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 nonzero real numbers. 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 ''x''
2 + 2''xy'' + ''y''
2 using
distributivity
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmeti ...
(commonly known by students 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
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
''p'', if ''x'' and ''y'' are members of a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
of
characteristic ''p'', then
(''x'' + ''y'')
''p'' = ''x''
''p'' + ''y''
''p''. In this more exotic type of arithmetic, the "mistake" actually gives the correct result, since ''p'' divides all the
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s apart from the first and the last, making all 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
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
.
Examples
*
, but
.
*
does not generally equal
. For example,
, which does not equal . In this example, the error is being committed with the exponent .
Prime characteristic
When
is a prime number and
and
are members of a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
of
characteristic , then
. This can be seen by examining the prime factors of the binomial coefficients: the ''n''th binomial coefficient is
:
The
numerator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
is ''p''
factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \ ...
, which is divisible by ''p''. However, when , both ''n''! and 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 ''n''th binomial coefficient is divisible by ''p'' and hence equal to 0 in the ring. We are left with the zeroth and ''p''th coefficients, which both equal 1, yielding the desired equation.
Thus in characteristic ''p'' the freshman's dream is a valid identity. This result demonstrates that exponentiation by ''p'' produces an
endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ...
, known as the
Frobenius endomorphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphis ...
of the ring.
The demand that the characteristic ''p'' be a prime number is central to the truth of the freshman's dream. A related theorem states that if ''p'' is prime then in the
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...