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REPRESENTING POSITIVE AND NEGATIVE NUMBERS
1998 Morgan Kaufmann Publishers
NEGATIVE NUMBERS

The sign (+/-) can be represented using an
additional bit known as the sign bit.

There are several different methods for
encoding the sign.

The simplest method is known as signmagnitude representation.
1998 Morgan Kaufmann Publishers
SIGN-MAGNITUDE REPRESENTATION
Let the high-order bit serve as the sign bit.
 A positive value has a sign bit of 0.
 A negative value has a sign bit of 1.


Can represent values from -(2(n-1)-1) to +(2(n-1)-1)

Exercise: signed-magnitude values for the integers
-7 to +7…
1998 Morgan Kaufmann Publishers
SIGN-MAGNITUDE
REPRESENTATION: 4 BITS
Decimal Signed magnitude
+7
0111b
+6
0110b
+5
0101b
+4
0100b
+3
0011b
+2
0010b
+1
0001b
+0
0000b
-0
1000b
-1
1001b
-2
1010b
-3
1011b
-4
1100b
-5
1101b
-6
1110b
-7
1111b
There are two possible
representations for zero.
Negative values have
sign bit set to 1
1998 Morgan Kaufmann Publishers
EXERCISE

2.11 What is the 8-bit signed-magnitude binary
representation for each of the following decimal
numbers?
(a) 23

(b) -23

(c) -48

1998 Morgan Kaufmann Publishers
See exercise 2.11 on page 39 of Computer Architecture by N. Carter
EXERCISE: SOLUTION


2.11 What is the 8-bit signed-magnitude binary representation for
each of the following decimal numbers?
(a) 23
00010111b

(b) -23
10010111b

(c) -48
10110000b
The sign bit is highlighted in red
1998 Morgan Kaufmann Publishers
See exercise 2.11 on page 39 of Computer Architecture by N. Carter
SIGN-MAGNITUDE REPRESENTATION

Can negate a number simply by inverting the sign bit.

Test if a value is positive or negative by checking the sign bit.

Easy to perform multiplication or division






Just perform unsigned multiplication or division
Set sign bit of the result based on sign bits of operands
positive x positive = positive
positive x negative = negative
negative x negative = positive
Addition and subtraction present a very difficult problem.
1998 Morgan Kaufmann Publishers
SIGN-MAGNITUDE REPRESENTATION

Multiply the numbers +7 and -5 using 6-bit signedmagnitude representation.


+7 = 000111b
-5 = 100101b

7x5 = 0100011b

positive x negative = negative so set the sign bit

Solution:
Answer = 1100011b

1998 Morgan Kaufmann Publishers
SIGN-MAGNITUDE REPRESENTATION

Try to directly add 8-bit signed-magnitude values for +10 and -4.
Solution:
+10 = 00001010b
-4 = 10000100b
00001010b
+ 10000100b
10001110b

The sum is 10001110b, which is interpreted as -14 (Wrong!)
1998 Morgan Kaufmann Publishers
SIGN-MAGNITUDE REPRESENTATION

A better solution:

Try a different representation for signed binary numbers?
1998 Morgan Kaufmann Publishers
PROPOSED SOLUTION: INVERT THE BITS
01010 = +9
10101 = -9
Leftmost bit is 1 indicating negative
Does addition of 9 + -9 give zero?
No?
Let’s try something different…
1998 Morgan Kaufmann Publishers
TWO’S COMPLEMENT REPRESENTATION

To obtain a two’s complement representation of a negative
number...
(1)
Find the unsigned binary integer representation
(2) Invert each bit
(3) Add 1 to the result
(4) Discard any overflow bits

Two’s complement for a positive number is the same as its
unsigned binary representation
1998 Morgan Kaufmann Publishers
EXAMPLE: TWO’S COMPLEMENT REPRESENTATION

What is the 6-bit two’s complement representation of -12?

