A googol is the large number 10¹⁰⁰. In decimal notation, it is written as the digit 1 followed by one hundred zeroes: 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

Etymology

The term was coined in 1920 by 9-year-old Milton Sirotta (1911–1981), nephew of U.S.

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

Edward Kasner. He may have been inspired by the contemporary comic strip character Barney Google. Kasner popularized the concept in his 1940 book '' Mathematics and the Imagination''. Other

names A name is a term used for identification by an external observer. They can identify a class or category of things, or a single thing, either uniquely, or within a given context. The entity identified by a name is called its referent. A persona ...

for this quantity include ''ten duotrigintillion'' on the

short scale The long and short scales are two of several naming systems for integer powers of ten which use some of the same terms for different magnitudes. For whole numbers smaller than 1,000,000,000 (109), such as one thousand or one million, the ...

, ''ten thousand sexdecillion'' on the

long scale The long and short scales are two of several naming systems for integer powers of ten which use some of the same terms for different magnitudes. For whole numbers smaller than 1,000,000,000 (109), such as one thousand or one million, the t ...

, or ''ten sexdecilliard'' on the Peletier long scale.

Size

A googol has no special significance in mathematics. However, it is useful when comparing with other very large quantities such as the number of subatomic particles in the visible universe or the number of hypothetical possibilities in a

chess Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dist ...

game. Kasner used it to illustrate the difference between an unimaginably large number and

infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions am ...

, and in this role it is sometimes used in teaching mathematics. To give a sense of how big a googol really is, the mass of an electron, just under , can be compared to the mass of the visible universe, estimated at between and . It is a ratio in the order of about 10⁸⁰ to 10⁹⁰, or at most one ten-billionth of a googol (0.00000001% of a googol). Another way of illustrating the immense size of a googol is to picture the

Frontier A frontier is the political and geographical area near or beyond a boundary. A frontier can also be referred to as a "front". The term came from French in the 15th century, with the meaning "borderland"—the region of a country that fronts ...

supercomputer A supercomputer is a computer with a high level of performance as compared to a general-purpose computer. The performance of a supercomputer is commonly measured in floating-point operations per second ( FLOPS) instead of million instructio ...

, which as of 2022 is the most powerful supercomputer in the world and measures 680 m² (7,300 sq ft), almost exactly the same size of a

basketball court In basketball, the basketball court is the playing surface, consisting of a rectangular floor, with baskets at each end. Indoor basketball courts are almost always made of polished wood, usually maple, with -high rims on each basket. Outdoor su ...

with run-offs and sidelines. The Frontier is capable of making 1,102,000 TFLOPs (1.1 quintillion calculations per second). Imagine if the supercomputer, which cost approximately US$600 million to build, was shrunk down to the size of an atom (for reference, a typical grain of sand might have 37 quintillion atoms). If every atom in the observable universe (~ atoms total) was as powerful as a Frontier supercomputer, it would take approximately 100 seconds of

parallel computing Parallel computing is a type of computation in which many calculations or processes are carried out simultaneously. Large problems can often be divided into smaller ones, which can then be solved at the same time. There are several different f ...

to manually add up all the digits like an

adding machine An adding machine is a class of mechanical calculator, usually specialized for bookkeeping calculations. In the United States, the earliest adding machines were usually built to read in dollars and cents. Adding machines were ubiquitous off ...

(instead of using shorthand calculations).

Carl Sagan Carl Edward Sagan (; ; November 9, 1934December 20, 1996) was an American astronomer, planetary scientist, cosmologist, astrophysicist, astrobiologist, author, and science communicator. His best known scientific contribution is research on ex ...

pointed out that the total number of elementary particles in the universe is around 10⁸⁰ (the Eddington number) and that if the whole universe were packed with

neutrons The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons behave ...

so that there would be no empty space anywhere, there would be around 10¹²⁸. He also noted the similarity of the second calculation to that of

Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...

in '' The Sand Reckoner''. By Archimedes's calculation, the universe of Aristarchus (roughly 2 light years in diameter), if fully packed with sand, would contain 10⁶³ grains. If the much larger observable universe of today were filled with sand, it would still only equal 10⁹⁵ grains. Another 100,000 observable universes filled with sand would be necessary to make a googol. The decay time for a supermassive

black hole A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can def ...

of roughly 1 galaxy-mass (10¹¹

solar mass The solar mass () is a standard unit of mass in astronomy, equal to approximately . It is often used to indicate the masses of other stars, as well as stellar clusters, nebulae, galaxies and black holes. It is approximately equal to the mass ...

es) due to Hawking radiation is on the order of 10¹⁰⁰ years. See in particular equation (27). Therefore, the

heat death Heat death may refer to: *Heat death of the universe, a proposed cosmological event ** Heat death paradox, a philosophical examination of the cosmological event *Hyperthermia, injury up to and including death, from excessive heat *Thermal shock, ...

of an expanding universe is lower-bounded to occur at least one googol years in the future. A googol is considerably smaller than a centillion.

Properties

A googol is approximately ''70!'' (

factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \ ...

of 70). Using an

integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...

binary numeral system A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" ( zero) and "1" (one). The base-2 numeral system is a positional notati ...

, one would need 333 bits to represent a googol, i.e., 1 googol =

2^

≈ 2^332.19280949. However, a googol is well within the maximum bounds of an

IEEE 754 The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found ...

double-precision Double-precision floating-point format (sometimes called FP64 or float64) is a floating-point number format, usually occupying 64 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point. F ...

floating point In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can ...

type, but without full precision in the mantissa. Using

modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his boo ...

, the series of residues (mod ''n'') of one googol, starting with mod 1, is as follows: :0, 0, 1, 0, 0, 4, 4, 0, 1, 0, 1, 4, 3, 4, 10, 0, 4, 10, 9, 0, 4, 12, 13, 16, 0, 16, 10, 4, 16, 10, 5, 0, 1, 4, 25, 28, 10, 28, 16, 0, 1, 4, 31, 12, 10, 36, 27, 16, 11, 0, ... This sequence is the same as that of the residues (mod n) of a googolplex up until the 17th position.

Cultural impact

Widespread sounding of the word occurs through the name of the company

Google Google LLC () is an American Multinational corporation, multinational technology company focusing on Search Engine, search engine technology, online advertising, cloud computing, software, computer software, quantum computing, e-commerce, ar ...

, with the name "Google" being an accidental misspelling of "googol" by the company's founders, which was picked to signify that the search engine was intended to provide large quantities of information. In 2004, family members of Kasner, who had inherited the right to his book, were considering suing Google for their use of the term "googol"; however, no suit was ever filed. Since October 2009, Google has been assigning domain names to its servers under the domain "1e100.net", the scientific notation for 1 googol, in order to provide a single domain to identify servers across the Google network. The word is notable for being the subject of the £1 million question in a 2001 episode of the British quiz show '' Who Wants to Be a Millionaire?'', when contestant Charles Ingram cheated his way through the show with the help of a confederate in the studio audience..

Notes

References

External links

* * * {{Large numbers Large integers Integers Units of amount 1920s neologisms