Googolplex
Encyclopedia
A googolplex is the number 10googol, i.e. 
.
In pure mathematics
, the magnitude
of a googolplex could be related to other forms of large number notation such as tetration
, Knuth's up-arrow notation
, Steinhaus-Moser notation, or Conway chained arrow notation
.
's nine-year-old nephew, Milton Sirotta, coined the term googol
, then proposed the further term googolplex to be "one, followed by writing zeroes until you get tired". Kasner decided to adopt a more formal definition "because different people get tired at different times and it would never do to have Carnera
be a better mathematician than Dr. Einstein
, simply because he had more endurance and could write for longer". It thus became standardized to
.
science program Cosmos: A Personal Voyage
, Episode 9: "The Lives of the Stars", astronomer
and television personality Carl Sagan
estimated that writing a googolplex in numerals (i.e., "10,000,000,000...") would be physically impossible, since doing so would require more space than the known universe provides.
An average book of 60 cubic inches can be printed with 5 zeroes (5 characters per word, 10 words per line, 25 lines per page, 400 pages), or 8.3 zeros per cubic inch. The observable (i.e. past light cone) universe contains 6 cubic inches (4/3 × π × (14 light years in inches)3). This math implies that if the universe is stuffed with paper printed with 0's, it could contain only 5.3 zeros—far short of a googol of zeros. In fact there are only about 2.5 elementary particle
s in the observable universe
so even if one were to use an elementary particle to represent each digit, one would run out of particles well before reaching a googol of digits.
Consider printing the digits of a googolplex in unreadable, one-point
font (0.353 mm per digit). It would take about 3.5 metres to write a googolplex in one-point font. The observable universe is estimated to be 8.80 meters, or 93 billion light-years, in diameter
, so the distance required to write the necessary zeroes is 4.0 times as long as the estimated universe.
The time it would take to write such a number also renders the task implausible: if a person can write two digits per second, it would take around about 1.51 years, which is about 1.1 times the age of the universe
, to write a googolplex.
A Planck space has a volume of a Planck length cubed, which is the smallest measurable volume, at approximately 4.222 m3 = 4.222 cm3. Thus 2.5 cm3 contain about a googol Planck spaces. There are only about 3 cubic metres in the observed universe, giving about 7.1 Planck spaces in the entire observed universe, so a googolplex is far larger than even the number of the smallest measurable spaces in the observed universe.
, the magnitude
of a googolplex could be related to other forms of large number notation such as tetration
, Knuth's up-arrow notation
, Steinhaus-Moser notation, or Conway chained arrow notation
, though neither googol nor googolplex are anywhere near the largest representable or even specifically named numbers. For example, Graham's number
, though finite, is unimaginably larger than other well-known large numbers such as a googol
, googolplex, and even larger than Skewes' number and Moser's number. Indeed, the observable universe
is far too small to contain an ordinary digital representation
of Graham's number, assuming that each digit occupies at least one Planck volume.
is presumed to be greater than the number of hydrogen
atoms in the observable universe
, which has been variously estimated to be between 1079 and 1081. A googol is also greater than the number of Planck time
s elapsed since the Big Bang
, which is estimated at about 8. Thus in the physical world it is difficult to give examples of numbers that compare to the vastly greater googolplex. In analyzing quantum states and black hole
s, physicist Don Page
writes that "determining experimentally whether or not information is lost down black holes of solar mass ... would require more than
measurements to give a rough determination of the final density matrix after a black hole evaporates". The end of the Universe via Big Freeze without proton decay
is subject to be
years into the future, which is still short of a googolplex.
In a separate article, Page shows that the number of states
in a black hole with a mass roughly equivalent to the Andromeda Galaxy
is in the range of a googolplex.
years (~ googolplex
years) — scale of an estimated Poincaré recurrence time
for the quantum state of a hypothetical box containing a black hole with the mass within the presently visible region of our universe. This time assumes a statistical model subject to Poincaré recurrence. A much simplified way of thinking about this time is in a model where our universe's history repeats itself
arbitrarily many times due to properties of statistical mechanics
, this is the time scale when it will first be somewhat similar (for a reasonable choice of "similar") to its current state again.
years (~ googolplex
years) — scale of an estimated Poincaré recurrence time for the quantum state of a hypothetical box containing a black hole with the estimated mass of the entire universe, observable or not, assuming a certain inflationary model with an inflaton whose mass is 10−6 Planck masses.
