The freshman's dream is a name sometimes given to the error (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. 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 (or 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 case, the "mistake" actually gives the correct result, due to p dividing all the binomial coefficients save the first and the last.
Read more about Freshman's Dream: Examples, Prime Characteristic, History and Alternate Names
Famous quotes containing the words freshman and/or dream:
“It is well known, that the best productions of the best human intellects, are generally regarded by those intellects as mere immature freshman exercises, wholly worthless in themselves, except as initiatives for entering the great University of God after death.”
—Herman Melville (1819–1891)
“There is nothing truer than myth: history, in its attempt to “realize” myth, distorts it, stops halfway; when history claims to have “succeeded” this is nothing but humbug and mystification. Everything we dream is “realizable.” Reality does not have to be: it is simply what it is.”
—Eugène Ionesco (b. 1912)