p-adic numbers – Quanta Magazine https://www.quantamagazine.org Illuminating science Mon, 16 Sep 2024 14:10:47 -0400 en-US hourly 1 https://wordpress.org/?v=6.2 A Question About a Rotating Line Helps Reveal What Makes Real Numbers Special https://www.quantamagazine.org/new-number-systems-point-geometry-problem-toward-a-real-solution-20220726/ https://www.quantamagazine.org/new-number-systems-point-geometry-problem-toward-a-real-solution-20220726/#respond Tue, 26 Jul 2022 14:05:04 +0000 https://www.quantamagazine.org/?p=117249 The post A Question About a Rotating Line Helps Reveal What Makes Real Numbers Special first appeared on Quanta Magazine

]]> The Kakeya conjecture sounds like a brain teaser. Place a needle flat on a table. How much area do you need in order to be able to turn it so that it points in all possible directions? The most obvious possible answer is a circle whose diameter is the length of the needle. But this is demonstrably wrong. And over the last century, the effort to understand the ways in which it’s wrong has revealed...

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]]> https://www.quantamagazine.org/new-number-systems-point-geometry-problem-toward-a-real-solution-20220726/feed/ 0 New Shape Opens ‘Wormhole’ Between Numbers and Geometry https://www.quantamagazine.org/with-a-new-shape-mathematicians-link-geometry-and-numbers-20210719/ https://www.quantamagazine.org/with-a-new-shape-mathematicians-link-geometry-and-numbers-20210719/#respond Mon, 19 Jul 2021 15:27:44 +0000 https://www.quantamagazine.org/?p=106834 The post New Shape Opens ‘Wormhole’ Between Numbers and Geometry first appeared on Quanta Magazine

]]> The grandest project in mathematics has received a rare gift, in the form of a mammoth 350-page paper posted in February that will change the way researchers around the world investigate some of the field’s deepest questions. The work fashions a new geometric object that fulfills a bold, once fanciful dream about the relationship between geometry and numbers. “This truly opens up a tremendous...

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]]> https://www.quantamagazine.org/with-a-new-shape-mathematicians-link-geometry-and-numbers-20210719/feed/ 0 Mathematicians Find Long-Sought Building Blocks for Special Polynomials https://www.quantamagazine.org/mathematicians-find-polynomial-building-blocks-hilbert-sought-20210525/ https://www.quantamagazine.org/mathematicians-find-polynomial-building-blocks-hilbert-sought-20210525/#respond Tue, 25 May 2021 15:05:58 +0000 https://www.quantamagazine.org/?p=104803 The post Mathematicians Find Long-Sought Building Blocks for Special Polynomials first appeared on Quanta Magazine

]]> Problems in mathematics often have a simple “yes or no” structure: Is this statement true or false? But the most enduring and interesting problems propagate through generations, the products of decades of work, like the medieval cathedrals that took centuries to build. The answers to these questions open new doors and provide novel structures on which to continue building. In the year 1900...

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]]> https://www.quantamagazine.org/mathematicians-find-polynomial-building-blocks-hilbert-sought-20210525/feed/ 0 An Infinite Universe of Number Systems https://www.quantamagazine.org/how-the-towering-p-adic-numbers-work-20201019/ https://www.quantamagazine.org/how-the-towering-p-adic-numbers-work-20201019/#respond Mon, 19 Oct 2020 14:05:36 +0000 https://www.quantamagazine.org/?p=94240 p-adics form an infinite collection of number systems based on prime numbers. They’re at the heart of modern number theory.

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]]> The rational numbers are the most familiar numbers: 1, -5, ½, and every other value that can be written as a ratio of positive or negative whole numbers. But they can still be hard to work with. The problem is they contain holes. If you zoom in on a sequence of rational numbers, you might approach a number that itself is not rational. This short-circuits a lot of basic mathematical tools...

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]]> https://www.quantamagazine.org/how-the-towering-p-adic-numbers-work-20201019/feed/ 0 A Master of Numbers and Shapes Who Is Rewriting Arithmetic https://www.quantamagazine.org/peter-scholze-becomes-one-of-the-youngest-fields-medalists-ever-20180801/ https://www.quantamagazine.org/peter-scholze-becomes-one-of-the-youngest-fields-medalists-ever-20180801/#respond Wed, 01 Aug 2018 12:55:30 +0000 https://www.quantamagazine.org/?p=61060 The post A Master of Numbers and Shapes Who Is Rewriting Arithmetic first appeared on Quanta Magazine

]]> In the run-up to the presentation of the Fields Medals — mathematics’ highest honor — the identity of the winners is always a closely guarded secret. But this time around, one name has been on everyone’s lips: Peter Scholze, of the University of Bonn in Germany. Today, those expectations proved justified. The 30-year-old Scholze was awarded the medal at the International Congress of Mathematicians...

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