Explore a Famous Large Number
Choose a number to see its definition, notation and mathematical background.
Googol
Definition: A 1 followed by 100 zeros.
Representation: 10^100
A googol is vastly larger than the estimated number of particles in the observable universe.
The number was introduced by mathematician Edward Kasner and popularized in the 20th century.
Enter Your Own Large-Number Expression
Enter a symbolic expression to inspect its notation. This tool validates a limited character set; it does not calculate arbitrary googology expressions.
Famous Googology Numbers
These examples come from different areas of mathematics. Their definitions are not all based on the same kind of notation or growth operation.
Googol
A googol is vastly larger than the estimated number of particles in the observable universe.
Googolplex
A googolplex has more digits than could be written out using the physical matter available in the observable universe.
Power Tower: 10 ↑↑ 4
Tetration repeats exponentiation, creating a power tower.
Skewes' Number
Skewes numbers are associated with the first sign change of the difference between the prime-counting function and the logarithmic integral.
Graham's Number
Graham’s number arose as an upper bound in a problem in Ramsey theory.
TREE(3)
TREE(3) is an extraordinarily large finite number from a theorem related to Kruskal’s tree theorem.
Busy Beaver Number
Busy Beaver functions grow faster than every computable function, under the standard definitions.
Googolplexian
The term is not as universally standardized as googol or googolplex.
Moser's Number
Moser’s number illustrates how rapidly repeated polygon operations can generate enormous values.
Rayo's Number
Rayo’s number is designed to exceed numbers describable by sufficiently short expressions in a specified formal language.
How Are Extremely Large Numbers Represented?
Ordinary decimal notation becomes impractical when numbers contain more digits than can be written or stored. Mathematicians therefore use compact symbolic systems to describe them.
1. Exponentiation
Exponentiation represents repeated multiplication. For example, \(10^{100}\) is a 1 followed by 100 zeros.
2. Tetration
Tetration represents repeated exponentiation in a power tower. For example:
3. Knuth Up-Arrow Notation
Knuth's notation extends exponentiation using additional arrows. One arrow represents exponentiation, two arrows represent tetration, and three arrows represent pentation.
3 ↑↑ 3 = 3^(3^3)
3 ↑↑↑ 3 = 3 ↑↑ (3 ↑↑ 3)
4. Recursive Definitions
Some famous numbers are defined using sequences in which each term determines the next. Graham's number is a well-known example. Such definitions can describe numbers that are impossible to expand into ordinary decimal notation.
5. Combinatorial and Logical Definitions
Numbers such as TREE(3), Busy Beaver values and Rayo's number come from different mathematical settings. Their definitions involve structures, computation or formal languages rather than simply adding more arrows to an exponentiation expression.
Comparing Famous Large Numbers
The table gives a conceptual overview, not a universal numerical ranking. Some entries are functions or families of numbers rather than one fixed value.
| Number | Representation | Mathematical area |
|---|---|---|
| Googol | 10100 | Powers of ten |
| Googolplex | 1010100 | Exponentiation |
| 10 ↑↑ 4 | Power tower of four tens | Tetration |
| Graham's number | Recursive up-arrow sequence | Ramsey theory |
| TREE(3) | Tree sequence | Combinatorics |
| BB(n) | Busy Beaver function | Computability theory |
| Rayo's number | Formal definability | Mathematical logic |
Do not interpret the row order as a rigorous ranking. For example, BB(n) depends on n and on the precise machine model, while Rayo-style constructions depend on their formal definition.
Related Large Number Calculators
Frequently Asked Questions
What is googology?
Googology is the informal study of extremely large numbers and the notation, functions and mathematical systems used to define them.
What is the difference between a googol and a googolplex?
A googol is 10100. A googolplex is 10 raised to the power of a googol, or 1010100.
Is Graham's number larger than a googolplex?
Yes. Graham's number is vastly larger than a googolplex. Its recursive up-arrow definition involves operations far beyond ordinary exponentiation.
Is TREE(3) larger than Graham's number?
Yes. TREE(3) is much larger than Graham's number. It comes from a combinatorial tree-sequence problem and has a very different definition.
Is Busy Beaver a single number?
BB(n) denotes a function whose value depends on the input n and the chosen formal machine model. Each input gives a particular finite value, but the function itself is not computable.
Can these numbers be written in decimal form?
Many are finite integers and therefore have decimal representations in principle. However, their decimal expansions can be too long to write or store. Symbolic and recursive definitions are much more practical.
Does this explorer calculate TREE(3) or Graham's number?
No. It explains their mathematical definitions and representations. The explorer is educational and does not attempt to expand these enormous values into decimal digits.