{{Short description|Mathematical fallacy}} 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).
In mathematics, the '''freshman's dream''',<ref name="Bastida 1984 CambridgeUP">{{Cite book |last=Bastida |first=Julio R. |title=Field Extensions and Galois Theory |date=1984 |publisher=Cambridge University Press |isbn=978-0-521-30242-5 <!-- or 978-0-521-17396-4 for paperback --> |series=Encyclopedia of Mathematics and its Applications, vol. 22 |pages=1–40 (see p. 8) |chapter=Chapter 1: Preliminaries on Fields and Polynomials |doi=10.1017/cbo9781107340749.007 |id={{EBSCOhost|589162}}, [https://research.ebsco.com/linkprocessor/plink?id=234808e1-8858-3404-80a3-e48607427a55 {{small|234808e1-8858-3404-80a3-e48607427a55}}]. |access-date=2025-12-04 |chapter-url=https://www.cambridge.org/core/books/field-extensions-and-galois-theory/preliminaries-on-fields-and-polynomials/F594CD1C96CD3A8CCE580203422BBAD7}}</ref><ref name="Maclagan 2011 ELGA">{{Cite AV media |url=https://www.youtube.com/watch?v=unjVp6HQVmc |title=Introduction to Tropical Algebraic Geometry ([class] 1 of 5) |date=August 1, 2011 |last=Maclagan |first=Diane |author-link=Diane Maclagan <!-- "Warwick, UK" --> |publisher=Difusión DM (Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires) |publication-place=Buenos Aires |series=ELGA (Escuela Latinoamericana de Geometría Algebraica y Aplicaciones) 2011<!-- see https://www.commalg.org/2011/08/01/elga-2011-argentina/ -->, by CIMPA, ICTP, UNESCO, MICINN, and Santaló<!-- see also https://www.cimpa.info/sites/default/files/Report_RS_11-E11_1.pdf --> |time=4m24s–5m23s |via=YouTube |publication-date=February 23, 2018}}</ref><ref name="Kalisnik 2019 FoCM">{{Cite journal |last=Kališnik |first=Sara <!-- Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany; Wesleyan University, Middletown, CT, USA --> |date=February 2019 |orig-date={{small|Received: 1 April 2016 / Revised: 6 June 2017 / Accepted: 22 December 2017 / Published online: 30 January 2018}} |others=<!-- Communicated by Herbert Edelsbrunner. --> |title=Tropical Coordinates on the Space of Persistence Barcodes |url=https://link.springer.com/article/10.1007/s10208-018-9379-y |journal=Foundations of Computational Mathematics |language=en |volume=19 |issue=1 |pages=101–129 (see p. 103) |doi=10.1007/s10208-018-9379-y |issn=1615-3375 |id={{Gale|A574342645}}.|arxiv=1604.00113 }}</ref><ref name="Fletcher 1978 MG">{{Cite journal |last=Fletcher |first=Colin R. <!-- University College of Wales --> |date=October 1978 |title=[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) |url= |journal=The Mathematical Gazette |volume=62 |issue=421 |pages=220–222 (see p. 221) |doi=10.2307/3616706 |issn=0025-5572 |jstor=3616706}}</ref> also known as '''freshman exponentiation''',<ref name="Fletcher 1978 MG" /><ref name="Fraleigh 1994 AddisonWesley">{{Cite book |last=Fraleigh |first=John B. |title=A First Course in Abstract Algebra |date=1993–1994 |publisher=Addison-Wesley Publishing Company |isbn=978-0-201-53467-2 |edition=5th |publication-place=Reading, Mass. |pages=283, 453 |lccn=93-1997 |id=Internet Archive [https://archive.org/details/firstcourseinabs0000fral_g1t8/page/452?q=freshman firstcourseinabs0000fral_g1t8].}} (See alternatively 6th ed. (1998), pp. 262 and 438.)</ref> the '''child's binomial theorem''',<ref name="Granville 2004 BullAMS">{{Cite journal |last=Granville |first=Andrew |date=September 30, 2004 <!-- "Article electronically published on September 30, 2004" --> |title=It is easy to determine whether a given integer is prime |url=https://www.ams.org/journals/bull/2005-42-01/S0273-0979-04-01037-7/S0273-0979-04-01037-7.pdf |journal=Bulletin of the American Mathematical Society |series=New Series |volume=42 |issue=1 |pages=3–38 (see pp. 8, 12) |doi=10.1090/s0273-0979-04-01037-7 <!-- "S 0273-0979(04)01037-7" --> |issn=0273-0979}}</ref> (rarely) the '''schoolboy binomial theorem''',<ref name="Clark 2018 UGA">{{Cite book |last=Clark |first=Pete L. |url=http://alpha.math.uga.edu/~pete/4400FULL2018.pdf |title=Number Theory: A Contemporary Introduction |date=April 19, 2018 <!-- PDF metadata "Create Date" and "Modify Date" are both "2018:04:19 14:14:00-04:00" --> |page=64 |chapter=Lemma 4.20. ("Schoolboy binomial theorem") |archive-url=https://web.archive.org/web/20241214183013/http://alpha.math.uga.edu/~pete/4400FULL2018.pdf |archive-date=2024-12-14 |url-status=dead |department=Chapter 4. Quadratic Reciprocity : 7. Proof of the Second Supplement}}</ref> or the '''Frobenius identity'''<ref name="Kalisnik 2019 FoCM" /> is the generally-false equation (''x'' + ''y'')<sup>''n''</sup> = ''x''<sup>''n''</sup> + ''y''<sup>''n''</sup>. Beginning students commonly make this error in computing the power of a sum of real numbers, falsely assuming powers distribute over sums.
