University:
Kentucky State University
Course:
COS 232 | Programming I
Academic year:

2023
Views:

340

Pages:

11
Author:

Hundebuchesn

2’s Complement & Subtraction using 2’s Complement How To Represent Signed Numbers • Plus and minus sign used for decimal numbers: 25 (or +25), -16, etc. • For computers, desirable to represent everything as bits. • Three types of signed binary number representations: signed magnitude, 1’s complement, 2’s complement. • In each case: left-most bit indicates sign: positive (0) or negative (1). Consider signed magnitude: 000011002 = 1210 Sign bit Magnitud e 100011002 = -12 10 Sign bit Magnitud e One’s Complement Representation • The one’s complement of a binary number involves inverting all bits. • 1’s comp of 00110011 is 11001100 • 1’s comp of 10101010 is 01010101 000011002 = 1210 Sign bit Magnitud e 11110011 2 = -12 10 Sign bit Magnitud e Two’s Complement Representation • The two’s complement of a binary number involves inverting all bits and adding 1. • 2’s comp of 00110011 is 11001101 • 2’s comp of 10101010 is 01010110 • To find negative of 2’s complement number take the 2’s complement. 000011002 = 1210 Sign bit Magnitud e 111101002 = -12 10 Sign bit Magnitud e Two’s Complement Shortcuts • Algorithm – Simply complement each bit and then add 1 to the result. • Finding the 2’s complement of (01100101)2 and of its 2’s complement… N = 01100101 [N] =10011011 10011010 01100100 + 1 + 1 ------------------------10011011 01100101 1’s Complement Addition • Using 1’s complement numbers, adding numbers is easy. • For example, suppose we wish to add +(1100)2 and +(0001)2. • Let’s compute (12)10 + (1)10. • (12)10 = +(1100) 2 = 011002 in 1’s comp. • (1)10 = +(0001) 2 Step 1: Add binary numbers Step 2: Add carry to low-order bit = 000012 in 1’s comp. Ad d Add carry Final Resul t 0 1 1 0 0 + 0 0 0 0 1 -------------0 0 1 1 0 1 0 -------------0 1 1 0 1 1’s Complement Subtraction • Using 1’s complement numbers, subtracting numbers is also easy. • For example, suppose we wish to subtract +(0001)2 from +(1100)2. 0 1 1 0 • Let’s compute (12)10 - (1) 10. • (12)10 = +(1100) 2 • (-1) 10 = -(0001)2 0 - 0 0 0 0 1 = 011002 in 1’s comp. = 111102 in 1’s comp . -------------1’s comp Step 1: Take 1’s complement of 2 nd operand Step 2: Add binary numbers Step 3: Add carry to low order bit Ad d Add carry Final Resul t 0 1 1 0 0 + 1 1 1 1 0 -------------1 0 1 0 1 0 1 -------------0 1 0 1 1 2’s Complement Addition • Using 2’s complement numbers, adding numbers is easy. • For example, suppose we wish to add +(1100)2 and +(0001)2. • Let’s compute (12)10 + (1)10. • • (12)10 = +(1100) 2 = 011002 in 2’s comp. = 000012 in 2’s comp. (1)10 = +(0001) 2 Step 1: Add binary numbers Step 2: Ignore carry bit Ad d Final Resul t 0 1 1 0 0 + 0 0 0 0 1 -------------0 0 1 1 0 1 Ignor e 2’s Complement Subtraction • Using 2’s complement numbers, follow steps for subtraction • For example, suppose we wish to subtract +(0001)2 from +(1100)2. 0 1 1 0 • Let’s compute (12)10 - (1) 10. - 0 0 0 0 • • 0 1 -------------- (12)10 = +(1100) 2 = 011002 in 2’s comp. = 111112 in 2’s comp. (-1) 10 = -(0001)2 2’s comp Step 1: Take 2’s complement of 2 nd operand Ad Step 2: Add binary numbers d Step 3: Ignore carry bit Final Resul t 0 1 1 0 0 + 1 1 1 1 1 -------------1 0 1 0 1 1 Ignor e Carry 2’s Complement Subtraction: Example #2 • Let’s compute (13)10 – (5) 10. • • = (01101)2 (13)10 = +(1101) 2 (-5) 10 = -(0101)2 = (11011)2 • Adding these two 5-bit codes… 0 1 1 0 1 + 1 1 0 1 1 carr -------------y 1 0 1 0 0 • Discarding the carry bit, the 0 sign bit is seen to be zero, indicating a correct result. Indeed, (01000)2 = +(1000)2 = +(8)10. 2’s Complement Subtraction: Example #3 • Let’s compute (5)10 – (12)10. • • = (10100)2 (-12)10 = -(1100)2 = +(0101) 2 = (00101)2 (5)10 • Adding these two 5-bit codes… 0 0 1 0 1 + 1 0 1 0 0 -------------1 1 0 0 1 • Here, there is no carry bit and the sign bit is 1. This indicates a negative result, which is what we expect. (11001)2 = -(7)10.

2’s Complement & Subtractionusing 2’s Complement

Related Documents

Recommended Documents

Description

Related Documents

Free up your schedule!

Take 5 seconds to unlock