The Magic of Numbers (Paperback)

by Benedict Gross (Author), Joe Harris (Author)

Editorial Reviews

Product Description
A math book for “non-math” people, this book—based on a popular course at Harvard University—communicates to readers some idea of the mathematical view of world, and what attracts people to math in the first place. It uses a sense of humor and lighthearted tone to give learners a positive, enjoyable experience of thinking in math, and an understanding of the increasingly important role that science and technology play in our lives. A four-part format presents readers with a fundamental introduction, shows how they can think about arithmetic in a new way—based on what they already know, introduces them to truly abstract mathematics, and teaches them about the special properties of modular arithmetic that are central to modern life. For anyone who has taken an algebra class in high school, and wants to take their understanding of math a little farther—and into everyday life.

From the Back Cover
A math book for “non-math” people, this book—based on a popular course at Harvard University—communicates to readers some idea of the mathematical view of world, and what attracts people to math in the first place. It uses a sense of humor and lighthearted tone to give learners a positive, enjoyable experience of thinking in math, and an understanding of the increasingly important role that science and technology play in our lives. A four-part format presents readers with a fundamental introduction, shows how they can think about arithmetic in a new way—based on what they already know, introduces them to truly abstract mathematics, and teaches them about the special properties of modular arithmetic that are central to modern life. For anyone who has taken an algebra class in high school, and wants to take their understanding of math a little farther—and into everyday life.

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Product Details

• Paperback: 304 pages
• Publisher: Prentice Hall (September 7, 2003)
• Language: English
• ISBN-10: 0131777211
• ISBN-13: 978-0131777217
• Product Dimensions: 9.1 x 7 x 0.8 inches
• Shipping Weight: 1.4 pounds
• Average Customer Review: 3.5 out of 5 stars See all reviews (2 customer reviews)
• Amazon.com Sales Rank: #383,258 in Books (See Bestsellers in Books)

Customer Reviews

Most Helpful Customer Reviews

14 of 14 people found the following review helpful:

4.0 out of 5 stars Excerpt from the Harvard Magazine, December 26, 2003

By A Customer

On Mathematical Imagination
From whole numbers to infinity--and beyond

by Robert Kaplan and Ellen Kaplan

The lunatic, the lover, and the poet, said Shakespeare, are of imagination all compact. He forgot the mathematician, whose daily concerns are shapes in 27 dimensions, series that converge after more terms than there are particles in the universe, numbers larger than infinity, and others infinitesimally small as well as surreal and hyperreal and shapes in 27 dimensions, series that converge after more terms than there are particles in the universe, numbers larger than infinity, and others infinitesimally small as well as surreal and hyperreal and imaginary. The most monstrous thing about these fantasies, of course, is that they turn out to describe how our one and only universe works.
The people who intermediate between lunatics and the world used to be called alienists; the go-betweens for mathematicians are called teachers. Many a student may rightly have wondered if the terms shouldn't be reversed. Those wonderful outworks of the imagination that mathematicians produce rarely make it to the classroom, where mind-numbing drill in what seems a dead language tends to rule instead.

We are, however, at a fulcrum moment in the history of math teaching, with the balance tipping at last from oases of horror in a desert of boredom to epiphanies that make mathematics as appealing as is its sister-art, music. Outstanding examples of this revolution are The Magic of Numbers, by Benedict H. Gross and Joseph D. Harris (based on their Core course in quantitative reasoning) and Imagining Numbers (particularly the square root of minus fifteen), by Gade University Professor Barry Mazur-two books intended to draw lay readers into the secrets of what mathematics is really like and how our fellow humans invent it.

Gross, who is Leverett professor of mathematics and dean of Harvard College, and Harris, who is Higgins professor of mathematics, have managed a miracle: addressing an audience they rightly assume to be resistant, they lead it gently from utter scratch (how many whole numbers are there from 1 to 10?) to counting with Catalan numbers (in how many ways can n pairs of parentheses appear in a sentence?)-and they do this with bonhomie, grace, and humor. Can you imagine any textbook speaking to you like this?

