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)
“Romeo. I dreamt a dream tonight.
Mercutio. And so did I.
Romeo. Well, what was yours?
Mercutio. That dreamers often lie.
Romeo. In bed asleep, while they do dream things true.
Mercutio. O then I see Queen Mab hath been with you.
She is the fairies’ midwife, and she comes
In shape no bigger than an agate stone
On the forefinger of an alderman,
Drawn with a team of little atomi
Over men’s noses as they lie asleep.”
—William Shakespeare (1564–1616)