Freshman's dream
The freshman's dream is a name given to the erroneous equation , where is a real number (usually a positive integer greater than 1) and 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.[1][2] When n = 2, it is easy to see why this is incorrect: (x + y)2 can be correctly computed as x2 + 2xy + y2 using distributivity (commonly known by students in the United States as the FOIL method). For larger positive integer values of n, the correct result is given by the binomial theorem.
The name "freshman's dream" also sometimes refers to the theorem that says that for a prime number p, if x and y are members of a commutative ring of characteristic p, then
(x + y)p = xp + yp. In this more exotic type of arithmetic, the "mistake" actually gives the correct result, since p divides all the binomial coefficients 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.[3]
, but .
does not equal . For example, , which does not equal 3 + 4 = 7. In this example, the error is being committed with the exponent n = 1/2.
History and alternate names[edit]
The history of the term "freshman's dream" is somewhat unclear. In a 1940 article on modular fields, Saunders Mac Lane quotes Stephen Kleene's remark that a knowledge of (a + b)2 = a2 + b2 in a field of characteristic 2 would corrupt freshman students of algebra. This may be the first connection between "freshman" and binomial expansion in fields of positive characteristic.[5] 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 quotes McBrien.[6] Alternative terms include "freshman exponentiation", used in Fraleigh (1998).[7] The term "freshman's dream" itself, in non-mathematical contexts, is recorded since the 19th century.[8]
Since the expansion of (x + y)n is correctly given by the binomial theorem, the freshman's dream is also known as the "child's binomial theorem"[4] or "schoolboy binomial theorem".