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Mathematics

Education & Academic

In Defense of Infinity

The Glencoe Algebra II study materials (p. 10) make an amazing claim (Reddit). This statement is in a math textbook, but it is horrifyingly wrong. A statement like "the letters A--Z cannot be matched up with the numbers 1--26" would be similarly wrong. These two sets of the same size (here, 26) can be matched up as A1, B2, C3, ..., Z26. Can the rational numbers be matched up with the integers? Both are infinite, which allows for the tricks of a technique called Hilbert's hotel, a hotel with infinite numbered rooms that can always make room for one more guest. The Glencoe claim asks if the cardinality of the integers and rationals is the same. Both are , or Aleph-0, which Georg Cantor proved in the 1870s.
Ed Pegg Jr, Scientific Information Editor, Scientific Information Group
Education & Academic

New Publications Using Wolfram Technologies

We're always on the lookout for new ideas and ways of using the Wolfram Language that our users produce and choose to write about in their books. In this quarter, we have applications that bridge the gap between art and geometry, and demonstrate intuitive data analysis. In addition to writing books, Wolfram welcomes authors to submit articles for publication in The Mathematica Journal, our very own in-house periodical.
Jenna Giuffrida, Content Administrator, Technical Communications and Strategy Group
Best of Blog

2 Pi or Not 2 Pi?

Three months ago the world (or at least the geek world) celebrated Pi Day of the Century (3/14/15...). Today (6/28) is another math day: 2π-day, or Tau Day (2π = 6.28319...). Some say that Tau Day is really the day to celebrate, and that τ(=2π) should be the most prominent constant, not π. It all started in 2001 with the famous opening line of a watershed essay by Bob Palais, a mathematician at the University of Utah: "I know it will be called blasphemy by some, but I believe that π is wrong." Which has given rise in some circles to the celebration of Tau Day—or, as many people say, the one day on which you are allowed to eat two pies. But is it true that τ is the better constant? In today's world, it's quite easy to test, and the Wolfram Language makes this task much simpler. (Indeed, Michael Trott's recent blog post on dates in pi—itself inspired by Stephen Wolfram's Pi Day of the Century post—made much use of the Wolfram Language.) I started by looking at 320,000 preprints from arXiv.org to see in practice how many formulas involve 2π rather than π alone, or other multiples of π. Here is a WordCloud of some formulas containing 2π:
Giorgia Fortuna, Advanced Research Group
Education & Academic

Dates Everywhere in Pi(e)! Some Statistical and Numerological Musings about the Occurrences of Dates in the Digits of Pi

In a recent blog post, Stephen Wolfram discussed the unique position of this year's Pi Day of the Century and gave various examples of the occurrences of dates in the (decimal) digits of pi. In this post, I'll look at the statistics of the distribution of all possible dates/birthdays from the last 100 years within the (first ten million decimal) digits of pi. We will find that 99.998% of all digits occur in a date, and that one finds millions of dates within the first ten million digits of pi. Here I will concentrate on dates than can be described with a maximum of six digits. This means I'll be able to uniquely encode all dates between Saturday, March 14, 2015, and Sunday, March 15, 1915—a time range of 36,525 days.
Michael Trott, Chief Scientist, Wolfram|Alpha Scientific Content
Education & Academic

John F. Nash, Jr. , In Memoriam

This past week, on May 23, 2015, the much loved and respected John F. Nash Jr., along with his wife, Alicia Nash, passed away in a tragic car accident while returning home from his receipt of the 2015 Abel Prize for his work in partial differential equations. The Nobel winner and his wife were the subject of the 2001 Academy Award winning film A Beautiful Mind. Nash's most famous contribution to mathematics and economics was in the field of game theory, which has enabled others to build on that work and was the focus of the film. Nash's long career as a mathematician was marked by both brilliant achievements and terrible struggles with mental illness. Despite his battle with schizophrenia, Nash inspired generations of mathematicians and garnered a stunning array of awards, including the 1994 Nobel Prize in economic sciences, the American Mathematical Society's 1999 Leroy P. Steele Prize for Seminal Contribution to Research, and the 1978 John von Neumann Theory Prize. We were personally honored in 2003 when Nash presented his work with Mathematica at the International Mathematica Symposium in London.
Wolfram Blog Team
Education & Academic

Biggest Little Polyhedron—New Solutions in Combinatorial Geometry

In many areas of mathematics, 1 is the answer. Squaring a number above or below 1 results in a new number that is larger or smaller. Sometimes, determining whether something is "big" is based on whether a largest dimension is greater than 1. For instance, with sides of length 13,800 km, Saturn's hexagon would be considered big. A "little polygon" is defined as a polygon where 1 is the maximum distance between vertices. In 1975, Ron Graham found the biggest little hexagon, which has more area than the regular hexagon, as shown below. The red diagonals have length 1. All other diagonals (not drawn) are smaller than 1.
Ed Pegg Jr, Scientific Information Editor, Scientific Information Group
Education & Academic

Why Alan Turing Has Already Won, No Matter How The Imitation Game Does at the Oscars

When I was invited to join the Turing Centenary Advisory Committee in 2008 by Professor Barry Cooper to prepare for the Alan Turing Year in 2012, I would have never imagined that just a few years later, Turing's life and work would have gained sufficient public attention to become the subject of a Hollywood-style feature film, nor that said movie would go on to earn eight Oscar nominations.
Hector Zenil, Editor of the Complex Systems journal
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