Contents
- Affix
- Argand (or Gauss) plane
- Biquadratic
- Blackboard bold
- Coffin Problems
- Cossic art, the
- Double factorial
- Omnific Integer
- Porism
- Quartic
- Surds
- Surreal Number
- Vinculum
- ZFC, or Zermelo-Fraenkel set theory
Affix
“Affix — from Wolfram MathWorld”. 2023. mathworld.wolfram.com. https://mathworld.wolfram.com/Affix.html.
“Affix of a complex number”. 2023. Encyclopedia of Mathematics. https://encyclopediaofmath.org/wiki/Affix_of_a_complex_number.
Argand (or Gauss) plane
“Argand Diagram — From Wolfram Mathworld”. 2023. mathworld.wolfram.com. https://mathworld.wolfram.com/ArgandDiagram.html.
“Lesson Explainer: Argand Diagram”. 2023. nagwa. https://www.nagwa.com/en/explainers/280109891548/.
“Polar Representation of Complex Number on a Argand Plane”. 2023. BYJUS. https://byjus.com/maths/argand-plane-and-polar-representation-of-complex-number/.
Biquadratic
⭐ “Biquadratic Equation”. 2023. planetmath.org. https://planetmath.org/biquadraticequation.
“Biquadratic Equation — from Wolfram MathWorld”. 2023. mathworld.wolfram.com. https://mathworld.wolfram.com/BiquadraticEquation.html.
“How to Solve Biquadratic Equations?”. 2023. unacademy. https://unacademy.com/content/upsc/study-material/mathematics/how-to-solve-biquadratic-equations/.
Blackboard bold
“Blackboard Bold – Wikipedia”. 2021. en.wikipedia.org. https://en.wikipedia.org/wiki/Blackboard_bold.
The table on this page shows all available Unicode blackboard bold characters. The first column shows the letter as typically rendered by the LaTeX markup system. The second column shows the Unicode code point. The third column shows the Unicode symbol itself (which will only display correctly on browsers that support Unicode and have access to a suitable font). The fourth column describes some typical usage in mathematical texts. Some of the symbols (particularly ℂ [complex], ℚ [rationals], ℝ [real] and ℤ [integers]) are nearly universal in their interpretation, while others are more varied in use.
“Doublestruck — From Wolfram MathWorld”. 2023. mathworld.wolfram.com. https://mathworld.wolfram.com/Doublestruck.html.
“Math Symbols | Blackboard Bold | OMC Math Blog”. 2021. Online Math Center. https://onlinemathcenter.com/blog/math/math-symbols-and-their-meanings-part-vi-blackboard-bold/.
Remembering all the mathematical symbols and operations can be difficult, especially when they double. At OMC, we created a fundamental Math Symbols and Their Meanings series to help you memorize the symbols.
Coffin Problems
⭐ Amit, Alon. “What Are Coffin Problems In Mathematics?”. 2023. Quora. https://qr.ae/pyooK7.
“Coffins”. 2023. tanyakhovanova.com. http://www.tanyakhovanova.com/coffins.html.
In the summer of 1975, while I was in a Soviet math camp preparing to compete in the International Math Olympiad on behalf of the Soviet Union, my fellow team members and I were approached for help by Valera Senderov, a math teacher in one of the best Moscow special math schools.
The Mathematics Department of Moscow State University, the most prestigious mathematics school in Russia, had at that time been actively trying to keep Jewish students (and other “undesirables”) from enrolling in the department. One of the methods they used for doing this was to give the unwanted students a different set of problems on their oral exam. These problems were carefully designed to have elementary solutions (so that the Department could avoid scandals) that were nearly impossible to find. Any student who failed to answer could be easily rejected, so this system was an effective method of controlling admissions. These kinds of math problems were informally referred to as “coffins”. “Coffins” is the literal translation from Russian; in English these problems are sometimes called “killer” problems.
These problems and their solutions were, of course, kept secret, but Valera Senderov and his friends had managed to collect a list. In 1975, they approached us to solve these problems, so that they could train the Jewish students in these mathematical ideas. We solved some of them. Here I present some of the “coffin” problems from my archive.
Cossic art, the
“Cossic; Cossical Meaning – Online Dictionary – Definitions Synonyms Words.” 2023. cubepost.red. Accessed October 16. https://cubepost.red/learn/app/online-dictionary/word/cossic;%20cossical.
KATZ, VICTOR J., and Bill Barton. “STAGES IN THE HISTORY OF ALGEBRA WITH IMPLICATIONS FOR TEACHING.” Educational Studies in Mathematics 66, no. 2 (2007): 185–201. http://www.jstor.org/stable/27822699.
Kerry, John. “A Treatise of the Elements of the Algebraical Art.” 2023. Maths History. Accessed October 16. https://mathshistory.st-andrews.ac.uk/Extras/Algebraical_art/.