Solution:
(1) unsigned 12
(2) invert bits
(3) add 1

-12 = 110100 in 6-bit two’s complement

As with signed magnitude, a 1 in the leftmost bit means
negative!



001100
110011
110100
1998 Morgan Kaufmann Publishers
TWO’S COMPLEMENT REPRESENTATION

What is the result of adding +12 and -12 in 6-bit two’s
complement?

Solution:
+12
001100
-12
+110100
----------------------------------------1 000000Discard the 7th (leftmost) overflow bit
So our answer is 0.
12 + -12 = 0. Now that’s much better.
1998 Morgan Kaufmann Publishers
RANGE OF REPRESENTED VALUES

Four-bit two’s complement: Write 0-7, Fill-in -1, -2, … -8

0 = 0000
1 = 0001
2 = 0010
3 = 0011
4 = 0100
5 = 0101
6 = 0110
7 = 0111

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
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

-8 = 1000
-7 = 1001
-6 = 1010
-5 = 1011
-4 = 1100
-3 = 1101
-2 = 1110
-1 = 1111
There is only one representation for zero
Range of values is -8 to +7 inclusive
-2^(n-1) to +(2^(n-1))-1
1998 Morgan Kaufmann Publishers
WHAT ARE MAX_INT AND MIN_INT IN 16 BITS?
1998 Morgan Kaufmann Publishers
WHAT ARE MAX_INT AND MIN_INT IN 16 BITS?
1998 Morgan Kaufmann Publishers
EXERCISES


2.12 What is the 8-bit two’s complement binary representation for the following
decimal numbers?
(a) 23
Answer:

(b) -23
Answer:

(c ) 57
Answer:

(d) -57
Answer:
Same as unsigned representation
1998 Morgan Kaufmann Publishers
See exercise 2.12 on page 39 of Computer Architecture by N. Carter
EXERCISES

2.12 What is the 8-bit two’s complement binary representation for the following
decimal numbers?

(a) 23
Answer: 00010111b Same as unsigned representation

(b) -23
Answer: 11101001b

(c ) 57
Answer: 00111001b

(d) -57
Answer: 11000111b
1998 Morgan Kaufmann Publishers
See exercise 2.12 on page 39 of Computer Architecture by N. Carter
NEGATION IN TWO’S COMPLEMENT

Given a binary number in two’s complement, form its
negative by inverting its bits and adding one.

This works regardless of whether the original two’s
complement binary number is positive or negative
(leftmost bit is 1).

Negate the 4-bit two’s complement representation of
+5 twice.



Begin with the two’s complement representation of +5
Negate it, this is the representation of -5
Negate it again, you should be back to +5
1998 Morgan Kaufmann Publishers
FROM TWO’S COMPLEMENT TO BASE TEN

(a) What is the base ten equivalent for this 6-bit two’s complement
value?
011001
Answer: 25

Sign bit is 0, read its value the usual way
(b) What is the decimal equivalent of this 6-bit two’s complement
value?
100011
Answer:
011100
+ 000001
011101
Sign bit is 1 so it’s a negative value.
How do we find its absolute value (magnitude)?
Negate it by inverting bits and adding 1
Magnitude is 29 but negative sign means ->
-29
1998 Morgan Kaufmann Publishers
EXERCISES

Convert these 5-bit two’s complement values into decimal.

(a) 11011
Answer:

(b) 10001
Answer:

(c) 11111
Answer:

(d ) 10000
Answer:
1998 Morgan Kaufmann Publishers
EXERCISES

Convert these 5-bit two’s complement values into decimal.

(a) 11011
Answer: -5

(b) 10001
Answer: -15

(c) 11111
Answer: -1

(d ) 10000
Answer: -16
1998 Morgan Kaufmann Publishers
ADDITION IN TWO’S COMPLEMENT

Addition is correctly computed by directly
adding the bits.

