the county I live in |
|
4,937,000 |
The number of citizens in the Roman Empire as |
|
36,132,147 |
The population of the state I live in |
|
45,000,000 |
The total population of the Roman Empire as of A.D. 1 |
|
250,000,000 |
The world population as of A.D. 1 |
|
300,000,000 |
The current population of the United States |
|
6,500,000,000 |
The current population of the world |
Speed
|
Words per Minute |
Activity Represented |
|
20 WPM |
The speed at which people can send text |
|
40 WPM |
The average typing speed of your typical web user source |
|
50 WPM |
The speed at which skilled operators can transmit |
|
100 WPM |
The speed at which extremely skilled typists or skilled Also the average speed at which slide presentations are given |
|
150 WPM |
The average speed at which books on tape are read |
|
200 WPM |
The average speed at which Americans speak in conversation |
|
250 WPM |
The fastest speed at which an auctioneer can speak |
|
300 WPM |
The fastest speed at which people can listen |
|
400 WPM |
The fastest speed at which people can read with |
|
600 WPM |
The fastest speed at which people can read with |
|
1000 WPM |
The fastest speed at which people can read with |
Most of these are fairly intuitive, people can write/speak more slowly than they can read/listen, and reading can happen faster than listening.
One thing we can't measure yet is how fast people can think, which could probably beat anything on that list.
Popularity
|
Numerical Value |
Magnitude Represented (loose estimates) |
|
17,304 |
The number of hits to all the pages on this site |
|
2 million |
The number of people that read the Wall Street Journal every day |
|
3.21 million |
The number of people that were watching CNN on 9/11 |
|
7.3 million |
The number of people that were watching CNN during |
|
8 million |
The number of people that played Halo 2 (also Half-Life) |
|
10 million |
The number of people that played Final Fantasy VII |
|
13 million |
The number of people that played the latest Grand Theft Auto Game |
|
18 million |
The number of people that played Super Mario Bros 3 |
|
19 million |
The number of people that bought The White Album, by the Beatles |
|
20 million |
The number of people that played one of the original |
|
27 million |
The number of people that bought Thriller, by Michael Jackson |
|
30 million |
The number of people that played Tetris |
|
40 million |
The number of people that played the original Super Mario Bros |
|
65 million |
The number of people that read the entire Harry Potter series so far |
|
100 million |
The number of people that read the entire Lord of the Rings Trilogy |
|
105 million |
The number of people that watched the series finale to MASH |
|
172 million |
The number of people that bought a ticket to the |
|
180 million |
The number of people that bought a ticket to the |
|
280 million |
The number of people that bought a ticket to Titanic |
|
~300 million |
The number of people that watched the World Cup |
|
6.7 billion |
The number of Bibles printed |
Box office figures are taken from boxofficemojo, dividing by the average ticket price for that year, best selling video games/albums/books were compiled from lists on wikipedia. Music and video games seem to be in comparable ranges for popularity, the best selling books ever trump both, and movies and television dominate for pure media exposure.
Even more big numbers
|
Numerical Value |
Magnitude Represented (loose estimates) |
|
29,200 |
The number of days you will probably live |
|
200,000 |
The number of English words in active use |
|
2,000,000,000 |
The number of operations your computer can perform in a second |
|
10^11 |
The number of possible seven character lower-case passwords, |
|
10^16 |
The number of planets in the universe |
|
4.247 * 10^17 |
The current estimated age of the universe in seconds |
|
3.1 * 10^20 |
The number of seconds before every star in the universe dies |
|
10^22 |
The number of stars in the universe |
|
10^43 |
The number of seconds before every proton in the universe decays |
|
10^50 |
The number of possible positions in a game of chess |
|
10^80 |
The number of atoms in the entire universe |
|
118^(10^80) |
The number of possible universes down to the atomic level |
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