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Googol
Googol
Googol
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A googol is the large number 10100 or ten to the power of one hundred. In decimal notation, it is written as the digit 1 followed by one hundred zeros: 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. Its systematic name is ten duotrigintillion (short scale) or ten sexdecilliard (long scale). Its prime factorization is 2100 × 5100.

Etymology

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The term was coined in 1920 by 9-year-old Milton Sirotta (1911–1981), nephew of American mathematician Edward Kasner.[1] He may have been inspired by the contemporary comic strip character Barney Google.[2] Kasner popularized the concept in his 1940 book Mathematics and the Imagination.[3] Other names for this quantity include ten duotrigintillion on the short scale (commonly used in English speaking countries),[4] ten thousand sexdecillion on the long scale, or ten sexdecilliard on the Peletier long scale.

Size

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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 game. Kasner used it to illustrate the difference between an unimaginably large number and infinity, and in this role it is sometimes used in teaching mathematics. To put in perspective the size of a googol, the mass of an electron, just under 10−30 kg, can be compared to the mass of the visible universe, estimated at between 1050 and 1060 kg.[5] 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).

Carl Sagan 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 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 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. The number would have to be 100,000 times bigger to make a googol.[6]

The decay time for a supermassive black hole of roughly 1 galaxy-mass (1011 solar masses) due to Hawking radiation is on the order of 10100 years.[7] Therefore, the heat death 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.[8]

Properties

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A googol is approximately equal to (factorial of 70). Using an integral, binary numeral system, one would need 333 bits to represent a googol, i.e., . However, a googol is well within the maximum bounds of an IEEE 754 double-precision floating point type without full precision in the mantissa.

Using modular arithmetic, 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, ... (sequence A066298 in the OEIS)

This sequence is the same as that of the residues (mod n) of a googolplex up until the 17th position.

Cultural impact

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Googol is a homophone of the company name Google, an accidental misspelling of "googol" by the company's founders;[9] it suggests that the search engine provides large quantities of information.[10] In 2004, Kasner's heirs considered suing Google over their use of "googol";[11] however, no suit was ever filed.[12]

Since October 2009, Google has used the domain "1e100.net", the scientific notation for 1 googol, to identify servers across its network.[13][14]

Googol was the £1 million answer in a 2001 episode of the British Who Wants to Be a Millionaire?, which the contestant allegedly won by cheating.[15]

See also

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References

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Googol

View on Grokipedia
from Grokipedia
A googol is a large number equal to 1010010^{100}, or 1 followed by 100 zeros in decimal notation. The term was coined in 1920 by nine-year-old Milton Sirotta, nephew of American mathematician (1878–1955), who asked his nephew to suggest a name for an unimaginably large but finite quantity during a family walk. Kasner, a at known for his work in and relativity, introduced the googol to make abstract concepts of scale accessible in popular mathematics. The name first appeared publicly in the 1940 book Mathematics and the Imagination, co-authored by Kasner and science writer James R. Newman, where it served as an example to illustrate the vastness of numbers beyond everyday comprehension. In the book, Kasner also defined the googolplex as 1010 raised to the power of a googol (101010010^{10^{100}}), a number so immense that it exceeds the estimated atoms in the by many orders of magnitude. The googol has since become a cultural touchstone for denoting extreme largeness in , , and popular media, often used to convey the limits of human intuition about quantity. It indirectly inspired the name of the search engine , founded in 1998 by and ; during brainstorming, Page favored "googol" to symbolize organizing the world's vast information, but a misspelling led to "Google" when registering the domain. The company's headquarters is named the , echoing Kasner's .

Definition and Origin

Definition

A googol is defined as the large integer equal to 1010010^{100}, which in decimal notation is written as the digit 1 followed by 100 zeros: 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000. In systematic naming conventions for , a googol is known as ten duotrigintillion on the short scale, which is the standard system used in . On the long scale, used traditionally in much of and in some British contexts, it is termed ten sexdecilliard. The googol is commonly expressed in as 1×101001 \times 10^{100} or simply in form as 1010010^{100}, highlighting its position as the 101st (starting from 100=110^0 = 1). This notation underscores its role as a benchmark for illustrating extremely large but finite quantities in .

Etymology

The term "googol" was coined in 1920 by nine-year-old Milton Sirotta, nephew of American mathematician , who had asked the boy to suggest a name for the enormous number 1010010^{100} during a family discussion on large numbers. This playful invention arose from a child's imagination, with no connection to prior mathematical nomenclature, and was selected to evoke a sense of whimsy and scale. Kasner first introduced the term to the public in his 1938 article "New Names in ," published in Scripta Mathematica, and further popularized it in his 1940 book and the Imagination, co-authored with James R. Newman, where it served as an accessible example to convey the immensity of certain quantities to a general audience. In the book, Kasner recounts the origin, quoting Sirotta's suggestion directly to highlight how even young minds could contribute to mathematical discourse. During the same conversation, Sirotta also proposed "googolplex" as an even larger number, initially described as a 1 followed by as many zeros as could be written before tiring, later formalized as 10googol10^{\text{googol}}. The linguistic roots of "googol" likely stem from childish or nonsensical sounds, possibly influenced by playful expressions like "," underscoring its non-technical, inventive character.

