p-adic number

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3-adic integers with dual colorings

p-adic numbers come from an alternate way of defining the distance between two rational numbers.[1][2]

p-adic numbers are based in modular arithmetic, which is a method of counting that loops back on itself, like a clock.[3] 2018 Fields Medalist Peter Scholze is an expert in this area.[4]

History[change | change source]

The p-adic numbers were invented at the beginning of the twentieth century by the German mathematician Kurt Hensel. The aim was to make the methods of power series expansions, which play such a dominant role in the theory of functions, available to the theory of numbers as well.[5]

Application[change | change source]

The p-adic absolute value gives us a new way to measure the distance between two numbers. The p-adic distance between two numbers x and y is the p-adic absolute value of the number x-y.[6]

References[change | change source]

  1. Gouvêa, Fernando Q. (1997), Gouvêa, Fernando Q. (ed.), "p-adic Numbers", p-adic Numbers: An Introduction, Universitext, Berlin, Heidelberg: Springer, pp. 43–85, doi:10.1007/978-3-642-59058-0_4, ISBN 978-3-642-59058-0, S2CID 118340827, retrieved 2022-05-04
  2. Koblitz, Neal (1984). "p-adic Numbers, p-adic Analysis, and Zeta-Functions". Graduate Texts in Mathematics. 58. doi:10.1007/978-1-4612-1112-9. ISBN 978-1-4612-7014-0. ISSN 0072-5285.
  3. Houston-Edwards, Kelsey (2020-10-19). "An Infinite Universe of Number Systems". Quanta Magazine. Retrieved 2022-05-04.
  4. Lamb, Evelyn. "The Numbers behind a Fields Medalist's Math". Scientific American Blog Network. Retrieved 2022-05-04.
  5. Neukirch, J. (1991), Ebbinghaus, Heinz-Dieter; Hermes, Hans; Hirzebruch, Friedrich; Koecher, Max (eds.), "The p-Adic Numbers", Numbers, New York, NY: Springer, pp. 155–178, doi:10.1007/978-1-4612-1005-4_7, ISBN 978-1-4612-1005-4, retrieved 2022-08-14
  6. Lamb, Evelyn. "The Numbers behind a Fields Medalist's Math". Scientific American Blog Network. Retrieved 2022-08-14.