How to Explain Number Theory at a Dinner Party

2017 ◽  
pp. 175-180
Keyword(s):  
Number Theory ◽  
Dinner Party

Mathematics without Apologies

2017 ◽  
Author(s):  
Michael Harris
Keyword(s):  
Number Theory ◽  
Financial Crisis ◽  
Pop Culture ◽  
First Century ◽  
2008 Financial Crisis ◽  
Dinner Party ◽  
Mathematical Life ◽  
Personal Experiences ◽  
The 2008 Financial Crisis ◽  
Guided Tour

What do pure mathematicians do, and why do they do it? Looking beyond the conventional answers, this book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twenty-first century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources. Drawing on the author's personal experiences as well as the thoughts and opinions of mathematicians from Archimedes and Omar Khayyám to such contemporary giants as Alexander Grothendieck and Robert Langlands, the book reveals the charisma and romance of mathematics as well as its darker side. In this portrait of mathematics as a community united around a set of common intellectual, ethical, and existential challenges, the book touches on a wide variety of questions, such as: Are mathematicians to blame for the 2008 financial crisis? How can we talk about the ideas we were born too soon to understand? And how should you react if you are asked to explain number theory at a dinner party? The book takes readers on an unapologetic guided tour of the mathematical life, from the philosophy and sociology of mathematics to its reflections in film and popular music, with detours through the mathematical and mystical traditions of Russia, India, medieval Islam, the Bronx, and beyond.

How to Explain Number Theory at a Dinner Party

2017 ◽  
Author(s):  
Michael Harris
Keyword(s):  
Number Theory ◽  
Dinner Party ◽  
First Session

This chapter begins with the author's account of a dinner party that he attended in the spring of 2008. One of the guests, an actress, had turned to him and asked, “What is it you do in number theory, anyway?” His response to the actress and the conversation that ensued lasted at most ten minutes. However, he has always felt that he could have done better. The chapter then presents what the author feels should have been his answer. The discussion has been broken up into short sessions in order to impede the accumulation of formulas, as the author works his way up to the guiding problem of the Birch–Swinnerton–Dyer conjecture. This chapter contains the first session.

How to Explain Number Theory at a Dinner Party

2017 ◽  
Author(s):  
Michael Harris
Keyword(s):  
Number Theory ◽  
Elliptic Curves ◽  
Approximate Solutions ◽  
Final Part ◽  
Dinner Party ◽  
Dollar Price ◽  
Bsd Conjecture

This chapter continues the discussion began in Chapter α‎. It presents the fourth and final part of author's response to the question, “What is it you do in number theory, anyway?” Here he deals with the Birch–Swinnerton–Dyer (BSD) conjecture. The BSD conjecture was the guiding problem for the first part of his career and has the distinction of being one of the seven Clay Millenium Prize Problems, whose solution carries a million dollar price tag. The BSD conjecture is an attempt to discern order in the apparently unpredictable nature of solutions to elliptic curves, following Hasse's theorem, which places strict limits on the numbers of approximate solutions for varying p.

How to Explain Number Theory at a Dinner Party

2017 ◽  
Author(s):  
Michael Harris
Keyword(s):  
Number Theory ◽  
Dinner Party ◽  
The Third

This chapter continues the discussion began in Chapter α‎. It presents the third part of author's response to the question, “What is it you do in number theory, anyway?” Working his way up to the guiding problem of the Birch–Swinnerton–Dyer conjecture, here he deals with congruences, specifically how to count solutions to congruences. The story of congruences is that a problem where the variables can take infinitely many values can be replaced by one in which the variables can take only finitely many values, and it is sometimes enough to solve the latter problem in order to solve the former.

How to Explain Number Theory at a Dinner Party

2017 ◽  
Author(s):  
Michael Harris
Keyword(s):  
Number Theory ◽  
Mathematical Theory ◽  
Transcendental Number ◽  
Dinner Party ◽  
Transcendental Numbers

This chapter discusses transcendental numbers. The first transcendental number was exhibited by Joseph Liouville in 1844; it was a number λ‎ he concocted for just that purpose, but it answers the question,“Are there any transcendental numbers? ”, and therefore certainly qualifies as a number. The statement that Liouville “exhibited” the number means that he wrote down a description and used this description to show that there is no polynomial f for which f(λ‎) = 0. If all transcendental numbers were as artificial as Liouville's λ‎, transcendence would never have grown into a mathematical theory. Methods were soon developed, however, to show that some familiar numbers are transcendental.

How to Explain Number Theory at a Dinner Party

2017 ◽  
Author(s):  
Michael Harris
Keyword(s):  
Number Theory ◽  
Quadratic Equation ◽  
Dinner Party ◽  
Square Roots ◽  
Renaissance Italy

This chapter continues the discussion began in Chapter α‎. It presents the second part of author's response to the question, “What is it you do in number theory, anyway?” Working his way up to the guiding problem of the Birch–Swinnerton–Dyer conjecture, here he deals with equations of degree 3 (or 4) in one variable or degree 2 in two variables. He says that if we are willing to allow square roots into our arithmetic, we can consider the quadratic equation a problem whose solution has been long understood (in some cases by the ancient Babylonians). Equations of degree 3 and 4, such as x3 − 2x2 + 14x + 9 and x4 + 5x3 + 11x2 + 17x − 29, were first solved in Renaissance Italy to great acclaim; the solutions are given by formulas involving cube roots and fourth roots.

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