If the entire volume of the observable universe (taken to be 3 × 1080 m3) were packed solid with fine dust particles about 1.5 micrometres in size, then the number of different ways of ordering these particles (that is, assigning the number 1 to one particle, then the number 2 to another particle, and so on until all particles are numbered) would be approximately one googolplex.
.In pure mathematics
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...
, the magnitude
Magnitude (mathematics)
The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....
of a googolplex could be related to other forms of large number notation such as tetration
Tetration
In mathematics, tetration is an iterated exponential and is the next hyper operator after exponentiation. The word tetration was coined by English mathematician Reuben Louis Goodstein from tetra- and iteration. Tetration is used for the notation of very large numbers...
, Knuth's up-arrow notation
Knuth's up-arrow notation
In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976. It is closely related to the Ackermann function and especially to the hyperoperation sequence. The idea is based on the fact that multiplication can be viewed as iterated...
, Steinhaus-Moser notation, or Conway chained arrow notation
Conway chained arrow notation
Conway chained arrow notation, created by mathematician John Horton Conway, is a means of expressing certain extremely large numbers. It is simply a finite sequence of positive integers separated by rightward arrows, e.g. 2 → 3 → 4 → 5 → 6.As with most...
.
History
In 1938, Edward KasnerEdward Kasner
Edward Kasner who studied under Cassius Jackson Keyser, was a prominent American mathematician who was appointed Tutor on Mathematics in the Columbia University Mathematics Department...
's nine-year-old nephew, Milton Sirotta, coined the term googol
Googol
A googol is the large number 10100, that is, the digit 1 followed by 100 zeros:The term was coined in 1938 by 9-year-old Milton Sirotta , nephew of American mathematician Edward Kasner...
, then proposed the further term googolplex to be "one, followed by writing zeroes until you get tired". Kasner decided to adopt a more formal definition "because different people get tired at different times and it would never do to have Carnera
Primo Carnera
Primo Carnera was an Italian boxer, nicknamed the Ambling Alp, who became the world heavyweight champion.-Biography:...
be a better mathematician than Dr. Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
, simply because he had more endurance and could write for longer". It thus became standardized to
.Size
In the PBSPublic Broadcasting Service
The Public Broadcasting Service is an American non-profit public broadcasting television network with 354 member TV stations in the United States which hold collective ownership. Its headquarters is in Arlington, Virginia....
science program Cosmos: A Personal Voyage
Cosmos: A Personal Voyage
Cosmos: A Personal Voyage is a thirteen-part television series written by Carl Sagan, Ann Druyan, and Steven Soter, with Sagan as presenter. It was executive-produced by Adrian Malone, produced by David Kennard, Geoffrey Haines-Stiles and Gregory Andorfer, and directed by the producers, David...
, Episode 9: "The Lives of the Stars", astronomer
Astronomer
An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...
and television personality Carl Sagan
Carl Sagan
Carl Edward Sagan was an American astronomer, astrophysicist, cosmologist, author, science popularizer and science communicator in astronomy and natural sciences. He published more than 600 scientific papers and articles and was author, co-author or editor of more than 20 books...
estimated that writing a googolplex in numerals (i.e., "10,000,000,000...") would be physically impossible, since doing so would require more space than the known universe provides.
An average book of 60 cubic inches can be printed with 5 zeroes (5 characters per word, 10 words per line, 25 lines per page, 400 pages), or 8.3 zeros per cubic inch. The observable (i.e. past light cone) universe contains 6 cubic inches (4/3 × π × (14 light years in inches)3). This math implies that if the universe is stuffed with paper printed with 0's, it could contain only 5.3 zeros—far short of a googol of zeros. In fact there are only about 2.5 elementary particle
Elementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
s in the observable universe
Observable universe
In Big Bang cosmology, the observable universe consists of the galaxies and other matter that we can in principle observe from Earth in the present day, because light from those objects has had time to reach us since the beginning of the cosmological expansion...
so even if one were to use an elementary particle to represent each digit, one would run out of particles well before reaching a googol of digits.
Consider printing the digits of a googolplex in unreadable, one-point
Point (typography)
In typography, a point is the smallest unit of measure, being a subdivision of the larger pica. It is commonly abbreviated as pt. The point has long been the usual unit for measuring font size and leading and other minute items on a printed page....
font (0.353 mm per digit). It would take about 3.5 metres to write a googolplex in one-point font. The observable universe is estimated to be 8.80 meters, or 93 billion light-years, in diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
, so the distance required to write the necessary zeroes is 4.0 times as long as the estimated universe.
