Calculus for mathematicians, computer scientists, and by Hwang A.

By Hwang A.

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One misnomer should be dispelled immediately: Though 2 is a “real” √ number while −1 is an “imaginary” number, each of these symbols represents a mathematical abstraction, and neither has an existence more or less 33 34 CHAPTER 2. NUMBERS √ “real” than the other. No physical quantity can definitively represent 2, because measurements cannot be made with arbitrary accuracy. If you take the square root of 2 on a calculator, you get a rational number whose√square is often noticeably not equal to 2.

B) Suppose the President answers “Yes”. Can the public conclude that the President has offered illegal drugs to visiting Heads of State? What if the answer is “No”? (c) Explain why both answers are embarrassing. 10 One striking peculiarity of the human brain is that it is “better” at seeing certain situations in an emotional light than it is at understanding an equivalent logical formulation. Here is an example. 5 Pronounced “moo” 32 CHAPTER 1. THE LANGUAGE OF MATHEMATICS (a) Each card in a deck is printed with a letter on one side and a number on the other.

Thus, 42 CHAPTER 2. NUMBERS if P (k) is true, then P (k + 1) is also true. By the Induction Principle, P (n) is true for all n ∈ N. ” To find the sum of the first 1000 odd numbers, it is not necessary to perform the addition, but merely to square 1000. ✷ Mathematicians are famous for reducing a problem to one they have already solved. In principle, the more difficult problem is then also solved; in practice, a complete solution may be extremely complicated, because the earlier problem may rely on solution of an even simpler problem, and so forth.

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