The correct result is given by the binomial theorem,<ref name="Fraleigh 1994 AddisonWesley" /> which has additional terms in the middle when ''n'' ≥ 2.<ref name="Bastida 1984 CambridgeUP" /> For example, when ''n'' = 2, the correct result is ''x''<sup>2</sup> + 2''xy'' + ''y''<sup>2</sup>, which can also be shown by multiplying (''x'' + ''y'')(''x'' + ''y'') by using the distributive property properly, or the FOIL method.
The freshman's dream is actually valid in commutative rings of characteristic ''p'', such as the finite field <math>\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}</math>, where ''p'' is a prime number, provided that the exponent ''n'' is ''p'' or more generally a power of ''p''. Equivalently, the Frobenius map of the ring is an endomorphism. One way to prove this is to show that ''p'' divides all the binomial coefficients except for the first and the last, so all the intermediate terms are equal to zero.<ref name="Bastida 1984 CambridgeUP" /><ref name="Fraleigh 1994 AddisonWesley" /> Another way to prove the common special case of this for <math>\mathbb{F}_p</math> is to use Fermat's little theorem that ''a<sup>p</sup>'' ≡ ''a'' mod ''p'' for all integers ''a''.<ref name="Granville 2004 BullAMS" /> (This can be iterated for powers of ''p'', using the property of exponentiation that taking a power of a power multiplies the exponents, and thereby proven in general using induction.)
The freshman's dream is valid for all ''n'' in tropical geometry<ref name="Maclagan 2011 ELGA" /><ref name="Kalisnik 2019 FoCM" /> (where multiplication is replaced with addition, so exponentiation becomes multiplication, and addition is replaced with minimum).
The freshman's dream equation is also true in some degenerate cases, such as when ''n'' = 1, when <math>n \ge 1</math> and at least one of ''x'' and ''y'' is zero, and when ''n'' is an odd integer and <math>y=-x</math>. These are all of the true cases for ''n'' ∈ {0, 1, 2, 3}, but when ''n'' ≥ 4 or ''n'' is negative or non-integer, there are generally additional pairs of complex numbers ''x'', ''y'' that satisfy the equation.
==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 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 {{nowrap|1=3 + 4 = 7}}. In this example, the error is being committed with the exponent {{nowrap|1=''n'' = {{sfrac|1|2}}}}.
==Prime characteristic==
When <math>p</math> is a prime number and <math>x</math> and <math>y</math> are members of a commutative ring of characteristic <math>p</math>, then <math>(x+y)^p=x^p+y^p</math>. 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''}}, both ''n''! and {{nowrap|(''p'' − ''n'')!}} 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, 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. A related theorem states that a number ''n'' is prime if and only if {{nowrap|1=(''x'' + 1)<sup>''n''</sup> = ''x<sup>n</sup>'' + 1}} in the polynomial ring <math>(\mathbb{Z}/n\mathbb{Z})[x]</math>. This theorem is a key fact in modern primality testing.<ref name="Granville 2004 BullAMS" />
==History==
<!-- note: this section is ordered chronologically -->
The history of the term "freshman's dream" is somewhat unclear.
