Signed numbers

Bit numbering

• bit (b) is the smallest unit of data in a computer

• Byte (B) is 8 bits

  • Nibble is 4 bits

• Word is the natural unit of data used by a particular processor design, (in our course, it is 32 bits=4 bytes)

  • Halfword is 16 bits

• Least significant bit (LSB) is the rightmost bit

• Most significant bit (MSB) is the leftmost bit

Signed numbers

Sign–magnitude

• if $0d_2{d}_3\dots d_n$ is a positive number, then $1d_2{d}_3\dots d_n$ is the negative of that number
• the most significant bit is the sign bit
• the range is $-2^{n-1}+1$ to $2^{n-1}-1$

One’s complement

• the range is $-2^{n-1}+1$ to $2^{n-1}-1$
• the negative of a number is obtained by inverting all bits
  • example: $5=0101⟹-5=1010$

Two’s complement

• the range is $-2^{n-1}$ to $2^{n-1}-1$
• If $k=0d_2{d}_3\dots d_n$ is a positive number, then we can find its negative $-k$ by:
  • invert (flip) each bit (One’s complement) and add 1 to the result:
    • example: $5=0101⟹-5=1010+1=1011$
  • subtract $k$ from $2^n$:
    • example: $5=0101,n=4⟹2^4-5=16-5=11⟹-5=1011$

signed binary numbers

4-bit

Binary	Unsigned	Two’s complement	Sign–magnitude	One’s complement
0000	0	0	0	0
0001	1	1	1	1
0010	2	2	2	2
0011	3	3	3	3
0100	4	4	4	4
0101	5	5	5	5
0110	6	6	6	6
0111	7	7	7	7
1000	8	-8	-0	-7
1001	9	-7	-1	-6
1010	10	-6	-2	-5
1011	11	-5	-3	-4
1100	12	-4	-4	-3
1101	13	-3	-5	-2
1110	14	-2	-6	-1
1111	15	-1	-7	-0

8-bit

Binary	Unsigned	Two’s complement	Sign–magnitude	One’s complement
00000000	0	0	0	0
00000001	1	1	1	1
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
01111110	126	126	126	126
01111111	127	127	127	127
10000000	128	-128	-0	-127
10000001	129	-127	-1	-126
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
11111110	254	-2	-126	-1
11111111	255	-1	-127	-0