How much larger is a googolplex than a trillion? Where does Graham's number fit? Explore famous enormous numbers, their notation, and the mathematical ideas used to describe quantities too large to write out.
Explore the scaleChoose two entries to see their relationship. Some comparisons are deliberately marked unknown when the selected definitions or parameters are insufficient.
Select two numbers and compare them. The explorer uses exact powers of ten where possible and established mathematical relationships for selected famous numbers.
Illustrative stages only, not a linear or quantitative scale.
Browse well-known examples and see how mathematicians represent quantities that exceed ordinary decimal notation.
One million is 1 followed by six zeros.
Notation: 10^6
A familiar benchmark used in population counts, finance, and statistics.
One billion is 1 followed by nine zeros.
Notation: 10^9
A billion is one thousand million.
One trillion is 1 followed by twelve zeros.
Notation: 10^12
A trillion is one thousand billion.
One quadrillion is 1 followed by fifteen zeros.
Notation: 10^15
A number that is already difficult to visualize in everyday contexts.
A googol is 10 raised to the power of 100.
Notation: 10^100
A googol has 101 decimal digits.
A googolplex is 10 raised to the power of a googol.
Notation: 10^(10^100)
Writing every decimal digit would require vastly more space than any physically available writing medium.
A historically famous enormous bound associated with the prime-counting function.
Notation: e^(e^(e^79))
The name refers to historical bounds in analytic number theory. Different versions of Skewes numbers use different assumptions and definitions.
A famous number arising as an upper bound in a problem in Ramsey theory.
Notation: Defined using Knuth up-arrows
Its construction uses an iterated sequence of numbers defined with increasingly many up-arrows.
A finite number defined through a tree-embedding problem.
Notation: TREE(3)
TREE(3) is vastly larger than Graham’s number. It comes from a finite combinatorial theorem related to Kruskal-type tree results.
A family of functions describing the maximum behavior of halting Turing machines with a fixed number of states.
Notation: BB(n)
The Busy Beaver function grows faster than every computable function. Exact values become extremely difficult to determine as n increases.
A number defined using a formal-language description-length challenge.
Notation: Rayo(10^100)
Its definition depends on a specified formal language and a bound on the number of symbols allowed in descriptions.
A conventional number line is useful for comparing small quantities, but it becomes impractical when numbers differ by dozens, hundreds, or vastly more orders of magnitude. A logarithmic scale offers a more useful way to understand many large numbers.
For example, the base-10 logarithm of a googol is 100. This means a googol is 10100. The logarithm of a googolplex is itself a googol, or 10100. For numbers such as Graham's number and TREE(3), even describing their logarithms requires more advanced notation.
A number is written as a coefficient multiplied by a power of ten.
Useful for very large but manageable exponents.
Powers express repeated multiplication compactly.
A googol is a simple example of a power-defined number.
Up-arrow notation extends exponentiation into repeated exponentiation and higher operations.
Graham's number is defined using an iterated sequence of up-arrow expressions.
Some huge numbers are defined by mathematical structures rather than decimal expansions.
The definition is compact even though the resulting number is extraordinarily large.
| Number | Representation | What makes it notable? |
|---|---|---|
| Googol | 10100 | One followed by 100 zeros. |
| Googolplex | 1010^100 | One followed by a googol zeros. |
| Graham's number | Iterated Knuth up-arrows | A famous upper bound from Ramsey theory. |
| TREE(3) | TREE(3) | A finite combinatorial number vastly larger than Graham's number. |
| Busy Beaver function | BB(n) | A non-computable function with extremely rapid growth. |
This table distinguishes individual numbers from functions. BB(n) is a function; a specific value requires a particular input n.
There is no single largest number among all possible numbers. TREE(3), Graham's number, and Rayo's number are examples of famous enormous finite numbers, each defined in a different mathematical context. The comparison depends on the exact definitions being considered.
Yes. A googol is 10100, while a trillion is 1012. The googol is enormously larger.
Yes. A googolplex is 10 raised to the power of a googol, whereas a googol is only 10 raised to the power of 100.
Yes. Graham's number is vastly larger than a googolplex. It is defined through a rapidly growing sequence of expressions using Knuth's up-arrow notation.
Yes. TREE(3) is vastly larger than Graham's number, despite both being finite integers defined by mathematical constructions.
A computer can store a compact symbolic representation such as 10^(10^100), but cannot store its complete decimal expansion in any physically realistic storage system.
Googology is the informal study and exploration of extremely large numbers, their notation, and the functions or mathematical structures used to define them.
No. A complicated-looking expression can represent a small number, while a short expression can define an enormous one. The mathematical definition, not the visual length of the notation, determines the value.