There aren't enough small numbers to meet the many demands made of them.
In other words, any given small number appears in far more contexts than may seem reasonable, leading to many apparently surprising coincidences in mathematics, simply because small numbers appear so often and yet are so few. Earlier (1980) this "law" was reported by Martin Gardner. Guy's paper gives 35 examples in support of this thesis. This can lead inexperienced mathematicians to conclude that these concepts are related, when in fact they are not.
Guy also formulated the Second Strong Law of Small Numbers:
"When two numbers look equal, it ain't necessarily so!"
Guy explains the latter law by the way of examples: he cites numerous sequences for which observing a small number of the first members may lead to a wrong guess about the generating formula or law for the sequence. Many of the examples are the observations of other mathematicians.
- Insensitivity to sample size
- Law of large numbers (unrelated, but the origin of the name)
- Mathematical coincidence
- Pigeonhole principle
- Representativeness heuristic
- Guy, Richard K. (1988). "The Strong Law of Small Numbers" (PDF). American Mathematical Monthly. 95 (8): 697–712. doi:10.2307/2322249. ISSN 0002-9890. JSTOR 2322249. Retrieved 2009-08-30.
- Gardner, M. "Mathematical Games: Patterns in Primes are a Clue to the Strong Law of Small Numbers." Sci. Amer. 243, 18-28, Dec. 1980.
- "Edge.org". www.edge.org. Retrieved 2019-05-07.
- Guy, Richard K. (1990). "The Second Strong Law of Small Numbers". Mathematics Magazine. 63 (1): 3–20. doi:10.2307/2691503. JSTOR 2691503.
- Caldwell, Chris. "Law of small numbers". The Prime Glossary.
- Weisstein, Eric W. "Strong Law of Small Numbers". MathWorld.
- Carnahan, Scott (2007-10-27). "Small finite sets". Secret Blogging Seminar, notes on a talk by Jean-Pierre Serre on properties of small finite sets.
- Amos Tversky; Daniel Kahneman (August 1971). "Belief in the law of small numbers". Psychological Bulletin. 76 (2): 105–110. CiteSeerX 10.1.1.592.3838. doi:10.1037/h0031322.
people have erroneous intuitions about the laws of chance. In particular, they regard a sample randomly drawn from a population as highly representative, I.e., similar to the population in all essential characteristics.
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