By Edmund Landau

Why does $2 \times 2 = 4$? What are fractions? Imaginary numbers? Why do the legislation of algebra carry? and the way can we turn out those legislation? What are the houses of the numbers on which the Differential and essential Calculus relies? In different phrases, What are numbers? And why have they got the homes we characteristic to them? because of the genius of Dedekind, Cantor, Peano, Frege and Russell, such questions can now accept a passable resolution. This English translation of Landau's well-known Grundlagen der Analysis-also to be had from the AMS-answers those very important questions.

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If not, why not? If so, does it “tell the whole truth”? Explain. 4 R. M. Waldo, the tallest documented human, was just under 9 feet in height. Assume for this question that he was the tallest human ever to live, and that he was exactly 9 feet tall. 30 CHAPTER 1. ” Is this claim true? If not, why not? If so, can you write it as an “If. . , then. . ” statement? ” Does this claim “tell the whole truth”? If so, in precisely what sense does it do so? (c) Does the assertion in part (a) imply the assertion in part (b), or vice versa?

It is left as an exercise to guess a formula for the number of transfers required to move a stack of n disks, and to prove this guess is correct by mathematical induction. It should be clear why recursive definitions are so useful; an immense amount of complexity can be encoded in a small set of recursive rules. Each person in the solution of the Tower of Hanoi needs to know only two trivial things, but by coordinated delegation of tasks they solve a complicated 44 CHAPTER 2. NUMBERS problem. However, the number of transfers needed essentially doubles with each additional disk.

CALCULUS AND THE “REAL WORLD” 29 theory, while giving interesting and substantial applications of these powerful mathematical techniques. Exercises Some of these exercises are fairly traditional, and assume you are familiar with standard mathematical notation. Others are designed to make you think about language, mental models, and semantics. The only way to learn to “speak Mathematics” is through practice; writing, reformulating, and thinking. Familiarity can be acquired through reading, but originality can only come through participation.