Introduction to Computer Systems Lecturer: Steve Maybank Department of Computer Science and Information Systems [email protected] http://www.dcs.bbk.ac.uk/~sjmaybank Autumn 2014 Week 3a: Number Representations 14 October 2014 Birkbeck College, U. London 1 Recap: Programs A program is a sequence of instructions. The instructions may refer to memory cells which store values such as integers. In Tom’s computer the memory cells are the boxes. Each memory cell has an address and contents. 14 October 2014 Birkbeck College, U. London 2 Recap: Running a Program The instructions of the program are executed one by one. When an instruction is executed, the values in the memory cells may change. When the program halts the output is usually the values of selected memory cells. 14 October 2014 Birkbeck College, U. London 3 Recap: Variables Here are two typical statements in a programming language: p = 0; q = 3+4; Left hand side: the name of a variable Right hand side: an expression Execution: evaluate the right hand side to obtain a number. Store the number in a memory location named by the variable. 14 October 2014 Birkbeck College, U. London 4 Exercise from Week 2 Sketch an algorithm that takes as input a strictly positive integer n and outputs an integer k such that 2 n and 2 k 14 October 2014 k 1 n Birkbeck College, U. London 5 Representations of Negative Integers Put a minus sign in front of the representation for a positive integer. Excess notation. Two’s Complement notation – the most popular representation for integers in computers. 14 October 2014 Brookshear, Section 1.6 6 Excess Notation Problem: represent a set of positive and negative integers using bit strings with a fixed length n. Represent 0 by 10…0 (n bits). Represent positive numbers by counting up from 10…0 in standard binary notation. Represent negative integers by counting down from 10…0 in standard binary notation. 14 October 2014 Brookshear, Section 1.6 7 Example of Excess Notation n=3 14 October 2014 111 110 101 3 2 1 100 0 011 010 001 000 -1 -2 -3 -4 Brookshear, Section 1.6 8 Examples Find the 6 bit excess notation for the decimal numbers 7 and -6. Which decimal number has the 5 bit excess notation 10101. 14 October 2014 Birkbeck College, U. London 9 Two’s Complement Notation Form the bit string 10…0 with n+1 bits. Represent 0 by the rightmost n bits of 10…0. Represent positive integers by counting up from 10…0 in standard binary notation and using the rightmost n bits. Represent negative integers by counting down from 10…0 in standard binary notation and using the rightmost n bits. 14 October 2014 Birkbeck College, U. London 10 Examples Four bit two's complement representation for 5 2 4 5 16 5 11 Decimal 11 Binary1011 Answer :1011 Four bit two's complement representation for 6 2 4 6 16 6 22 Decimal 22 Binary10110 Answer : 0110 14 October 2014 Birkbeck College, U. London 11 Example of Two’s Complement Notation 0111 0110 0101 0100 0011 0010 0001 0000 7 6 5 4 3 2 1 0 1111 1110 1101 1100 1011 1010 1001 1000 -1 -2 -3 -4 -5 -6 -7 -8 n=4 The left most bit indicates the sign. 14 October 2014 Brookshear, Section 1.6 12 Addition and Subtraction In the two’s complement system subtraction reduces to addition. E.g. to evaluate 6-5 in 4 bit two’s complement notation, add the tc bit strings for 6 and –5, then take the four rightmost bits. 0110 1011 === 10001 14 October 2014 6 -5 == 1 Brookshear, Section 1.6 13 Explanation The bit strings for TC[6] and TC[-5] are the rightmost four bits of Binary[24+6] and Binary[24 -5], respectively. The bit strings TC[6], TC[-5] are added as if they were binary numbers. The rightmost four bits of the result equal the rightmost four bits of Binary[(24+6)+(24-5)]= Binary[24+24 +1]. The right most four bits of Binary[24+24 +1] are the bit string for TC[1]. 14 October 2014 Brookshear, Section 1.6 14 Why Use Two’s Complement Addition and subtraction require one circuit for addition and one circuit for negation. This is more efficient than having a circuit for addition and a circuit for subtraction. 14 October 2014 Brookshear, Section 1.6 15 Two’s Complement Notation for m and -m Suppose TC[m] = s || 1 || t, where t is a string of zeros. Then TC[-m]=Complement[s]||1||t. Proof: the rightmost n bits of TC[m]+TC[-m] are all zero. Example: n=4, TC[3]=0011, TC[-3]=1101. 14 October 2014 Brookshear, Section 1.6 16 Example Find the 5 bit two’s complement representations for the decimal integers 5 and -5. 14 October 2014 Birkbeck College, U. London 17
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