Bits, Data types, and Operations: Chapter 2

1. 1 Bits, Data types, and Operations: Chapter 2 COMP 2610 Dr. James Money COMP 2610

2. Signed Integers • Thus, for simplicity of hardware we use the two’s complement representation • How do we compute two’s complement? • Write the magnitude of the number in binary • Compute the one’s complement, that is flip the bits • Add one to the result

3. Signed Integers • How do we reverse two’s complement? • We apply it again! • That is two’s complement(two’s complement(N)) = N

4. Signed Integers • If the following 6 bits number is in two’s complement form, what is the number? (101001)2 Compute one’s complement: (010110)2 Add one: (010111)2 = (23)10 • Thus, the number is -23

5. Binary to Decimal Conversion • We now systematically look at convert from binary to decimal for two’s complement stored integers • We assume the binary number is in the form M=(bn bn-1 … b2 b1 b0)2 where bi is either 0 or 1.

6. Binary to Decimal Conversion • Consider the leading bit bn. If it is zero, the integer is positive or zero. If it is one, the integer is negative. • If bn=1, then compute the two’s complement of bn-1…b1b0. Then bn-1x2n-1 + … + b2x22 + b1x21 + b0x20 is the magnitude of the value • If the leading bit bn was 1, then add the negative sign.

7. Binary to Decimal Conversion • Convert the following two’s complement representations to their decimal values: • 1100 0011 • 0011 0110 • 1111 1110

8. Decimal to Binary Conversion • Take the magnitude of the decimal number • Convert the magnitude of the number to binary form • If the number is negative, compute the two’s complement of the number

9. Review of Unsigned Integers • To convert from positive decimal number M to binary: • Set x=M, k=0 • Compute x÷2 = y R bk • Set x=y, k=k+1 • Go to 2 if x≠0 and repeat • The bits are the remainders listed right to left …b3b2b1b0

10. Decimal to Binary Conversion • Convert the following decimal numbers to their two’s complement forms for 8 bits: • 45 • -34 • -2 • 0