Compute X-Y by computing X + (-Y) using
negation of Y.
1998 Morgan Kaufmann Publishers
EXERCISES

Add the values +3 and -4 in two’s complement
notation using 4 bits.

Compute -3 - 4 in two’s complement notation
using 5-bit numbers.
1998 Morgan Kaufmann Publishers
EXERCISES: SOLUTION

Negate the 4-bit two’s complement representation of +5 twice.
0101 (+5)  1010 + 0001 = 1011 (-5)  0100 + 0001 = 0101 (+5)
Negating twice results in the original number as we would expect

Add the values +3 and -4 in two’s complement notation.
0011 (+3)
0100 (+4)  1011 + 0001 = 1100 (-4)
+1100 (-4)
1111 (-1)

Compute -3 - 4 in two’s complement notation.
11101 (-3)
+11100 (-4)
11001 (-7)
11001  00110 + 00001 = 00111 (+7)
 Means to negate the two’s complement value
1998 Morgan Kaufmann Publishers
USEFUL PROPERTIES OF TWO’S COMPLEMENT

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
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Sign is determined by examining the high-order bit.
Negating a number twice results in the original number.
Addition is correctly computed by directly adding the bits.
Compute X-Y by computing X+(-Y) using negation of Y.
Represents values in range -(2(n-1)) to +(2(n-1)-1)
Only one representation for zero.
Due to these properties, two’s complement notation is used
in all modern computers.
1998 Morgan Kaufmann Publishers
PROPERTIES OF TWO’S COMPLEMENT

Multiplication in two’s complement is more complex than with signedmagnitude notation.

Addition and subtraction in two’s complement is easier than with
signed-magnitude notation.

Why would two’s complement still be favored?

Answer: Addition and subtraction operations tend to be more
frequent than multiplication. Designers choose the fastest way to
perform the more frequent computation.
1998 Morgan Kaufmann Publishers
SIGN EXTENSION

Given two binary numbers of differing numbers of bits.

Example: Four bit number and Eight bit number

1001b and 01101100b

Sometimes necessary to convert the shorter number to
the same number of bits as the larger number before
doing arithmetic

If the two numbers are unsigned, then simply append
extra 0 bits to the high-order end of the shorter number

Example: Append an extra four 0 bits to extend to 8 bits

0000  1001b = 00001001b
1998 Morgan Kaufmann Publishers
SIGN EXTENSION FOR SIGN-MAGNITUDE
Given a binary number expressed in sign-magnitude format
 Extend the number by appending the extra bits set to 0
 Copy the given sign bit into the high-order bit position
 Set the former sign bit to 0

Example: Sign extend the 4-bit sign-magnitude number
1001b to 8 bits. This is -1 (base 10) in sign
magnitude.
1001b
1000 1001b
Copy original sign bit
into new high-order bit
10000001b
Clear former sign bit
10000001b (-1 in sign magnitude)
1998 Morgan Kaufmann Publishers
SIGN EXTENSION FOR SIGN-MAGNITUDE

Problem: What is the 16-bit sign-magnitude
representation of the 8-bit sign-magnitude 10000111 (7)?

Solution:
00000000 10000111 // Preppend extra 0 bits
10000000 00000111 // Move original sign bit into new
// high-order bit position
1000000000000111
1998 Morgan Kaufmann Publishers
SIGN EXTENSION FOR TWO’S COMPLEMENT

Given 2’s complement binary number



Extend the number by appending the extra bits
Set all of the extra bits to the original sign bit value
Example: Sign-extend the 4-bit two’s complement number
1001b to 8 bits.
1001b
11111001b
1998 Morgan Kaufmann Publishers
SIGN EXTENSION FOR TWO’S COMPLEMENT

Problem: What is the 16-bit sign extension of the 8-bit
two’s complement value 10010010 (-110)?

Solution:
00000000 10010010 // Preppend 8 extra bits
11111111 10010010 // Set extra bits to original sign
// bit value
11111111 10010010
1998 Morgan Kaufmann Publishers