Bits, Data types, and Operations: Chapter 2

COMP 2610Dr. James Money

COMP 2610

1

Review of Unsigned Integers

We can write an unsigned integer

M=(ak ak-1 … a2 a1 a0)10

a binary number of the form

M=(bn bn-1 … b2 b1 b0)2

using simple arithmetic

Review of Unsigned Integers

To convert from binary to decimal, we compute the appropriate bit times the correct power of 2 and add:

bnx2n + bn-1x2n-1 + … + b2x22 + b1x21 + b0x20

For example,

(0101)2 = 0x23 + 1x22 + 0x21 + 1x20=4+1=(5)10

Review of Unsigned Integers

To convert from decimal number M to binary:

1. Set x=M, k=0

2. Compute x÷2 = y R bk

3. Set x=y, k=k+1

4. Go to 2 if x≠0 and repeat

5. The bits are the remainders listed right to left …b3b2b1b0

Review of Unsigned Integers

For example, (11)10:

11 ÷2 = 5 R1

5 ÷ 2 = 2 R1

2 ÷2 = 1 R0

1 ÷2 = 0 R1

Thus, (11)10 = (1011)2

Review of Unsigned Integers

Addition: add the same way as you would for long addition for decimals, except that (1+1)2 =

(10)2 and you put down 0 and carry the one.

Example: 0101 5

+ 0111 + 7

1100 12

Signed Integers

We need to be able to work with signed integers as well, if for not other reason than to perform subtraction

We can divide the bit codes up into two ranges, one for positive values and one for negative values

For example, 1 to 15, and -1 to -15, plus 0

Signed Integers

How do we do this? First way is signed magnitude – this uses the first

bit as a sign. 0-positive, 1 – negative The remaining bits are interpreted as the magnitude

of the value

For example, -4 for 5 bits is (10100)2 and 4 is

(00100)2

Signed Integers

The second approach flips all the bits. Thus for (4)10 = (00100)2 we have that -4 is

represented by

(11011)2

This is called ones complement.

Signed Integers

The final approach is to use twos complement for the representation.

That is, we compute the ones complement and then add 1.

For example, we have (-4)10 is (11100)2

Signed IntegersRepresentation Signed

Magnitude1’s Complement

2’s Complement

0000 0 0 0

0001 1 1 1

….

0111 7 7 7

1000 -0 -7 -8

1001 -1 -6 -7

1010 -2 -5 -6

1011 -3 -4 -5

1100 -4 -3 -4

1101 -5 -2 -3

1110 -6 -1 -2

1111 -7 -0 -1

Signed Integers

Now, we can use any, but only two’s complements works with arithmetic unchanged from unsigned integers.

In addition, in two’s complement has only one representation for zero.

Signed Integers

Consider 4+-2=2 in each representation and using normal arithmetic– Signed Magnitude: 0100 + 1010 = 1110 = (-6)10

– Ones Complement: 0100 + 1101 = 0001 = (1)10

– Twos Complement: 0100 + 1110 = 0010 = (2)10

Signed Integers

Thus, for simplicity of hardware we use the two’s complement representation

How do we compute two’s complement?1. Write the magnitude of the number in binary

2. Compute the one’s complement, that is flip the bits

3. Add one to the result

Signed Integers

For example, for -9 in 6 bits we have:

(9)10 = (001001)2

One’s complement: (110110)2

Adding one: (-9)10 = (110111)2

Signed Integers

How do we reverse two’s complement? We apply it again! That is

two’s complement(two’s complement(N)) = N

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