By Terence Tao
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Additional resources for Analysis I (Volume 1)
10. In other words, we can divide a natural number n by a positive number q to obtain a quotient m (which is another natural number) and a remainder r (which is less than q). This algorithm marks the beginning of number theory, which is a beautiful and important subject but one which is beyond the scope of this text. 36 Proof. 5. 2. 11 (Exponentiation for natural numbers). Let m be a natural number. To raise m to the power 0, we define m 0 := 1. Now suppose recursively that mn has been defined for some natural number n, then we define mn++ := mn x m.
But it is difficult to quantify what we mean by "can be obtained from" without already using the natural numbers, which we are trying to define. 2 for more details. 22 2. 5 (Principle of mathematical induction). Let P(n) be any property pertaining to a natural number n. Suppose that P(O) is true, and suppose that whenever P(n) is true, P(n++) is also true. Then P( n) is true for every natural number n. 10. We are a little vague on what "property" means at this point, but some possible examples of P(n) might be "n is even"; "n is equal to 3"; "n solves the equation (n + 1) 2 = n 2 + 2n + 1"; and so forth.
Now suppose inductively that we have defined how to add n to m. Then we can add n++ tom by defining (n++) + m := (n + m)++. Thus 0 + m is m, 1 + m = (0++) + m is m++; 2 + m (1++) +m = (m++ )++; and so forth; for instance we have 2+3 = (3++ )++ = 4++ = 5. From our discussion of recursion in the previous section we see that we have defined n + m for every integer n. Here we are specializing the previous general discussion to the setting where an = n + m and fn(an) =an++· Note that this definition is asymmetric: 3 + 5 is incrementing 5 three times, while 5 + 3 is incrementing 3 five times.
Analysis I (Volume 1) by Terence Tao