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Summing it up : from one plus one to modern number theory

Author: Avner Ash; Robert Gross
Publisher: Princeton : Princeton University Press, 2016.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
We use addition on a daily basis--yet how many of us stop to truly consider the enormous and remarkable ramifications of this mathematical activity? Summing It Up uses addition as a springboard to present a fascinating and accessible look at numbers and number theory, and how we apply beautiful numerical properties to answer math problems. Mathematicians Avner Ash and Robert Gross explore addition's most basic  Read more...
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Genre/Form: Popular works
Document Type: Book
All Authors / Contributors: Avner Ash; Robert Gross
ISBN: 9780691170190 0691170193
OCLC Number: 1001699559
Description: xiv, [4], 220 s. : illustrations
Contents: PART ONE. FINITE SUMS --
1. Proem --
2. Sums of two squares --
3. Sums of three and four squares --
4. Sums of higher powers: waring's problem --
5. Simple sums --
6. Sums of powers, using lots of algebra --
PART TWO. INFINITE SUMS --
7. Infinite series --
8. Cast of characters --
9. Zeta and Bernoulli --
10. Count the ways --
PART III. MODULAR FORMS AND THEIR APPLICATIONS --
11. The upper half-plane --
12. Modular forms --
13. How many modular forms are there? --
14. Congruence groups --
15. Partitions and sums of squares revisited --
16. More theory of modular forms --
17. More things to do with modular forms: applications.
Responsibility: Avner Ash and Robert Gross.

Abstract:

We use addition on a daily basis--yet how many of us stop to truly consider the enormous and remarkable ramifications of this mathematical activity? Summing It Up uses addition as a springboard to present a fascinating and accessible look at numbers and number theory, and how we apply beautiful numerical properties to answer math problems. Mathematicians Avner Ash and Robert Gross explore addition's most basic characteristics as well as the addition of squares and other powers before moving onward to infinite series, modular forms, and issues at the forefront of current mathematical research. Ash and Gross tailor their succinct and engaging investigations for math enthusiasts of all backgrounds. Employing college algebra, the first part of the book examines such questions as, can all positive numbers be written as a sum of four perfect squares? The second section of the book incorporates calculus and examines infinite series--long sums that can only be defined by the concept of limit, as in the example of 1+1/2+1/4+. . .=? With the help of some group theory and geometry, the third section ties together the first two parts of the book through a discussion of modular forms--the analytic functions on the upper half-plane of the complex numbers that have growth and transformation properties. Ash and Gross show how modular forms are indispensable in modern number theory, for example in the proof of Fermat's Last Theorem.

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"Offers a clear and beautiful progression from addition to modern number theory."--Math-Blog "The authors did a remarkable job in making some aspects of modern number theory very accessible to Read more...

 
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