1 Discrete Structures – CNS2300 Text Discrete Mathematics and Its Applications Kenneth H. Rosen (5 th Edition) Chapter 2 The Fundamentals: Algorithms, the Integers, and Matrices

2 Section 2.5 Integers and Algorithms

3 Lemma (A short theorem not used for much except proving another theorem) Let a = bq + r, where a,b,q and r are integers. Then gcd(a,b) = gcd(b,r)

4 Euclidean Algorithm procedure gcd(a,b: positive integers) x := a y := b while y > 0 begin r := x mod y x := y y := r end {gcd(a,b) is x}

5 Example 18 = (1)(12) = (2)(6) + 0 gcd(12,18)

6 Example 18 = (1)(12) = (2)(6) + 0 gcd(12,18)

7 Example gcd(123,277) 277 = (2)(123) = (3)(31) = (1)(30) = (30)(1) + 0

8 Representation of Integers Let b be a positive integer greater than 1. Then if n is a positive integer, it can be expressed uniquely in the form where k is a nonnegative integer, a 0,a 1,…,a k are nonnegative integers less than b, and Which means?????

9 You can change bases 351 = ( ) = ( 537) = (15F) = (253) 12

10 People There are only 10 types of people. Those who understand binary numbers and those who don’t.

11 How? 351 = (29)(12) = (2)(12) = (0)(12) + 2 The remainders tell us the number in base = Convert 351 to base 12

12 Convert 351 to base 16 (hex) 351 = 21(16) + 15 (F in base 16) 21 = 1(16) +5 Therefore 351 = 15F 16 1 = 0(16) +1

13 How About the Other Way? = = 351

14 Special Relationships

15 Special Relationships

16 Special Relationships

17 Special Relationships

18 Special Relationships = 537 8

19 Try it Again

20 This time, group by 4 bits F

21 Special Relationships F5

22 Special Relationships F51

23 Special Relationships F = 15F 16

E M range

25 Addition of Integers procedure add(a,b: positive integers) c := 0 for j := 0 to n-1 begin d := (a j + b j + c)/2 s j := a j + b j + c - 2d c := d end s n := c

26 Multiplying Integers procedure multiply(a,b: positive integers) for j := 0 to n-1 begin if b j = 1 then c j := a shifted j places else c j := 0 end p := 0 for j := 0 to n - 1 p := p + c j

27 finished