Googol

Key takeaways

Googol belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Googol to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Googol from memory before moving on to harder problems.

Reference excerpt

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 The term was coined in 1920 by nine-year-old Milton Sirotta (1911–1981), nephew of American mathematician 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 for this quantity include ten duotrigintillion on the short scale (commonly used in English–speaking countries), ten thousand sexdecillion on the long scale, 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 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. 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 observable 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. 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. 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.

Properties A googol is approximately equal to 70 ! ≈ 1.1979 × 10 100 {\displaystyle 70!\approx 1.1979\times 10^{100}} (factorial of 70). Using an integral, binary numeral system, one would need 333 bits to represent a googol, i.e., 10 100 = 2 ( 100 / l o g 10 2 ) ≈ 2 332.19280949 {\displaystyle 10^{100}=2^{(100/\mathrm {log} _{10}2)}\approx 2^{332.19280949}} . 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 Googol is a homophone of the company name Google, an intentional misspelling of "googol" by the company's founders; it suggests that the search engine provides large quantities of information. In 2004, Kasner's heirs considered suing Google over their use of "googol"; however, no suit was ever filed. Since October 2009, Google has used the domain "1e100.net", "1e100" being E notation for 1 googol, to identify servers across its network. "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. A 1976 Richie Rich comic strip featured "The Googol", a masked villain so named because he had once been a US pilot pursued by 100 Zeros in the Second World War.

See also Googolplex Graham's number Infinity Names of large numbers Orders of magnitude (numbers) Skewes' number

References

External links

Weisstein, Eric W. "Googol". MathWorld. Googol at PlanetMath. Padilla, Tony; Symonds, Ria. "Googol and Googolplex". Numberphile. Brady Haran. Archived from the original on 2014-03-29. Retrieved 2013-04-06.

Worked examples

Example 1 — a first encounter with Googol

Start with the simplest possible case. Write down what Googol claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Googol before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Googol ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Googol

In research: Googol appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Googol in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Googol is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1920s neologisms, Integers, Large integers, so understanding it makes those chapters shorter.
In everyday life: Look for Googol outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Googol in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Googol means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Googol out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Googol in simple terms?

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.

Why does Googol matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Googol?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Googol.