The phrase "freshman's dream" is recorded in non-mathematical contexts since at least the 1840s.<ref>{{Cite magazine |date=March 1846 |title=Notices to Correspondents |department=Editors' Table (pp. 213–216) |magazine=Nassau Monthly [Nassau Literary Magazine] |publisher=[Open Court Publishing Co] |pages=214–216 (see p. 215) |publication-place=[Princeton] |volume=V |issue=VI |id=Internet Archive [https://archive.org/details/sim_nassau-literary-magazine_1846-03_5_6/page/215?q=%22freshman%27s+dream%22 sim_nassau-literary-magazine_1846-03_5_6] (canister [https://archive.org/search?query=source%3A%22IA1641630-03%22 IA1641630-03], sim_pubid 4839, [https://ark.archive.org/ark:/13960/t9q36m07x ark:/13960/t9q36m07x]). {{ProQuest|137473839}} (in American Periodicals Series II). |quote=A prose article claiming for itself the dignity of blank verse, entitled, "The Freshman's Dream," and signed "Minnow," next turns up its woful face to notice.}}</ref><ref>{{Cite periodical |date=1849 |orig-date=February <!-- according to the American Antiquarian Society --> |title=Poetry ''versus'' Science, a Freshman's Dream |magazine=Bentley's Miscellany |publisher=Richard Bentley, New Burlington Street |pages=176–184 (see also pp. iii,<!-- ToC --> 651<!-- index -->) |publication-place=London |volume=XXVI <!-- labeled "twenty-sixth" on p. 652 --> |lccn=05014033 <!-- listed on EBSCOhost as "5-14033//r83" --> |id={{EBSCOhost|47217143}}. {{Gale|HNTCQJ330608311|CY0106821297}}. Google Books [https://www.google.com/books/edition/Bentley_s_Miscellany/dt4RAAAAYAAJ?gbpv=1&dq=%22freshman's%20dream%22&pg=PA176 dt4RAAAAYAAJ]. HathiTrust [https://babel.hathitrust.org/cgi/pt?id=njp.32101076368255&seq=192&q1=%22freshman%27s+dream%22 njp.32101076368255], [https://babel.hathitrust.org/cgi/pt?id=nyp.33433081753042&seq=192 nyp.33433081753042]. Internet Archive [https://archive.org/details/bentleysmiscell06cruigoog/page/180?q=%22freshman%27s+dream%22 bentleysmiscell06cruigoog], [https://archive.org/details/sim_bentleys-miscellany_1849-07_26?q=%22freshman%27s+dream%22 sim_bentleys-miscellany_1849-07_26]. {{ProQuest|1310865214}}.}} The issues of ''The Literary Gazette, and Journal of the Belles Lettres, Arts, Sciences, &c.'' (no. 1697, p. 558), ''The Athenæum'' (no. 1135, p. 754), ''The Spectator'' ([vol. 22], no. 1100, p. 714), and ''The Examiner'' (no. 2165, p. 480) for Saturday 28 July 1849 (available via HathiTrust, Internet Archive, and sometimes elsewhere), as well as various newspapers in England and Scotland through 4 August<!-- e.g. the Exeter and Plymouth Gazette -->, contain an advertisement stating that "On Monday will be published, [...] the August Number, [...] of ''Bentley's Miscellany''", followed by a list of contents indicating that this poem is "By the Author of 'The Caliph's Daughter.'"</ref>
On September 6, 1938, ''The New York Sun'' published a 16-line poem by Harold Willard Gleason titled «"Dark and Bloody Ground---" (''The Freshman's Dream'')» that bears some resemblance to this equation. It begins with "In minuends of Algebra / Wild corollaries twine;" and ends with "Or you shall factor cubes, for terms / Of infinite progression!" It mentions "binomial" and "parenthesis" and cautions to "Remove the brackets, radicals [...] with discretion". However, it has no context or explanation to confirm or refute whether it actually refers to this equation. This poem was reproduced by other periodicals over the following two months, including the ''National Mathematics Magazine'' published by the Mathematical Association of America (MAA).<ref>Manuscript in: {{Cite archive|collection=Harold Willard Gleason Papers|institution=Special Collections Research Center, Syracuse University|collection-url=https://library.syracuse.edu/digital/guides/g/gleason_hw.htm|item=Typescript poems 1938–1943|box=2}}<p>First publication: {{Cite news |last=Gleason |first=Harold Willard |date=September 6, 1938 |title="Dark and Bloody Ground---" (''The Freshman's Dream'') |newspaper=The Sun |publication-place=New York, N.Y. |volume=CVI |issue=4}}</p><p>Reproduced in: <!