Quasi Fermi level: Difference between revisions

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rephrase in terms of conduction band and valence band, since that's what is really going on
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[[File:Freshman's Dream.svg|right|thumbnail|An illustration of the Freshman's dream in two dimensions. Each side of the square is X+Y in length. The area of the square is the sum of the area of the yellow region (=X<sup>2</sup>), the area of the green region (=Y<sup>2</sup>), and the area of the two white regions (=2×X×Y).]]
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The '''freshman's dream''' is a name sometimes given to the error (''x''&nbsp;+&nbsp;''y'')<sup>''n''</sup>&nbsp;=&nbsp;''x''<sup>''n''</sup>&nbsp;+&nbsp;''y''<sup>''n''</sup>, where ''n'' is a real number (usually a positive integer greater than 1). Beginning students commonly make this error in computing the [[exponentiation|power]] of a sum of real numbers.<ref>Julio R. Bastida, ''Field Extensions and Galois Theory'', Addison-Wesley Publishing Company, 1984, p.8.</ref><ref>Fraleigh, John B., ''A First Course in Abstract Algebra'', Addison-Wesley Publishing Company, 1993, p.453, ISBN 0-201-53467-3.</ref> When ''n'' = 2, it is easy to see why this is incorrect: (''x''&nbsp;+&nbsp;''y'')<sup>2</sup> can be correctly computed  as ''x''<sup>2</sup>&nbsp;+&nbsp;2''xy''&nbsp;+&nbsp;''y''<sup>2</sup> 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_(algebra)|characteristic]] ''p'', then
(''x''&nbsp;+&nbsp;''y'')<sup>''p''</sup>&nbsp;=&nbsp;''x''<sup>''p''</sup>&nbsp;+&nbsp;''y''<sup>''p''</sup>. In this case, the "mistake" actually gives the correct result, due to ''p'' dividing all the [[binomial coefficient]]s save the first and the last.
 
==Examples==
*<math>(1+4)^2 = 5^2 = 25</math>, but <math>1^2+4^2 = 17</math>.  
*<math>\sqrt{x^2+y^2}</math> does not generally equal <math>\sqrt{x^2}+\sqrt{y^2}=|x|+|y|</math>. For example, <math>\sqrt{9+16}=\sqrt{25}=5</math>, which does not equal 3+4=7. In this example, the error is being committed with the exponent ''n''&nbsp;=&nbsp;{{frac|1|2}}.
 
==Prime characteristic==
 
When ''p'' is a prime number and ''x'' and ''y'' are members of a [[commutative ring]] of [[characteristic_(algebra)|characteristic]] ''p'', then (''x''&nbsp;+&nbsp;''y'')<sup>''p''</sup>&nbsp;=&nbsp;''x''<sup>''p''</sup>&nbsp;+&nbsp;''y''<sup>''p''</sup>. This can be seen by examining the prime factors of the binomial coefficients: the ''n''th binomial coefficient is
 
:<math>\binom{p}{n} = \frac{p!}{n!(p-n)!}.</math>
 
The [[numerator]] is ''p'' [[factorial]], which is divisible by ''p''. However, when {{nowrap|0 < ''n'' < ''p''}}, neither ''n''! nor {{nowrap|(''p'' &minus; ''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 ''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]], 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 a number ''n'' is prime [[if and only if]] {{nowrap|(''x''+1)<sup>''n''</sup> ≡ ''x<sup>n</sup>'' + 1 (mod ''n'')}} in the [[polynomial ring]] <math>\mathbb{Z}_n[x]</math>. This theorem is a direct consequence of [[Fermat's Little Theorem]] and it is a key fact in modern primality testing.<ref name=Granville>A. Granville, ''[http://www.ams.org/bull/2005-42-01/S0273-0979-04-01037-7/S0273-0979-04-01037-7.pdf It Is Easy To Determine Whether A Given Integer Is Prime]'', Bull. of the AMS, Volume 42, Number 1 (Sep. 2004), Pages 3–38.</ref>
 
==History and alternate names==
The history of the term "freshman's dream" is somewhat unclear. In a 1940 article on [[modular field]]s, [[Saunders Mac Lane]] quotes [[Stephen Cole Kleene|Stephen Kleene]]'s remark that a knowledge of (''a'' + ''b'')<sup>2</sup>= ''a''<sup>2</sup> + ''b''<sup>2</sup> in a [[field (mathematics)|field]] of characteristic 2 would corrupt freshman students of [[abstract algebra|algebra]]. This may be the first connection between "freshman" and binomial expansion in fields of positive characteristic.<ref>Colin R. Fletcher, Review of '''Selected papers on algebra, edited by [[Susan Montgomery]], Elizabeth W. Ralston and others. Pp xv, 537. 1977. SBN 0 88385 203 9 (Mathematical Association of America)''', ''The Mathematical Gazette'', Vol. 62, No. 421 (Oct., 1978), The Mathematical Association. p. 221.</ref> 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 [[Thomas W. Hungerford|Hungerford's]] undergraduate algebra textbook (1974), where he quotes McBrien.<ref>Thomas W. Hungerford, ''Algebra,'' Springer, 1974, p. 121; also in ''Abstract Algebra: An Introduction'', 2nd edition. Brooks Cole, July 12, 1996, p. 366.</ref> Alternative terms include "'''freshman exponentiation'''", used in Fraleigh (1998).<ref>John B. Fraleigh, ''A First Course In Abstract Algebra'', 6th edition, Addison-Wesley, 1998. pp. 262 and 438.</ref> The term "freshman's dream" itself, in non-mathematical contexts, is recorded since the 19th century.<ref>[http://www.google.com/search?tbo=p&tbm=bks&q=%22freshman%27s+dream%22&tbs=,cdr:1,cd_min:Jan%201_2%201800,cd_max:Dec%2031_2%201900&num=10 Google books 1800–1900 search for "freshman's dream"]: [http://books.google.com/books?id=3XNHAAAAYAAJ&pg=PA176&dq=%22freshman%27s+dream%22 Bentley's miscellany, Volume 26, p. 176], 1849</ref>
 
Since the expansion of (''x'' + ''y'')<sup>''n''</sup> is correctly given by the [[binomial theorem]], the freshman's dream is also known as the "'''Child's Binomial Theorem'''" <ref name=Granville/> or "'''Schoolboy Binomial Theorem'''".
 
==See also==
*[[Primality test]]
*[[Sophomore's dream]]
 
==References==
<references/>
 
[[Category:Algebra]]
[[Category:Mathematics education]]

Revision as of 13:30, 9 February 2014

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