Search Loci: Convergence:
The errors of definitions multiply themselves according as the reckoning proceeds; and lead men into absurdities, which at last they see but cannot avoid, without reckoning anew from the beginning.
In J. R. Newman (ed.), The World of Mathematics, New York: Simon and Schuster, 1956.
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The Square Root of 2
The Square Root of 2: A Dialogue Concerning a Number and a Sequence. David Flannery.2006, xii, 260pp., 31 illustrations, $25, hardcover, ISBN 0-387-20220-X New York: Copernicus Books (Springer), www.springer.com.
In my early days of teaching high school mathematics, I recommended Polya’s How to Solve It to a particularly bright senior. In six weeks she returned the book and a sheaf of papers containing solutions to all the problems. She went on to graduate work in math, earning a Ph.D. at Boulder, Colorado, and has enjoyed mathematics and teaching mathematics ever since. David Flannery’s book, The Square Root of 2, is the kind of book to recommend to a particularly bright high school senior, not to ignore a frosh in college. From page 1 through its conclusion, it is a masterful dialogue, in the order of Socrates and the Slave Boy. Both dialogues start with the same problem, the search for a square that is twice another, although their goals are different. Where Socrates (or was it really Plato?) sought to establish the existence of innate ideas, Flannery seeks to arouse a cool passion for mathematics in his student. Let us consider what happens in the five chapters.
While the task of creating a square of area twice that of another is simple, finding its measure is not so simple. If the side of the smaller square is 1, then the side of its double is called √2, the square root of two. But what is this? Coming to an understanding of this measure is the goal of the dialogue. The teacher suggests leads that the student examines and uses to draw projections. The student quickly realizes that what can be constructed geometrically is not transferable to numbers. There is no square number that is twice another. Regardless, the introduction of algebra produces m2 = 2n2. An investigation of columns of numbers m in the operation that takes m2 to 2n2leads to the relationship m2 = 2n2 ± 1. From here the student finds the way to m/n = √2 and begins a study of m/n. This produces the sequence 1/1, 3/2, 7/5, 17/12, 41/29 . . ., each successive term of the sequence being nearer to the value of √2. In a close inspection of the terms of the sequence, the student discovers (and later proves) the first of several provocative formulas. Then, and I am skipping over a lot of “good stuff” here, Flannery and his student find more powerful formulas for getting as close as is practicable to a decimal representation of the value of √2. Cautiously, the student is led to prove each step of the progress. Flannery quotes a quatrain proof of the irrationality of √2 as an introduction to the student’s completion of a four-fold charge arising from √2. A floral arrangement of √n, n a natural number, brings an enjoyable reading to a close. Not to be overlooked is the good history in the story, as occasions for progress in the dialogue. The contributions of the ancient Hindus and Greeks (Hippasus of Metapontum, Euclid, Hero of Alexandria, and Pappus), Christoff Rudolff, John Pell, Bertrand Russell, G.H. Hardy, and Ramanujan are recognized, with adequate credit to modern writers in the references.
Flannery has woven an engaging dialogue from history and theory that offers the student insights into the thinking mind of the working mathematician. He also laid out a series of vignettes that a resourceful teacher might translate into discovery exercises.
Barnabas Hughes O.F.M., Professor Emeritus, California State University, Northridge