Magnitude

Numerical Representation

A googol is expressed in decimal notation as the integer 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, or more succinctly, 1 followed by 100 zeros, comprising a total of 101 digits. This full expansion is impractical to write out by hand in a continuous line due to its length, though it fits easily on a single printed page with standard formatting. In practice, the number is almost always represented using as 1010010^{100} to avoid the verbosity of the explicit digit string. In binary, a googol requires exactly 333 bits for its representation, determined by the formula ⌊log⁡2(10100)⌋+1=⌊100log⁡210⌋+1\lfloor \log_2 (10^{100}) \rfloor + 1 = \lfloor 100 \log_2 10 \rfloor + 1, where log⁡210≈3.321928\log_2 10 \approx 3.321928. The binary form begins with a leading 1 followed by a specific of 332 bits, but it is rarely written out explicitly due to the same length constraints as the version. For (base-16) representation, it spans 84 digits, computed similarly as ⌊100log⁡1610⌋+1≈⌊100×0.83048⌋+1=84\lfloor 100 \log_{16} 10 \rfloor + 1 \approx \lfloor 100 \times 0.83048 \rfloor + 1 = 84. In higher bases, such as base 10,000, the googol simplifies to 1 followed by 25 zeros, since (104)25=10100(10^4)^{25} = 10^{100}, reducing the digit count to 26 while preserving the value. These alternative base representations highlight the efficiency of for but underscore why exponential form remains the standard for a googol, given its 101-digit length exceeds typical manual or compact storage needs.

Scale Comparisons

To grasp the enormity of a googol, consider everyday-scale analogies that pale in comparison. The total number of grains of on Earth's beaches is estimated at approximately 7.5 × 10^{18}. Similarly, the number of in the is about 10^{24}. A googol surpasses these figures by 82 orders of magnitude in the case of sand grains and by 76 orders in the case of stars, underscoring how profoundly larger it is than tangible, human-scale or even cosmic counts of discrete objects. On astronomical scales, the googol remains vastly superior. The contains roughly 10^{80} baryonic particles, such as protons and neutrons. Estimates for the total number of atoms—primarily and —fall in a similar range of about 10^{80} to 10^{82}. Even these colossal tallies, representing the fundamental building blocks of all visible matter, are dwarfed by a googol, which exceeds them by 18 to 20 orders of magnitude. In cosmological contexts, the googol evokes hypothetical timescales beyond comprehension. For instance, a solar-mass black hole would take approximately 10^{67} years to fully evaporate through —a duration exceeding the current (about 1.4 × 10^{10} years) by 57 orders of magnitude. A timescale of a googol years would correspond to the evaporation lifetime of an extraordinarily massive black hole, far larger than any observed in the universe, highlighting the googol's placement on the farthest edges of physical possibility. Hypothetical packing scenarios further emphasize the scale. At the Planck density—the theoretical maximum of about 5 × 10^{96} kg/m³—the volume required to accommodate a googol of fundamental particles (one per Planck volume of roughly 10^{-105} m³) would be on the order of 10^{-5} m³, roughly the volume of a small grapefruit. This compactness illustrates how a googol, while immense in count, could theoretically fit into an extraordinarily confined space under quantum gravitational limits, yet such a configuration defies known physics for ordinary matter.

Mathematical Properties

Arithmetic Characteristics

The googol, defined as 1010010^{100}, possesses straightforward arithmetic characteristics stemming from its structure as a pure , which simplifies many basic operations. Its prime is 2100×51002^{100} \times 5^{100}, making it divisible by any power of 2 or 5 up to the exponent 100, such as 2502^{50} or 5755^{75}, but not by any other primes. This shows that the googol is composed solely of the primes 2 and 5, with (100+1)(100+1)=10,201(100 + 1)(100 + 1) = 10,201 positive divisors, arising from independently choosing exponents for 2 and 5 from 0 to 100. Powers of the googol follow the standard rules for exponents with base 10. For instance, the square of the googol is (10100)2=10200(10^{100})^2 = 10^{200}, a 1 followed by 200 zeros. Higher powers, such as the cube 1030010^{300}, similarly result in escalating exponents without altering the base. Roots of the googol are equally : the nnth root is 10100/n10^{100/n}, where for nn dividing 100, the result is an power of 10. A representative example is the 10th root, which equals 101010^{10}, or 10 billion. Multiplication by small integers preserves the power-of-10 form in a scaled manner. Multiplying the googol by 2 yields 2×101002 \times 10^{100}, simply 2 followed by 100 zeros, while multiplication by 3 produces 3×101003 \times 10^{100}. In general, for any positive integer k<10k < 10, k×10100k \times 10^{100} is kk followed by 100 zeros. Addition and subtraction involving the googol are less simplifying; adding 1 results in 10100+110^{100} + 1, which is a 1 followed by 99 zeros and ending in another 1, a form without a simpler closed mathematical expression beyond the explicit sum. Subtraction of 1 gives 10100−110^{100} - 1, consisting of 100 nines, again a direct but non-power-of-10 result.

Approximations and Relations

The googol, defined as 1010010^{100}, finds a close approximation among factorials through Stirling's formula, an asymptotic expansion for large n!n! given by n!≈2πn(ne)nn! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n
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