# Freshman's dream

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{{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](/source/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](/source/child)'s binomial [theorem](/source/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](/source/student) 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](/source/equation) (''x''&nbsp;+&nbsp;''y'')<sup>''n''</sup>&nbsp;=&nbsp;''x''<sup>''n''</sup>&nbsp;+&nbsp;''y''<sup>''n''</sup>. Beginning students commonly make this error in computing the [power](/source/exponentiation) of a [sum](/source/summation) of [real number](/source/real_number)s, falsely assuming powers [distribute](/source/distributive_property) over sums.

The correct result is given by the [binomial theorem](/source/binomial_theorem),<ref name="Fraleigh 1994 AddisonWesley" /> which has additional [terms](/source/addition) in the middle when ''n''&nbsp;≥&nbsp;2.<ref name="Bastida 1984 CambridgeUP" /> For example, when ''n''&nbsp;=&nbsp;2, the correct result is ''x''<sup>2</sup>&nbsp;+&nbsp;2''xy''&nbsp;+&nbsp;''y''<sup>2</sup>, which can also be shown by [multiplying](/source/multiplication) (''x''&nbsp;+&nbsp;''y'')(''x''&nbsp;+&nbsp;''y'') by using the distributive property properly, or the [FOIL method](/source/FOIL_method).

The freshman's dream is actually valid in commutative [ring](/source/Ring_(mathematics))s of [characteristic](/source/characteristic_(algebra)) ''p'', such as the [finite field](/source/finite_field) <math>\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}</math>, where ''p'' is a [prime number](/source/prime_number), provided that the exponent ''n'' is ''p'' or more generally a power of ''p''.  Equivalently, the [Frobenius map](/source/Frobenius_endomorphism) of the ring is an [endomorphism](/source/endomorphism). One way to prove this is to show that ''p'' [divides](/source/divisor) all the [binomial coefficient](/source/binomial_coefficient)s except for the first and the last, so all the intermediate terms are equal to [zero](/source/0).<ref name="Bastida 1984 CambridgeUP" /><ref name="Fraleigh 1994 AddisonWesley" /> Another way to prove the common [special case](/source/special_case) of this for <math>\mathbb{F}_p</math> is to use [Fermat's little theorem](/source/Fermat's_little_theorem) that ''a<sup>p</sup>''&nbsp;[≡](/source/congruence_relation)&nbsp;''a''&nbsp;[mod](/source/modulo)&nbsp;''p'' for all integers ''a''.<ref name="Granville 2004 BullAMS" /> (This can be [iterated](/source/iteration) for powers of ''p'', using the [property of exponentiation](/source/exponentiation) that taking a power of a power multiplies the exponents, and thereby proven in general using [induction](/source/mathematical_induction).)

The freshman's dream is valid for all ''n'' in [tropical geometry](/source/tropical_geometry)<ref name="Maclagan 2011 ELGA" /><ref name="Kalisnik 2019 FoCM" /> (where multiplication is replaced with [addition](/source/addition), so exponentiation becomes multiplication, and addition is replaced with [minimum](/source/minimum_(mathematics))).

The freshman's dream equation is also true in some [degenerate](/source/degeneracy_(mathematics)) cases, such as when ''n''&nbsp;=&nbsp;1, when <math>n \ge 1</math> and at least one of ''x'' and ''y'' is zero, and when ''n'' is an [odd](/source/parity_(mathematics)) integer and <math>y=-x</math>. These are all of the true cases for ''n''&nbsp;∈&nbsp;{0, 1, 2, 3}, but when ''n''&nbsp;≥&nbsp;4 or ''n'' is [negative](/source/negative_number) or non-[integer](/source/integer), there are generally additional pairs of [complex number](/source/complex_number)s ''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](/source/commutative_ring) of [characteristic](/source/characteristic_(algebra)) <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](/source/numerator) is ''p'' [factorial](/source/factorial)(!), which is divisible by ''p''. However, when {{nowrap|0 < ''n'' < ''p''}}, both ''n''! and {{nowrap|(''p'' &minus; ''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](/source/endomorphism), known as the [Frobenius endomorphism](/source/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](/source/if_and_only_if) {{nowrap|1=(''x'' + 1)<sup>''n''</sup> = ''x<sup>n</sup>'' + 1}} in the [polynomial ring](/source/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](/source/terminology) "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](/source/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](/source/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](/source/Google_Books) [https://www.google.com/books/edition/Bentley_s_Miscellany/dt4RAAAAYAAJ?gbpv=1&dq=%22freshman's%20dream%22&pg=PA176 dt4RAAAAYAAJ]. [HathiTrust](/source/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](/source/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.](/source/The_Literary_Gazette)'' (no. 1697, p. 558), ''[The Athenæum](/source/The_Athenaeum_(British_magazine))'' (no. 1135, p. 754), ''[The Spectator](/source/The_Spectator)'' ([vol. 22], no. 1100, p. 714), and ''[The Examiner](/source/The_Examiner_(1808%E2%80%931886))'' (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](/source/The_Sun_(New_York_City))'' published a 16-line [poem](/source/poetry) by [Harold Willard Gleason](/source/Harold_Willard_Gleason) titled «"Dark and Bloody Ground---" (''The Freshman's Dream'')» that bears some resemblance to this equation. It begins with "In [minuends](/source/subtraction) of [Algebra](/source/Algebra) / Wild [corollaries](/source/corollary) twine;" and ends with "Or you shall [factor](/source/factorization) [cubes](/source/cube_(algebra)), for terms / Of [infinite](/source/infinity) progression!" It mentions "binomial" and "[parenthesis](/source/bracket)" and cautions to "Remove the brackets, [radicals](/source/nth_root) [...] with [discretion](/source/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](/source/periodical_literature) over the following two [month](/source/month)s, including the ''[National Mathematics Magazine](/source/Mathematics_Magazine)'' published by the [Mathematical Association of America](/source/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](/source/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](/source/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](/source/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](/source/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](/source/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](/source/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](/source/Saunders_Mac_Lane) delivered an [address](/source/public_speaking) to the MAA in [Columbus, Ohio](/source/Columbus%2C_Ohio), wherein he explained the theorem for fields of prime characteristic, then stated that "As [S. C. Kleene](/source/Stephen_Cole_Kleene) has remarked, a [knowledge](/source/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, [author](/source/author)s of [undergraduate](/source/undergraduate_education) algebra texts took note of the common error.

In 1974, in a [textbook](/source/textbook) about [algebra](/source/algebra) for [graduate students](/source/postgraduate_education), [Thomas W. Hungerford](/source/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](/source/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](/source/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](/source/Pons_asinorum)
*[Primality test](/source/Primality_test)
*[Sophomore's dream](/source/Sophomore's_dream)
*[Frobenius endomorphism](/source/Frobenius_endomorphism)

==References==
{{reflist|2}}

Category:Algebra education
Category:Mathematical fallacies
Category:Theorems in ring theory
Category:Prime numbers

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Adapted from the Wikipedia article [Freshman's dream](https://en.wikipedia.org/wiki/Freshman's_dream) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Freshman's_dream?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.