Let us clarify the word Cossic which Kersey sometimes uses. A ‘thing’ in Latin is Causa, in Italian is Cosa and in German is Coss, and the ‘thing’ became the word for the ‘unknown’. Those trying to find the ‘unknown’ were known as Cossists in early times and algebra was often called the cossic art. Let us also note that, of course, Renates des Cartes is René Descartes.
“The History of Algebra: Al-Khwarizmi.” Developmental Stages Of Algebra. 2023. CUEMATH. Accessed October 16. https://www.cuemath.com/learn/mathematics/algebra-history-of-algebra/#P004.
Double factorial
“Double Factorial.” 2022. GeeksforGeeks. GeeksforGeeks. July 11. https://www.geeksforgeeks.org/double-factorial/.
“Double Factorial.” 2023. From Wolfram MathWorld. Accessed December 22. https://mathworld.wolfram.com/DoubleFactorial.html.
“Double Factorials and Multifactorials.” 2023. Brilliant Math & Science Wiki. Accessed December 22. https://brilliant.org/wiki/double-factorials-and-multifactorials/.
Omnific Integer
Joyce, David. “How do you show that √ω is an Omnific Integer?”. 2023. Quora. https://qr.ae/pyl0I5.
Omnific integers are certain kinds of surreal numbers that correspond to the usual integers for real numbers. In fact, the only real numbers that are also omnific integers are the usual integers.
A surreal number x is an omnific integer when it is the case that
x = { x−1 ∣ x+1 },
that is to say, it’s the simplest surreal number between one less than itself and one greater than itself.
⭐ Lang, Jonathan. “Why is ω√2 an Omnific Integer?”. 2023. Quora. https://qr.ae/pyl0kv.
Omnific integers are a “subset” of the surreal numbers, which in turn have the ordinal numbers as a backbone. So it helps to have an understanding of the ordinal numbers. Ordinal numbers are an extension of the natural numbers. You start with 0 as the first ordinal and use what’s known as a finite recursion rule to say that every ordinal has a next ordinal: the next ordinal after 0 is 1; the next ordinal after 1 is 2, and so on. In this way, you generate all of the natural numbers. In effect, each use of finite recursion is a step forward through the natural numbers. The ordinal numbers extend beyond the natural numbers by introducing a second rule, of transfinite recursion. With transfinite recursion, you take an unending series of ordinals and “leap past” them to define a limit ordinal which serves as a “least upper bound” of that series: the smallest ordinal that’s larger than every ordinal in the series. For the natural numbers, this is ω, the first transfinite ordinal.
Porism
“Porism – Wikipedia”. 2021. en.wikipedia.org. https://en.wikipedia.org/wiki/Porism.
A porism is a mathematical proposition or corollary. It has been used to refer to a direct consequence of a proof, analogous to how a corollary refers to a direct consequence of a theorem. In modern usage, it is a relationship that holds for an infinite range of values but only if a certain condition is assumed, such as Steiner’s porism. The term originates from three books of Euclid that have been lost. A proposition may not have been proven, so a porism may not be a theorem or true.
“Porism — From Wolfram MathWorld”. 2023. mathworld.wolfram.com. https://mathworld.wolfram.com/Porism.html.
The term “porism” is an archaic type of mathematical proposition whose historical purpose is not entirely known. It is used instead of “theorem” by some authors for a small number of results for historical reasons. However, two meanings predominate in nonhistorical usage. The first is “corollary,” a usage now mostly superseded by that term itself. The second (which may now be considered the “modern” usage) is, “A proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate, or capable of innumerable solutions” (Playfair 1792). Unfortunately, this definition is slightly inaccurate, because the proposition actually states the conditions, rather than affirming the possibility of finding them.
Quartic
Peterson, Dave. 2023. “Factoring a Quartic Polynomial – The Math Doctors”. themathdoctors.org. https://www.themathdoctors.org/factoring-a-quartic-polynomial/.
“Quartic Equation – Art Of Problem Solving”. 2023. artofproblemsolving.com. https://artofproblemsolving.com/wiki/index.php/Quartic_Equation.
“Quartic Equation — from Wolfram MathWorld”. 2023. mathworld.wolfram.com. https://mathworld.wolfram.com/QuarticEquation.html.
Tavora, Marco. “Solving Cubic and Quartic Equations”. 2020. Medium. https://towardsdatascience.com/solving-cubic-and-quartic-equations-f024d4f9e37d.
In this math tutorial, we’ll show you how to tackle quartic polynomials with confidence using the Double-Cross Factoring Method. This powerful technique involves factoring out the greatest common factor, grouping terms, and using a double-cross diagram to simplify and solve complex equations. With our easy-to-follow guide and step-by-step instructions, you’ll be able to approach even the most challenging quartic equations with ease. So whether you’re a math student or professional, this tutorial is a must-watch for anyone looking to improve their problem-solving skills.