The time it would take to write such a number also renders the task implausible: if a person can write two digits per second, it would take around about 1.51 years, which is about 1.1 times the age of the universe
Age of the universe
The age of the universe is the time elapsed since the Big Bang posited by the most widely accepted scientific model of cosmology. The best current estimate of the age of the universe is 13.75 ± 0.13 billion years within the Lambda-CDM concordance model...
, to write a googolplex.
A Planck space has a volume of a Planck length cubed, which is the smallest measurable volume, at approximately 4.222 m3 = 4.222 cm3. Thus 2.5 cm3 contain about a googol Planck spaces. There are only about 3 cubic metres in the observed universe, giving about 7.1 Planck spaces in the entire observed universe, so a googolplex is far larger than even the number of the smallest measurable spaces in the observed universe.
In pure mathematics
In pure mathematicsPure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...
, the magnitude
Magnitude (mathematics)
The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....
of a googolplex could be related to other forms of large number notation such as tetration
Tetration
In mathematics, tetration is an iterated exponential and is the next hyper operator after exponentiation. The word tetration was coined by English mathematician Reuben Louis Goodstein from tetra- and iteration. Tetration is used for the notation of very large numbers...
, Knuth's up-arrow notation
Knuth's up-arrow notation
In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976. It is closely related to the Ackermann function and especially to the hyperoperation sequence. The idea is based on the fact that multiplication can be viewed as iterated...
, Steinhaus-Moser notation, or Conway chained arrow notation
Conway chained arrow notation
Conway chained arrow notation, created by mathematician John Horton Conway, is a means of expressing certain extremely large numbers. It is simply a finite sequence of positive integers separated by rightward arrows, e.g. 2 → 3 → 4 → 5 → 6.As with most...
, though neither googol nor googolplex are anywhere near the largest representable or even specifically named numbers. For example, Graham's number
Graham's number
Graham's number, named after Ronald Graham, is a large number that is an upper bound on the solution to a certain problem in Ramsey theory.The number gained a degree of popular attention when Martin Gardner described it in the "Mathematical Games" section of Scientific American in November 1977,...
, though finite, is unimaginably larger than other well-known large numbers such as a googol
Googol
A googol is the large number 10100, that is, the digit 1 followed by 100 zeros:The term was coined in 1938 by 9-year-old Milton Sirotta , nephew of American mathematician Edward Kasner...
, googolplex, and even larger than Skewes' number and Moser's number. Indeed, the observable universe
Observable universe
In Big Bang cosmology, the observable universe consists of the galaxies and other matter that we can in principle observe from Earth in the present day, because light from those objects has had time to reach us since the beginning of the cosmological expansion...
is far too small to contain an ordinary digital representation
Numerical digit
A digit is a symbol used in combinations to represent numbers in positional numeral systems. The name "digit" comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 number system, i.e...
of Graham's number, assuming that each digit occupies at least one Planck volume.
In the physical universe
One googolGoogol
A googol is the large number 10100, that is, the digit 1 followed by 100 zeros:The term was coined in 1938 by 9-year-old Milton Sirotta , nephew of American mathematician Edward Kasner...
is presumed to be greater than the number of hydrogen
Hydrogen
Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...
atoms in the observable universe
Observable universe
In Big Bang cosmology, the observable universe consists of the galaxies and other matter that we can in principle observe from Earth in the present day, because light from those objects has had time to reach us since the beginning of the cosmological expansion...
, which has been variously estimated to be between 1079 and 1081. A googol is also greater than the number of Planck time
Planck time
In physics, the Planck time, , is the unit of time in the system of natural units known as Planck units. It is the time required for light to travel, in a vacuum, a distance of 1 Planck length...
s elapsed since the Big Bang
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...
, which is estimated at about 8. Thus in the physical world it is difficult to give examples of numbers that compare to the vastly greater googolplex. In analyzing quantum states and black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
s, physicist Don Page
Don Page (physicist)
Don N. Page is a Canadian theoretical physicist at the University of Alberta, Canada.His work focuses on quantum cosmology and black holes, and he is noted for being a doctoral student of the eminent Professor Stephen Hawking, in addition to publishing several journal articles with him...
writes that "determining experimentally whether or not information is lost down black holes of solar mass ... would require more than
measurements to give a rough determination of the final density matrix after a black hole evaporates". The end of the Universe via Big Freeze without proton decayProton decay
In particle physics, proton decay is a hypothetical form of radioactive decay in which the proton decays into lighter subatomic particles, such as a neutral pion and a positron...
