Discrete Mathematics with Graph Theory (3rd Edition) 224

Discrete Mathematics with Graph Theory (3rd Edition) 224 -...

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222 Solutions to Exercises Exercises 8.2 1. [BB] Let addition be the basic operation. We need n subtractions, n squarings (each equivalent to two basic operations), n - 1 additions, and one square root (the equivalent of 20 basic operations). Hence, f(n) = n + 2n + n -1 + 20 = 4n + 19. 2. The multiplication of a = (a n -Ia n -2 ... alao)lO by the single digit b requires initially the multiplica- tion of band ao, one operation. Then we multiply b and al and perhaps add a carry from the first step. Since two single-digit additions may be needed in this carry (e.g. 89 x 7), at most three operations are required in the multiplication of band al and, similarly, in the multiplication of b and each ai, i ;:::: 1. An upper bound for the total number of operations required is 1 + 3(n - 1) = 3n - 2. 3. [BB] Consider the division of a number a, which has at most two digits, by a single-digit number b, where a < lOb. Since the quotient q has just one digit, the division counts as one operation, the product qb counts as another, and since a - qb < b, the difference a - qb has just a single digit, so the subtraction requires a third operation.
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