The freshman's dream is a name sometimes given to the erroneous equation (x + y)n = xn + yn, where n is a real number (usually a positive integer greater than 1). 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 x2 + 2xy + y2 using distributivity (commonly known 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 intermediate terms equal to zero.
When p is a prime number and x and y are members of a commutative ring of characteristic p, then (x + y)p = xp + yp. This can be seen by examining the prime factors of the binomial coefficients: the nth binomial coefficient is
The numerator is p factorial, which is divisible by p. However, when 0 < n < p, both n! and (p − n)! 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 nth binomial coefficient is divisible by p and hence equal to 0 in the ring. We are left with the zeroth and pth coefficients, which both equal 1, yielding the desired equation.
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. 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. 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.