-- Title and author would be redundant to display, so they are left out, and the inevitable error messages are hidden. --></p>{{#invoke:String|replace|pattern=<span class="cs1-visible-error citation-comment">|replace=<span style="display: none;">|source=<ul><li>{{Cite news |no-tracking=y |date=September 9, 1938 |newspaper=The Ottawa Journal |publication-place=Ottawa, Ontario, Canada |volume=LIII |issue=229 |page=6 |id=Newspapers.com [https://www.newspapers.com/image/45993022/?match=1&terms=%22freshman%27s%20dream%22%20algebra 45993022]. }}</li><li>{{Cite news |no-tracking=y |date=September 9, 1938 |newspaper=Waterbury Democrat |publication-place=Waterbury, Connecticut, United States |page=8 |id=NewspaperArchive [https://newspaperarchive.com/waterbury-democrat-sep-09-1938-p-8/ waterbury-democrat-sep-09-1938-p-8] ([https://access.newspaperarchive.com/us/connecticut/waterbury/waterbury-democrat/1938/09-09/page-8 library access]). }}</li><li>{{Cite news |no-tracking=y |date=September 9, 1938 |newspaper=The Bangor Daily News |publication-place=Bangor, Maine, United States |page=14 |id=Newspapers.com [https://www.newspapers.com/image-view/663268201/?match=1&terms=%22wild%20corollaries%20twine%22 663268201]. }}</li><li>{{Cite news |no-tracking=y |date=September 15, 1938 |newspaper=The Moncton Transcript |publication-place=Moncton, Moncton Parish, New Brunswick, Canada |volume=LVII |issue=92 |page=[4] |id=Newspapers.com [https://www.newspapers.com/image/1104615164/?terms=%22freshman%27s%20dream%22%20algebra&match=1 1104615164]. }}</li><li>{{Cite news |no-tracking=y |date=September 29, 1938 |newspaper=The Boston (Daily) Globe |publication-place=Boston, Massachusetts, United States |volume=CXXXIV |issue=91 |page=28 |id=Newspapers.com [https://www.newspapers.com/image/431880838/?match=1&terms=%22freshman%27s%20dream%22%20algebra 431880838], [https://www.newspapers.com/image/431883159/?match=1&terms=%22freshman%27s%20dream%22%20algebra 431883159]. {{ProQuest|847896830}}. }}</li><li>{{Cite news |no-tracking=y |date=October 3, 1938 |newspaper=The Reporter |publication-place=Lansdale, Pennsylvania, United States |page=3 |id=Newspapers.com [https://www.newspapers.com/image-view/992183188/?match=1&terms=%22wild%20corollaries%20twine%22 992183188]. }}</li><li>{{Cite journal |no-tracking=y |date=October 1938 |journal=National Mathematics Magazine |volume=13 |issue=1 |page=50 |issn=1539-5588 |jstor=3028376 }}</li></ul>}}</ref>
On December 30, 1939, Saunders Mac Lane delivered an address to the MAA in Columbus, Ohio, wherein he explained the theorem for fields of prime characteristic, then stated that "As S. C. Kleene has remarked, a knowledge of the case ''p''=2 of this equation would corrupt freshman students of algebra!"<ref name="Mac Lane 1940 AMM">{{Cite journal |last=Mac Lane |first=Saunders |date=May 1940 |orig-date=An address delivered before the Mathematical Association of America at Columbus, Ohio, December 30, 1939. |title=Modular Fields |journal=The American Mathematical Monthly |volume=47 |issue=5 |pages=259–274 |doi=10.2307/2302685 |issn=0002-9890 |jstor=2302685}}</ref> This may be the first connection between "freshman" and binomial expansion in fields of positive characteristic.<ref name="Fletcher 1978 MG" /> Since then, authors of undergraduate algebra texts took note of the common error.
In 1974, in a textbook about algebra for graduate students, Thomas W. Hungerford published an exercise with a title of "The Freshman's Dream" with a footnote stating, "Terminology due to [[Vincent O. McBrien|V[incent] O. McBrien]]."<ref name="Hungerford 1974 HRW">{{Cite book |last=Hungerford |first=Thomas W. |title=Algebra |date=1974 |publisher=Holt, Rinehart and Winston |isbn=978-0-03-086078-2 |page=121 (see also pp. ix, 498 for full name) |language=en |id=Google Books [https://www.google.com/books/edition/Algebra/KvruAAAAMAAJ?gbpv=1&bsq=McBrien KvruAAAAMAAJ].}} (See alternatively softcover reprint: {{Cite book |last=Hungerford |first=Thomas W. <!-- Cleveland State University --> |url=https://link.springer.com/book/10.1007/978-1-4612-6101-8 |title=Algebra |date=2012 <!-- Google Books says 2012-12-06 --> |publisher=Springer |isbn=978-1-4612-6103-2 |series=Graduate Texts in Mathematics (GTM), vol. 73 |page= |pages=xiv, 121, 498 |language=en |doi=10.1007/978-1-4612-6101-8 |lccn=73-15693 |id=SPIN 11013129. Google Books [https://www.google.com/books/edition/Algebra/e-YlBQAAQBAJ?gbpv=1&pg=PA121 e-YlBQAAQBAJ].}}) (Also in: {{Cite book |title=Abstract Algebra: An Introduction |date=July 12, 1996 |publisher=Brooks Cole |edition=2nd |page=366}})</ref> ==See also== *Pons asinorum *Primality test *Sophomore's dream *Frobenius endomorphism
==References== {{reflist|2}}
Category:Algebra education Category:Mathematical fallacies Category:Theorems in ring theory Category:Prime numbers