is subject to be
years into the future, which is still short of a googolplex.In a separate article, Page shows that the number of states
State function
In thermodynamics, a state function, function of state, state quantity, or state variable is a property of a system that depends only on the current state of the system, not on the way in which the system acquired that state . A state function describes the equilibrium state of a system...
in a black hole with a mass roughly equivalent to the Andromeda Galaxy
Andromeda Galaxy
The Andromeda Galaxy is a spiral galaxy approximately 2.5 million light-years from Earth in the constellation Andromeda. It is also known as Messier 31, M31, or NGC 224, and is often referred to as the Great Andromeda Nebula in older texts. Andromeda is the nearest spiral galaxy to the...
is in the range of a googolplex.
years (~ googolplex years) — scale of an estimated Poincaré recurrence timePoincaré recurrence theorem
In mathematics, the Poincaré recurrence theorem states that certain systems will, after a sufficiently long time, return to a state very close to the initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence. The result applies to physical systems in which...
for the quantum state of a hypothetical box containing a black hole with the mass within the presently visible region of our universe. This time assumes a statistical model subject to Poincaré recurrence. A much simplified way of thinking about this time is in a model where our universe's history repeats itself
Loschmidt's paradox
Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics...
arbitrarily many times due to properties of statistical mechanics
Ergodic hypothesis
In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a particle in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equiprobable over a...
, this is the time scale when it will first be somewhat similar (for a reasonable choice of "similar") to its current state again.
years (~ googolplex years) — scale of an estimated Poincaré recurrence time for the quantum state of a hypothetical box containing a black hole with the estimated mass of the entire universe, observable or not, assuming a certain inflationary model with an inflaton whose mass is 10−6 Planck masses.If the entire volume of the observable universe (taken to be 3 × 1080 m3) were packed solid with fine dust particles about 1.5 micrometres in size, then the number of different ways of ordering these particles (that is, assigning the number 1 to one particle, then the number 2 to another particle, and so on until all particles are numbered) would be approximately one googolplex.
Other uses in popular culture
- The "GoogleplexGoogleplexThe Googleplex is the corporate headquarters complex of Google, Inc., located at 1600 Amphitheatre Parkway in Mountain View, Santa Clara County, California, United States, near San Jose. "Googleplex" is a portmanteau of Google and complex, and a reference to googolplex, the name given to the large...
" is the Google company headquarters, located at 1600 Amphitheatre Parkway in Mountain View, Santa Clara CountySanta Clara County, CaliforniaSanta Clara County is a county located at the southern end of the San Francisco Bay Area in the U.S. state of California. As of 2010 it had a population of 1,781,642. The county seat is San Jose. The highly urbanized Santa Clara Valley within Santa Clara County is also known as Silicon Valley...
, CaliforniaCaliforniaCalifornia is a state located on the West Coast of the United States. It is by far the most populous U.S. state, and the third-largest by land area...
, near San JoseSan Jose, CaliforniaSan Jose is the third-largest city in California, the tenth-largest in the U.S., and the county seat of Santa Clara County which is located at the southern end of San Francisco Bay...
. - Doc Brown mentions to the barbed wireBarbed wireBarbed wire, also known as barb wire , is a type of fencing wire constructed with sharp edges or points arranged at intervals along the strand. It is used to construct inexpensive fences and is used atop walls surrounding secured property...
peddlar in Back to the Future Part IIIBack to the Future Part IIIBack to the Future Part III is a 1990 American science fiction comedy Western film. It is the third installment of the Back to the Future trilogy. The film was directed by Robert Zemeckis and starred Michael J. Fox, Christopher Lloyd, Mary Steenburgen, Thomas F. Wilson and Lea Thompson. The film...
, that Clara Clayton is "one in a Googolplex".
See also
- Large numbersLarge numbersThis article is about large numbers in the sense of numbers that are significantly larger than those ordinarily used in everyday life, for instance in simple counting or in monetary transactions...
- Names of large numbersNames of large numbersThis article lists and discusses the usage and derivation of names of large numbers, together with their possible extensions.The following table lists those names of large numbers which are found in many English dictionaries and thus have a special claim to being "real words"...
- Orders of magnitude (numbers)Orders of magnitude (numbers)This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantity and probabilities. Each number is given a name in the short scale, which is used in English speaking countries, as well as a name in the long scale, which is used in some of the...