Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, sophomore's dream is a name occasionally used for the identities (especially the first) discovered in 1697 by Johann Bernoulli. The name is in contrast to the "freshman's dream" which is given to the incorrect (unless one is working over a field or unital commutative ring of characteristic n) equation (x + y)n = xn + yn. (The correct result is given by...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, sophomore's dream is a name occasionally used for the identities (especially the first) discovered in 1697 by Johann Bernoulli. The name is in contrast to the "freshman's dream" which is given to the incorrect (unless one is working over a field or unital commutative ring of characteristic n) equation (x + y)n = xn + yn. (The correct result is given by the binomial theorem.) The sophomore's dream has a similar too-good-to-be-true feel, but is in fact true. We prove the second identity; the first is completely analogous.
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