Let's suppose you climb out of bed one morning, still somewhat groggy from the night before. You grope your way to your closet, where you discover that your cache of clean clothes has been reduced to four shirts and three pairs of pants. It's far too early to exercise any aesthetic judgment whatsoever: any shirt will go with any pants; you only need something that will get you as far as the dining hall and that blessed, life-giving cup of coffee. The question is,

How many different outfits can you make out of your four shirts and three pairs of pants?

Admittedly the narrative took a sharp turn toward the bizarre with that last sentence. Why on earth would you or anyone care how many outfits you can make? Well, bear with us while we try to answer it anyway.

This is a far cry from the standard word-problems that the Canadian humorist Stephen Leacock made such wonderful fun of: dashing old A rowing the best boat while B, as always, has all he can do to keep his leaky tub afloat-and you ask who won?

The Magic of Numbers takes us from counting to the heart of mathematics-number theory-where we feel the pulse and see the valves opening and closing. Like medical students, the lucky people in the Gross and Harris class, and now all readers, are shown the deepening layers of anatomy, the sense the systems make, and how they function; then, in carefully calibrated exercises, they have a chance to do probing of their own. What was impersonal becomes part of each reader's personality. The motto here could well be what medical students are told about operations: "Watch one, do one, teach one."

The book leads you so far toward the frontiers of mathematics that no one could be blamed for losing sight of everything else and waking up 20 years later in its mountains. But just when our heads begin to break through the clouds, Gross and Harris return us to a heightened reality by showing how relevant what we have learned is to a vital practical issue: public-key cryptography. Here the primes, whose tantalizing mysteries Gross and Harris have explored with their students, act as the covert keys to encrypted messages that are openly transmitted along with a public key: the product of two very large primes known only to the receiver. Why, then, can't a spy decipher the message? Because to work back from the product to its prime factors would take more time (even with the fastest computers in the world) than the enemy has to spare-and so a problem in the surreal world of espionage is solved by recourse to the more than real world of mathematics. We find ourselves initiates now in what, at the book's beginning, we knew about only by rumor.

Gross and Harris close their book by saying, "If you've stayed with us till now, you know a tremendous amount of real mathematics-more than all but a small fraction of people walking around.... [N]ow it's time for...you to close the book and get on with your life." What they don't say is that your life will be wholly different. Pindar praised the winners of Olympic events by telling them they had managed, once only and briefly, to touch the bronze sky. Those who win through to the end of The Magic of Numbers will be for the rest of their lives in touch with the accessible mystery of things.

3.0 out of 5 stars Would be excellent if exercises were not "worthless"., October 14, 2008

By	Athaedrielle "Amazonian Denizen" (Greater San Francisco Bay Area, CA, USA) - See all my reviews

This book is VERY readable -- and easily approachable -- BUT, suffers from a major defect for the SELF-LEARNER. There are *tons* of exercises that are recommended -- which would be great except doing the exercises is a complete waste of time -- you don't know if you got the exercise right or wrong! Yup. Ok, Now you do it: 1. what's 2^5 mod 11?, 2. 2^17 mod 23?....etc? Nice BUT if you do the exercises, you don't know if you got the right answer or not, so its a waste of time.
Worse -- the authors emphasize the need to do the exercises -- but if you don't know if you did them right or wrong, what's the point? There are a fair number of examples that do give answers, but these are about 1/5th - 1/6th the problems and the answers are given in the text of the problem. So while you know the answer, you can't really do the example, then see if you got the right answer -- because it's also there in front of you.
Having an answer key -- even if it was for every even or odd exercise would be helpful. Non-answered questions could be for those classes where the teacher doesn't trust the student to learn, but just copy answers, but if that's the case, why bother taking a class? Just copying answers doesn't help much in the real world nor on exams (which one would presumably have in a classroom setting).

But I bought this book for educational/recreational reading (just like some people like to participate in sports or go to a gym to exercise a body, many find mental exercise equally valuable for one's "head").

I would NOT recommend *against* the book, but it may go slowly, as the paucity answered problems make for slower going in terms of comprehension and retention (I find I more often have to reread prior chapter I understood, but didn't get enough practice to retain in order to progress onto more complex subjects).

For self-learning, FEEDBACK is required -- no feedback, then no learning if you are doing it right or wrong, but the format, typesetting, and conversational style on this book make it very readable. Just wish they had the answers (or at least odd numbered answers in the back).