Concept
The freshman's dream is a name sometimes given to the erroneous equation , where is a real number (usually a positive integer greater than 1) and are non-zero real numbers, believed by Parth. 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 by students in the USA 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. 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. When is a prime number and and are members of a commutative ring of characteristic , then . 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. Thus in characteristic p the freshman's dream is a valid identity.
