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Summing It Up: From One Plus One to Modern Number Theory
The power and properties of numbers, from basic addition and sums of squares to cutting-edge theory
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.
Appropriate for numbers novices as well as college math majors, Summing It Up delves into mathematics that will enlighten anyone fascinated by numbers.
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.
Appropriate for numbers novices as well as college math majors, Summing It Up delves into mathematics that will enlighten anyone fascinated by numbers.
- GenresMathematicsScience
228 pages, Hardcover
Published May 17, 2016
31 people are currently reading
129 people want to read
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Displaying 1 - 4 of 4 reviews
July 25, 2016
This book is one of the set of math books where the claims on the DJ are not justified by the contents. The last paragraph on the front flap of the DJ is as follows:
“Appropriate for numbers novices as well as college math majors, Summing It Up delves into mathematics that will enlighten anyone fascinated by numbers.”
Although this makes it appear as if the book is one of popular mathematics, there is nothing novice about it.
The first section on finite sums contains material found in math major level courses in number theory. Topics such as sums of squares, figurate numbers and quadratic residues appear. The following section moves into much more advanced topics such as lengthy sums and series, double and telescoping sums, Bernoulli numbers and complex analytic functions. There is no sparing of the formula or proof of the theorems here, when mathematics is needed to completely explain a concept, it is used. The last section deals with modular forms and takes the reader into topics such as elliptical curves Hecke operators and L-functions.
While this is a very good book for people interested in advanced number theory, much of the material is beyond the coverage of a math major course in number theory. It will definitely not enlighten the person that has only a fascination with numbers.
This book was made available for free for review purposes
“Appropriate for numbers novices as well as college math majors, Summing It Up delves into mathematics that will enlighten anyone fascinated by numbers.”
Although this makes it appear as if the book is one of popular mathematics, there is nothing novice about it.
The first section on finite sums contains material found in math major level courses in number theory. Topics such as sums of squares, figurate numbers and quadratic residues appear. The following section moves into much more advanced topics such as lengthy sums and series, double and telescoping sums, Bernoulli numbers and complex analytic functions. There is no sparing of the formula or proof of the theorems here, when mathematics is needed to completely explain a concept, it is used. The last section deals with modular forms and takes the reader into topics such as elliptical curves Hecke operators and L-functions.
While this is a very good book for people interested in advanced number theory, much of the material is beyond the coverage of a math major course in number theory. It will definitely not enlighten the person that has only a fascination with numbers.
This book was made available for free for review purposes
July 7, 2021
Amazing.
Ash has written a trilogy of math books on subjects related to elliptic curves, culminating in this excursion into number theory. Introducing Riemann zeta function, Bernoulli series, and the divisor function (by means of various standard generating function techniques). Ash and Gross start in Chapter 8 and 11 with their mission proper, the upper half plane, q, linear fractional transfromations, SL2(Z) and notions of modular form. Providing just enough proofs, they are a splendid introduction to the wonders of a field for a reasonably diligent mathematically willing reader to stop and admire the many blossoms of the number theory flower garden, as Freeman Dyson put it somewhere.
I found it convenient to start my reading at the modular form chapter and go backward and forward to sample the other delights.
Easily a serious math writer for the semi popular or semi serious math educated public all in a expository league by himself.
Ash has written a trilogy of math books on subjects related to elliptic curves, culminating in this excursion into number theory. Introducing Riemann zeta function, Bernoulli series, and the divisor function (by means of various standard generating function techniques). Ash and Gross start in Chapter 8 and 11 with their mission proper, the upper half plane, q, linear fractional transfromations, SL2(Z) and notions of modular form. Providing just enough proofs, they are a splendid introduction to the wonders of a field for a reasonably diligent mathematically willing reader to stop and admire the many blossoms of the number theory flower garden, as Freeman Dyson put it somewhere.
I found it convenient to start my reading at the modular form chapter and go backward and forward to sample the other delights.
Easily a serious math writer for the semi popular or semi serious math educated public all in a expository league by himself.
November 10, 2025
Don't be misled by the introduction: the ideal reader for this book is someone who has taken, or is currently taking, an undergraduate course in complex analysis. A good background in algebra is also required, as evidenced by the book taking about 5 pages to summarize all the necessary group theory.
This book is an excellent introduction to the theory of modular forms. Their intrinsic beauty and usefulness are properly examined with context and clear, motivated explanations. The last 2-4 chapters are quite technical and brief and probably serve as inspiration rather than explanation.
This book is an excellent introduction to the theory of modular forms. Their intrinsic beauty and usefulness are properly examined with context and clear, motivated explanations. The last 2-4 chapters are quite technical and brief and probably serve as inspiration rather than explanation.
August 22, 2021
Excellent book to introduce modular form, the math in the last part is difficult to follow, especially if you are not familiar with elementary complex analysis. However, it’s worth for you to follow all the details and you will learn how important modular form is in advanced number theory.
Displaying 1 - 4 of 4 reviews