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)
“A novel is what you dream in your night sleep. A novel is
not waking thoughts although it is written and thought
with waking thoughts. But really a novel goes as
dreams go in sleeping at night and some dreams are like
anything and some dreams are like something and some
dreams change and some dreams are quiet and some dreams
are not. And some dreams are just what any one would
do only a little different always just a little
different and that is what a novel is.”
—Gertrude Stein (1874–1946)