How do I Solve this Equation? Look at the Symmetries! – The Idea behind Galois Theory

Originating author is Timo Leuders.

There are some questions that accompany the development of mathematics through cultures and ages. One of these questions is how to find an unknown quantity x of which one knows some relations such as – in today’s algebraic notation:

    \[x^2 =x+5\]

Finding solutions to such quadratic equations are essentially known since Babylonian
times and are core content school mathematics:

    \[x^2-x-5=0 \, \Rightarrow \, x=\frac{1}{2}+\frac{1}{2} \sqrt{21} \, \vee \, x=\frac{1}{2}-\frac{1}{2} \sqrt{21}\]

But how about x^5 = x + 5 , which looks only slightly different? Are there also straightforward ways to calculate the solutions? Do the solutions also look symmetric in a similar way?

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Mathematical Impressions

SIMONSJune’s Site of the Month is: Mathematical Impressions

A series of video presentations of mathematical phenomena, and discussion of their properties.
These videos by geometer George Hart are part of the Simons Foundation scientific outreach. George Hart’s work can be seen in other places (e.g. the American Mathematical Society website).

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On-Line Encyclopaedia of Integer Sequences

OEISMay’s Site of the Month is the On-Line Encyclopaedia of Integer Sequences.
The wiki-page is available in over 50 languages.

This site of over a quarter of a million sequences allows you to enter the first integers of any sequence and search for information on that sequence. It has links to famous, puzzle, classic and “hot” sequences, and is very fully referenced. There is a movie of the first 1000 terms of 1000 sequences. The site also invites you to contribute your own sequence.

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Metamagical Themas

matemagicalMay’s Book of the Month is Metamagical Themas by Douglas Hofstadter. First published in 1985 by Basic Books, available in hardback and paperback. Also available in French under the title Ma Thémagie (InterEditions, 1988).

Our second ever Book of the Month was Martin Gardner’s Mathematical Puzzles and Diversions — that author’s collected columns from Scientific American. Hofstadter followed Gardner writing this column, and the current month’s book is also a collection of those columns. Hofstadter also wrote Gödel, Escher, Bach which is another featured Book of the Month.

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TEDEd Lessons Worth Sharing

TEDEd2April’s Site of the Month is: TEDEd Lessons Worth Sharing

TEDEd Lessons are short (~4mins) video clips, and the ones on mathematics include several riddles, plus unusual uses of mathematics. There are also some conventional clips such as “The Secrets of Pascals Triangle” (which does acknowledge prior appearances in other cultures) and “The Complex Geometry of Islamic Design”. If you have an idea for a mathematical TEDEd Lesson then you can get assistance making the animated video.

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How To Lie With Statistics

StatiApril’s Book of the Month is How To Lie With Statistics by Darrell Huff. Norton reissued this 1954 book in 1993, and this edition is available on Kindle. There is a 1991 Penguin edition and it has been translated into many languages.

This book may be over 60 years old, (and therefore the examples are dated), and statistics has certainly moved on since it was first published, but the volume remains a best-seller and is a humorously illustrated light introduction to many key ideas in statistics. It informs us about statistics through showing how statistical analyses can be used to fool and create misunderstanding.

For other translations look here: Indonesian, Thai, Czech, Turkish, Greek

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The Exact Sciences in Antiquity

exactSciencesMarch’s Book of the Month is: The Exact Sciences in Antiquity by Otto Neugebauer. Second edition published in 1957 by Brown University Press, now republished by Dover in 1969.

Commentary adapted from the Amazon website, see this link for further information: Based on a series of lectures to non-experts 1949, this is the standard non-technical coverage of Egyptian and Babylonian mathematics and astronomy. It reveals that the Babylonian strength in algebraic and numerical work is comparable to the mathematics of the early Renaissance in Europe. In the realm of astronomy, it describes a sophistication which is interpreted less as the result of millennia of observations (as used to be the interpretation) than as a competent mathematical system. An Appendix discusses certain aspects of Greek astronomy and the indebtedness of the Copernican system to Ptolemaic and Islamic methods.

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culturMATHMarch’s Site of the Month is: CultureMATH

CultureMATH is a French language website dedicated to secondary mathematics teacher education. Published articles present mathematical ideas at several levels. The topics are often related to the curriculum, to present them more deeply. Other resources are also available: some pedagogical experiences, examples of problem solving (in order to present the diversity of procedures), and a series of puzzles.

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Mathematical Etudes

01_301_2February’s Site of the Month: Mathematical Etudes

This is a Russian site, but has an English and a French version, use “en” or “fr” at the end of the URL.

The site contains several 3D animated films that illustrate theorems or ideas in mathematics, and much more. For example, in the Apps for iPhones there are 90 cryptarithms which are available in English, French, German, Italian, Russian, and Spanish.

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The Story of a Number

storyofanumberFebruary’s Book of the Month is: The Story of a Number by Eli Manor. First published in 1993 and available in paperback in the Princeton Science Library.

From the “Goodreads” website. See here for more details.
“…. Geared to the reader with only a modest background in mathematics, the book describes the story of e from a human as well as a mathematical perspective. In a sense, it is the story of an entire period in the history of mathematics, from the early seventeenth to the late nineteenth century, with the invention of calculus at its center. Many of the players who took part in this story are here brought to life. …. The unifying theme throughout the book is the idea that a single number can tie together so many different aspects of mathematics …. The book ends with an account of the discovery of transcendental numbers ….”

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