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COMS 161 Introduction to Computing Title: Numeric Processing Date: October 27, 2004 Lecture Number: 25 1 Announcements • Homework 7 – Due Wednesday, 11/03/2004 2 Review • Numeric Processing • Integers – Magnitude representation – Sign-magnitude representation 3 Outline • Numeric Processing • Integers – Ones’s complement representation – Two’s complement representation • Real numbers 4 Sign-Magnitude • Mathematical operations sometimes give an incorrect result 4 – 3 = 4 + -3 = 1 4 +(-3) 1 0100 +1011 1 111 -7 5 Ones compliment • Positive integers have the most significant bit (leftmost) equal to 0 – The magnitude of positive numbers is the same as the signed magnitude representation • Negative integers have the most significant bit equal to 1 – The magnitude of a negative number is the same as the magnitude representation of a different number N 2 1 N n 6 Ones compliment N 2n 1 N -3 = 23 – 1 – 3 = 8 – 4 = 4 -2 = 23 – 1 – 2 = 8 – 3 = 5 -1 = 23 – 1 – 1 = 8 – 2 = 6 -0 = 23 – 1 – 0 = 8 – 1 = 7 Unsigned decimal 0 1 2 3 4 5 6 7 Ones complement 000 001 010 011 100 101 110 111 Signed decimal 0 1 2 3 -3 -2 -1 -0 7 Ones compliment • Example: let n = 8, and N = 5 N 2n 1 N 5 2 8 1 5 256 6 250 1111 1010 – There is something special here and in the previous table 5 0000 0101 - 5 1111 1010 – Negatives are made from the positive by inverting each bit! – Simpler hardware for arithmetic operations – Still two representations of zero 8 Ones complement Decimal Ones Complement 5 4 3 2 1 0 -0 -1 -2 -3 -4 -5 0000 0101 0000 0100 0000 0011 0000 0010 0000 0001 0000 0000 1111 1111 1111 1110 1111 1101 1111 1100 1111 1011 1111 1010 9 Ones complement • Mathematical operations sometimes give an incorrect result using this representation 4 – 3 = 4 + -3 = 1 4 +(-3) 1 0100 +1100 00 00 4 – 2 = 4 + -2 = 2 0 4 +(-2) 2 • Two representations of zero 0100 +1101 00 01 1 10 Twos Complement • Positive integers have the most significant bit (leftmost) equal to 0 – The magnitude of positive numbers is the same as signed magnitude and ones complement representations • Negative integers have the most significant bit (leftmost) equal to 1 – Negative numbers can be computed as: N 2 N n 11 Two’s Complement • Example: let n = 8, and N = 5 N 2n N 5 2 8 5 256 5 251 1111 1011 – There is something special here • ones’ complement N 2 1 N n N 2 N • two’s complement • It is simple to determine the representation of a negative number in ones complement given the positive • It is easy to convert a ones complement representation to a twos complement representation by simply adding 1 n 12 Twos Complement Decimal Sign/mag 5 4 3 2 1 0 -0 -1 -2 -3 -4 -5 0000 0101 0000 0100 0000 0011 0000 0010 0000 0001 0000 0000 0000 0000 1000 0001 1000 0010 1000 0011 1000 0100 1000 0101 Ones’ Two’s 0000 0101 0000 0100 0000 0011 0000 0010 0000 0001 0000 0000 1111 1111 1111 1110 1111 1101 1111 1100 1111 1011 1111 1010 0000 0101 0000 0100 0000 0011 0000 0010 0000 0001 0000 0000 ------------1111 1111 1111 1110 1111 1101 1111 1100 1111 1011 13