The Art and Craft of Problem Solving
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The newly revised Second Edtion of this distinctive text uniquely blends interesting problems with strategies, tools, and techniques to develop mathematical skill and intuition necessary for problem solving. Readers are encouraged to do math rather than just study it. The author draws upon his experience as a coach for the International Mathematics Olympiad to give students an enhanced sense of mathematics and the ability to investigate and solve problems.
penultimate step of an argument (or sub-argument), you want to mark this off to your audience clearly. The abbreviations TS and ISTS ("to show" and "it is sufficient to show") are particularly useful for this purpose. 5. A nice bit of notation, borrowed from computer science and slowly becoming more common in mathematics, is ":=" for "is defined to be." For example, A := B + C introduces a new variable A and defines it to be the sum of the already defined variables B and C. Think of the colon as
240 Miscellaneous Instructive Examples 24 7 Can a Polynomial Always Output Primes? 247 If You Can Count It, It's an Integer 249 A Combinatorial Proof of Fermat's Little Theorem Sums of Two Squares Chapter 8 250 Geo metry for Americans 8.1 Three "Easy" Problems 8.2 Survival Geometry I 256 256 258 Points, Lines, Angles, and Triangles 259 249 xvii xviii CONTENTS Parallel Lines 260 Circles and Angles 264 Circles and Triangles 8.3 266 Survival Geometry II Area 2 70 270
exponentiation. Indeed, Euler's formula states that 4.2.7 Cis e = e iO , where e = 2.7 1 828 . is the familiar natural logarithm base that you have encountered in calculus. This is a useful notation, somewhat less cumbersome than Cis e , and quite profound, besides. Most calculus and complex analysis textbooks prove Euler's formula using the power series for e , sin x, and cosx, but this doesn 't really give much insight as to why it is true. This is a deep and interesting issue that is beyond
What happens as you move counterclockwise, starting from - 1 along the unit circle? Roots of Unity The zeros of the equation y;!l = 1 are called the nth roots of unity. These numbers have many beautiful properties that interconnect algebra, geometry and number theory. One reason for the ubiquity of roots of unity in mathematics is symmetry: roots of unity, in some sense, epitomize symmetry, as you will see below. (We will be assuming some knowledge about polynomials and summation. If you are
way. As r � 00, we get the limiting case, a beautiful identity of generating functions: 1 - 1)/2, • Our final example is from the theory of partitions of integers, a subject first inves tigated by Euler. Given a positive integer n, a partition of n is a representation of n as a sum of positive integers. The order of the summands does not matter, so they are conventionally placed in increasing order. For example, and are two different partitions of 1 + 1 +3 1 + 1 + 1 +2 5. Example 4.3.7