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.
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