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The freshman's dream is a name given to the erroneous equation (x+y)^n=x^n+y^n, where n is a real number (usually a positive integer greater than 1) and x,y are non-zero real numbers. Beginning students commonly make this error in computing the power of a sum of real numbers, falsely assuming powers distribute over sums. When ''n'' = 2, it is easy to see why this is incorrect: (''x'' + ''y'')2 can be correctly computed as ''x''2 + 2''xy'' + ''y''2 using
distributivity In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary ...
(commonly known by students in the
United States The United States of America (USA), also known as the United States (U.S.) or America, is a country primarily located in North America. It is a federal republic of 50 U.S. state, states and a federal capital district, Washington, D.C. The 48 ...
as the FOIL method). For larger positive integer values of ''n'', the correct result is given by the
binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and a ...
. The name "freshman's dream" also sometimes refers to the theorem that says that for a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
''p'', if ''x'' and ''y'' are members of a
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
of characteristic ''p'', then (''x'' + ''y'')''p'' = ''x''''p'' + ''y''''p''. In this more exotic type of arithmetic, the "mistake" actually gives the correct result, since ''p'' divides all the
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s apart from the first and the last, making all the intermediate terms equal to zero. The identity is also actually true in the context of tropical geometry, where multiplication is replaced with addition, and addition is replaced with
minimum In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given range (the ''local'' or ''relative ...
.


Examples

*(1+4)^2 = 5^2 = 25, but 1^2+4^2 = 17. *\sqrt does not equal \sqrt+\sqrt=, x, +, y, . For example, \sqrt=\sqrt=5, which does not equal . In this example, the error is being committed with the exponent .


Prime characteristic

When p is a prime number and x and y are members of a
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
of characteristic p, then (x+y)^p=x^p+y^p. This can be seen by examining the prime factors of the binomial coefficients: the ''n''th binomial coefficient is :\binom = \frac. The numerator is ''p''
factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), 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 ...
(!), which is divisible by ''p''. However, when , both ''n''! and are coprime with ''p'' since all the factors are less than ''p'' and ''p'' is prime. Since a binomial coefficient is always an integer, the ''n''th binomial coefficient is divisible by ''p'' and hence equal to 0 in the ring. We are left with the zeroth and ''p''th coefficients, which both equal 1, yielding the desired equation. Thus in characteristic ''p'' the freshman's dream is a valid identity. This result demonstrates that exponentiation by ''p'' produces an
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
, known as the
Frobenius endomorphism In commutative algebra and field theory (mathematics), field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative Ring (mathematics), rings with prime number, prime characteristic (algebra), ...
of the ring. The demand that the characteristic ''p'' be a prime number is central to the truth of the freshman's dream. A related theorem states that if ''p'' is prime then in the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
\mathbb_p /math>. This theorem is a key fact in modern primality testing.A. Granville,
It Is Easy To Determine Whether A Given Integer Is Prime
', Bull. of the AMS, Volume 42, Number 1 (Sep. 2004), Pages 3–38.


History and alternate names

In 1938, Harold Willard Gleason published a poem titled «"Dark and Bloody Ground---" (''The Freshman's Dream'')» in ''
The New York Sun ''The New York Sun'' is an American Conservatism in the United States, conservative Online newspaper, news website and former newspaper based in Manhattan, Manhattan, New York. From 2009 to 2021, it operated as an (occasional and erratic) onlin ...
'' on September 6, which was subsequently reprinted in various other newspapers and magazines. It consists of 2 stanzas, each containing 8 lines with alternating indentation; it has an ABCB rhyming scheme. Words and phrases that hint that it might be related to this concept include: "
Algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
", "Wild
corollaries In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
twine", " surds", "of plus and minus sign", " binomial", " quadratic", "
parenthesis A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. They come in four main pairs of shapes, as given in the box to the right, which also gives their n ...
", " exponents", "in terms of x and y", "remove the brackets, radicals, and do so with discretion", and "factor cubes". The history of the term "freshman's dream" is somewhat unclear. In a 1940 article on modular fields,
Saunders Mac Lane Saunders Mac Lane (August 4, 1909 – April 14, 2005), born Leslie Saunders MacLane, was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near w ...
quotes
Stephen Kleene Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of ...
's remark that a knowledge of in a field of characteristic 2 would corrupt freshman students of
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
. This may be the first connection between "freshman" and binomial expansion in fields of positive characteristic. Since then, authors of undergraduate algebra texts took note of the common error. The first actual attestation of the phrase "freshman's dream" seems to be in Hungerford's graduate algebra textbook (1974), where he states that the name is "due to" Vincent O. McBrien. Alternative terms include "freshman exponentiation", used in Fraleigh (1998). The term "freshman's dream" itself, in non-mathematical contexts, is recorded since the 19th century.Google books 1800–1900 search for "freshman's dream"Bentley's miscellany, Volume 26, p. 176
1849 Since the expansion of is correctly given by the
binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and a ...
, the freshman's dream is also known as the "child's binomial theorem" or "schoolboy binomial theorem".


See also

* Pons asinorum *
Primality test A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating wheth ...
* Sophomore's dream *
Frobenius endomorphism In commutative algebra and field theory (mathematics), field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative Ring (mathematics), rings with prime number, prime characteristic (algebra), ...


References

{{reflist, 2 Algebra education Mathematical fallacies Theorems in ring theory Prime numbers