Agnijo's Mathematical Treasure Chest banner
Home > Number theory

Large numbers

The term "large numbers" has different meanings depending on what you consider large. A million seems large enough for some people. However, in economics for example, we can go much further. Bill Gates has a net worth of over 70 billion dollars. The GDP of countries can often be measured in trillions of pounds. We hit a brick wall at our next statistic: the GDP of Earth is in the ballpark of a quadrillion pounds. To go further, let's change the subject to astronomy. This is the reason why unfathomably huge numbers are often termed "astronomical". A light-year is roughly 10 quadrillion metres. The Milky Way is roughly 100,000 light years or (using scientific notation) 1021 metres. The diameter of the observable universe is approximately 2.7*1026 metres. To go further, better change the units to hydrogen nuclei. We get roughly 1041. OK, Planck lengths. We get roughly 1.7*1071 Planck lengths. There are approximately 1080 fundamental particles in the observable universe. We can calculate the volume of the observable universe, which, assuming a Euclidean universe, is about 2.5*10213 cubic Planck lengths.

There is one final number.Using ^ to represent exponentiation, it is 10^10^10^10^10^1.1. Note that exponents are resolved right-to-left. This is the maximum Poincare recurrence time for a certain type of universe, as calculated by Don Page. It doesn't matter whether you measure it in years, Planck times, or universe ages, because it''s virtually identical either way.

Going back down for a moment, a googol is 10100. Many people consider it a very large number, and there is even an art/science (depending on how you view it) called googology, which is the naming and defining of very, very large numbers. Even children can have a lot of fun doing this, as the googol shows. Mathematician Edward Kasner was using 10100 as an example of a large number, when his nine-year old nephew Milton Sirotta named it a googol. They then proceeded to name 10^10^100 a googolplex.

Donald Knuth defined a notation for describing large numbers. a^^b = a^a^...a with b as. For example, 3^^3 = 3^3^3 = 3^27 which is roughly 7.6 trillion. Also, 10^^100 = 10^10^10...10 with 100 10s. This is called a giggol and was defined by Jonathan Bowers. Also, a^^^b = a^^a^^a...a with b as. Thus 3^^^3 = 3^^3^^3 = 3^3^3^3^...3 with about 7.6 trillion 3s. This number is too big to comprehend. It is called tritri. Also, 10^^^100 is known as gaggol. Similarly, a^^^^b = a^^^a^^^...a with b as. 3^^^^3 = 3^^^3^^^3 = 3^3^3...3 where the number of 3s is 3^3^3...3 where the number of 3s is 3^3^3...3 where... where the number of 3s is 3. The number of sections separated by the words "where the number of 3s is" is tritri, which is already unfathomable. This is G1. Ronald Graham defined a truly massive number known as Graham's number. I will use the notation a{c}b = a^^...^b with c up-arrows. This notation is due to Jonathan Bowers. G1 = 3{4}3. The next step is unbelievable. G2 = 3{G1}3 = 3^^...^3 with G1 up-arrows. Likewise, G3 = 3{G2}3. Graham's number is G64. It is an upper bound to a problem in Ramsey theory. This has been reduced to less than 9^^^4. The lower bound is currently 13. We can go much further. Define a{1,2}b = a{a{a...{a}...a}a}a with b nestings. This notation is due to me, but Jonathan Bowers prefers a{{1}}b. Now, a{2,2}b = a{1,2}a{1,2}...a with b as. Likewise, a{3,2}b = a{2,2}a{2,2}...a with b as. We can go beyond that in one huge step: a{1,3}b = a{a{a...a{a,2}a,2}a,2}...a,2}a with b nestings. Likewise, we can also have a{2,3}b and a{3,3}b. This allowed Jonathan Bowers to define tetratri, which is 3{3,3}3.

Of course, there will always be bigger and bigger numbers. Sometimes people hold "large number contests" where the aim is to name the largest number. Of course, entries like gazillion are forbidden as they do not represent a definite number - unless you define them. Also forbidden are infinity and any infinite number, and entries like "one plus the opponent's number". One such contest between Agustin Rayo and Adam Elga ended at Rayo's number, which is so large that you need over a googol symbols of first-order set theory to define it. It is defined as being the smallest number greater than any definable in a googol or less symbols of first-order set theory. For any computable number you can name that can be written (using any computable notation whatsoever) within the observable universe, it will take less than a googol symbols of ZFC set theory to define it. Rayo's number will always be greater, unless you write an uncomputable number.

Related entries

   • Fast growing hierarchy

   • Why infinity is unreachable

   • Graham's number