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Computer Organization
Integers and Arithmetic Operations
Terrence Mak
Counting to 9, oh no, 1 please
• Binary numbers (0, 1) are commonly used in
computers as they are easily represented as
on/off electrical signals
– Other related radix systems are Octal & Hexadecimal
• Different number (representation) systems are
used in computers
• Let's assume n bits (binary digits) are used in the
following discussions
Only 0 and 1 in a Sequence
• Numbers are represented as binary vectors
B = bn-1 bn-2 … b1 b0
• MSB = Most Significant Bit = bn-1
i.e. leftmost digit in a binary vector
• LSB = Least Significant Bit = b0
i.e. rightmost digit in a binary vector
Representing Non-negative Numbers
• Unsigned (non-negative) numbers are in
range 0 to 2n – 1
• Represented by
Value(B) = bn-12n-1 + … + b121 + b020
if B is a binary vector representing an unsigned integer
Representing Signed Numbers
• In written decimal system, a signed number is
"usually" represented by a "dash" or "plus"
sign and followed by the magnitude
e.g. –73, –215, +349
• In binary system, we have several choices:
– Sign-and-magnitude
– 1’s complement
– 2’s complement
Representing Signed Numbers
• Sign-and-magnitude
– MSB determines sign, remaining unsigned bits represent
magnitude
– MSB: 0 means "+", 1 means "–"
• 1’s complement
– MSB determines sign
– To change sign from unsigned to negative, invert all the bits
• 2’s complement (commonly used)
– MSB determines sign
– To change sign from unsigned to negative, invert all the bits
and then add 1
– This is equivalent to subtracting the positive number from 2n
Signed Number Systems
B (n = 4)
b 3 b 2 b1 b 0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
0
0
1
1
0
0
0
0
1
1
0
0
1
1
1
0
1
0
1
0
1
0
0
1
0
1
0
1
0
1
Values represented
Sign and
magnitude
1's complement
+7
+6
+5
+4
+3
+2
+1
+0
- 0
- 1
- 2
- 3
- 4
- 5
- 6
- 7
+7
+6
+5
+4
+3
+2
+1
+0
-7
-6
-5
-4
-3
-2
- 1
-0
2's complement
+
+
+
+
+
+
+
+
-
7
6
5
4
3
2
1
0
8
7
6
5
4
3
2
1
2’s Complement
• 2's complement numbers actually make sense since they follow
normal modulo arithmetic except when they overflow
• Range is –2(n–1) to +2(n–1) – 1
e.g. –8 to +7 for n = 4
N- 1
N-2
0
0000
1111
1
1110
2
1101
1100
-1
0010
+1
- 2
- 3
-4
-5
1011
+2
+3
0011
Overflow + 4
cut-off + 5
- 6
+6
- 7 - 8 +7
1010
1001
(a) Circle representation of integers mod N
0
0001
1000
0100
0101
0110
0111
(b) Mod 16 system for 2's-complement numbers
for n = 4, N = 2n = 24 = 16
1-bit Addition
0
+
0
0
1
+
0
1
0
+
1
1
1
+
1
10
Carry-out
n-bit Addition/ Subtraction
• X+Y
– Use 1-bit addition propagating carry to the
most significant bit
• X–Y
 X + (– Y)
– Add X to the 2’s complement of Y
4-bit Addition/ Subtraction
of numbers represented in 2’s Complement
(a)
(c)
(e)
0100
+1010
(+ 4)
( - 6)
1110
( - 2)
0111
+1101
(+7)
(- 3)
(- 7)
0100
(+4)
(- 3)
(- 7 )
1101
+0111
0010
+0011
( + 2)
( + 3)
0101
( + 5)
1011
+1110
(- 5)
(- 2)
1001
1101
- 1001
(b)
(d)
0100
(f)
0010
- 0100
(+ 2)
(+4)
(+4)
0010
+1100
1110
(- 2)
4-bit Addition/ Subtraction
of numbers represented in 2’s Complement
(g)
0110
- 0011
(+ 6 )
(+ 3 )
0110
+1101
0011
(h)
1001
- 1011
( - 7)
( - 5)
1001
+0101
1110
(i)
1001
- 0001
( -7)
( +1 )
0010
- 1101
( +2)
( - 3)
( - 2)
1001
+1111
1000
(j)
(+3)
( - 8)
0010
+0011
0101
(+ 5)
Sign Extension
• Given a 4-bit 2’s complement number
• Make it into an 8-bit 2's complement number
– Positive number
• add 0’s to LHS
• e.g. 0111  00000111
– Negative number
• add 1’s to LHS
• e.g. 1010  11111010
– c.f. circle representation, i.e. extend the MSB
Overflow
• In 2’s complement arithmetic
– Addition of opposite sign numbers never overflow
– If the numbers are the same sign and the result is the
opposite sign, overflow has occurred
• e.g. 0111 + 0100 = 1011 = –5!
• positive + positive = negative?!
• In unsigned arithmetic
– Carry out signals an overflow
Exercise
• Using 4-bit 2’s complement number system,
–
–
–
–
What is the binary for -2?
Calculate 2+3
Calculate -2-3
Calculate 5+5
• Using 5-bit number system
– What is the largest 2’s complement number?
– What is the smallest 2’s complement number?
– What is the largest unsigned number?
• Convert 56 to unsigned binary
• What is the decimal value of 10110101 in 2’s
complement?
• What is the decimal value of unsigned number
10110101?
Characters - 8-bit ASCII representation