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「一年生の夢」を参照

一年生の夢の視覚的表現。正方形の一辺の長さがX+Y。色のついた部分が一年生の夢が正しかった場合、この図形の面積は黄色の部分(=X²)と緑色の部分(=Y²)の和となるが、実際には白い部分(=2×X×Y)も含めた和がこの正方形の面積となる。

数学において、一年生の夢(英:freshman's dream) とは、nが実数(実際には1より大きい自然数を指すことが多い)のとき、(x + y)ⁿ = xⁿ + yⁿとする誤りにつけられた名前である。学び始めの学生(=一年生)がよく間違えるとされる、実数の和の累乗である。例えばn = 2のとき、何が間違っているかを知ることは容易い。(x + y)²は正しくは分配法則(またはFOIL method)を用いることによって、x² + 2xy + y²と計算される。また、2以上の自然数がnの場合は、正しい結果は二項定理によって与えられる。

また、「一年生の夢」という呼称は refers to the theorem that says that for a 素数p, if x and y are members of a commutative ring of characteristic p, then (x + y)^p = x^p + y^p. In this case, the "mistake" actually gives the correct result, due to p dividing all the binomial coefficients save the first and the last.

Examples

• $(1+4)^{2}=5^{2}=25$, but $1^{2}+4^{2}=17$.
• ${\sqrt {x^{2}+y^{2}}}$ does not generally equal ${\sqrt {x^{2}}}+{\sqrt {y^{2}}}=|x|+|y|$. For example, ${\sqrt {9+16}}={\sqrt {25}}=5$, which does not equal 3+4=7. In this example, the error is being committed with the exponent n = 1⁄2.

Prime characteristic

When p is a prime number and x and y are members of a commutative ring of characteristic p, then (x + y)^p = x^p + y^p. This can be seen by examining the prime factors of the binomial coefficients: the nth binomial coefficient is

${\binom {p}{n}}={\frac {p!}{n!(p-n)!}}.$

The numerator is p factorial, which is divisible by p. However, when 0 < n < p, neither n! nor (p − n)! is divisible by p since all the terms 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. This result demonstrates that exponentiation by p produces an endomorphism, known as the Frobenius endomorphism of the ring.

The demand that the characteristic p be a prime number is central to the truth of the freshman's dream. In fact, a related theorem states that if a number n is prime then (x+1)ⁿ ≡ xⁿ + 1 (mod n) in the polynomial ring $\mathbb {Z} _{n}[x]$. This theorem is a direct consequence of Fermat's Little Theorem and it is a key fact in modern primality testing.

History and alternate names

The history of the term "freshman's dream" is somewhat unclear. In a 1940 article on modular fields, Saunders Mac Lane quotes Stephen Kleene's remark that a knowledge of (a + b)²= a² + b² in a field of characteristic 2 would corrupt freshman students of algebra. This may be the first connection between "freshman" and binomial expansion in fields of positive characteristic. Since then, authors of undergraduate algebra texts took note of the common error. The first actual attestation of the phrase "freshman's dream" seems to be in Hungerford's undergraduate algebra textbook (1974), where he quotes McBrien. Alternative terms include "freshman exponentiation", used in Fraleigh (1998). The term "freshman's dream" itself, in non-mathematical contexts, is recorded since the 19th century.

Since the expansion of (x + y)ⁿ is correctly given by the binomial theorem, the freshman's dream is also known as the "Child's Binomial Theorem" or "Schoolboy Binomial Theorem".

See also

• Primality test
• Sophomore's dream
• Frobenius endomorphism