Surds
“Surds – Mathematics GCSE Revision”. 2023. revisionmaths.com. https://revisionmaths.com/gcse-maths-revision/number/surds.
“Surds (Maths): Definition, Examples & Rules | StudySmarter”. 2023. StudySmarter US. https://www.studysmarter.us/explanations/math/pure-maths/surds/.
⭐ “Surds – Definition, Types, Rules, And Problems”. 2023. BYJUS. https://byjus.com/maths/surds/.
⭐ “Surds And Indices – Definition, Types, Rules, And Practice Problems”. 2023. CUEMATH. https://www.cuemath.com/numbers/surds/.
Surreal Number
⭐ Simons, Jim. “MEET THE SURREAL NUMBERS”. 2017. m-a.org.uk. https://www.m-a.org.uk/resources/downloads/4H-Jim-Simons-Meet-the-surreal-numbers.pdf.
G. H. Hardy wrote in A Mathematician’s Apology, “A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent that theirs, it is because they are made with ideas.” And, I would add, because they are less culturally dependent. I heard a radio play long ago, possibly around the time of the Cuban missile crisis in 1962, about a world in which many millions of years after a nuclear war wiped out humanity, a species of intelligent lizard evolved, whose archeologists eventually unearthed remains of human civilisation. Would the lizards appreciate Mozart, Rembrandt or Shakespeare? Probably not. But would the mathematicians amongst them appreciate Conway’s surreal numbers – yes I’m pretty sure they would, unless of course they had already discovered them themselves, in which case they would be very remarkable lizards indeed. Hardy also wrote that “The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful . . . There is no permanent place in the world for ugly mathematics.” The surreal numbers must be amongst the most beautiful patterns that mankind has yet produced, and that is why everyone who possibly can should study them. The Curious Mind of John Horton Conway tells us what people have said about the surreal numbers. Conway himself said “I walked around for about six weeks after discovering the surreal numbers in a sort of permanent daydream, in danger of being run over.” This sense of reverie overtakes others who study them. Martin Kruskal, a mathematician of wide-ranging achievements, spent some of his later years studying the surreal numbers, and he wrote “The usual numbers are very familiar, but at root they have a very complicated structure. Surreals are in every logical, mathematical and aesthetic sense better.” Of the quite magical way in which the numbers are created, Martin Gardner wrote “An empty hat rests on a table made of a few axioms of standard set theory. Conway waves two simple rules in the air, then reaches into almost nothing and pulls out an infinitely rich tapestry of numbers.”
“Surreal Number”. 2023. archive.lib.msu.edu. https://archive.lib.msu.edu/crcmath/math/math/s/s883.htm.
The most natural collection of numbers which includes both the real numbers and the infinite Ordinal Numbers of Georg Cantor. They were invented by John H. Conway in 1969. Every Real Number is surrounded by surreals, which are closer to it than any Real Number.
⭐ “Surreal Number”. 2023. planetmath.org. https://planetmath.org/surrealnumber.
“Surreal Number — From Wolfram MathWorld”. 2023. mathworld.wolfram.com. https://mathworld.wolfram.com/SurrealNumber.html.
Vinculum
Joyce, David. “What on Earth Is ‘√’?” 2023. Quora. https://qr.ae/pKRq8I.
It’s a radical sign. It’s made out of two symbols run together. The first is a stylized letter r. The second is a horizontal line called a vinculum.
Sheehy, Sara & Ryan. 2023. “About.” Math Mum – Enrich Your Child’s Math Ability. April 29. https://mathmum.com/vinculum-in-math-definition-and-examples/.
The vinculum is most commonly used to denote a radical expression, repeating decimals, a unit, and in other contexts such as to denote the line segment joining two points, to indicate the complex conjugate, or to negate a logical expression.
“Vinculum (Symbol).” 2023. Wikipedia. Wikimedia Foundation. https://en.wikipedia.org/wiki/Vinculum_(symbol).
ZFC, or Zermelo-Fraenkel set theory
See the Zermelo-Fraenkel Set Theory (ZFC) web page.
Clapham, Christopher, and James Nicholson. 2023. “The Concise Oxford Dictionary Of Mathematics”. Oxford University Press. https://www.oxfordreference.com/display/10.1093/acref/9780199235940.001.0001/acref-9780199235940;jsessionid=44F8961D71ECBCF233D1594FABBF6518.
“Illustrated Mathematics Dictionary”. 2023. mathsisfun.com. https://www.mathsisfun.com/definitions/.
“Mathwords”. 2023. mathwords.com. https://www.mathwords.com/.
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