1. Lecture 4: Digital Systems & Binary Numbers (4)
CPE 201 Digital Design
Lecture 4:
Digital Systems & Binary Numbers (4)

2. Lecture Outline
Signed numbers
Encoding techniques

3. Signed Binary Numbers Cannot use the “minus” sign we use on paper
Signed-magnitude
MSB keeps the sign: 0 – positive, 1 – negative
Hard to use with arithmetic operations
The user determines if the number is signed/not
(unsigned binary) or +9 (signed binary)
11001
Other examples:
= =
= =
= = -0
Signed complement used for computer arithmetic
Signed 1’s complement and signed 2’s complement
25 (unsigned binary)
or -9 (signed binary)

4. Signed Complement Negative number is indicated by its complement
Positive numbers start with 0  complement will always start with 1 (negative number)
2’s complement is most common
E.g.: represent -9 on eight bits
Represent +9:
Signed magnitude:
Signed 1’s complement:
Signed 2’s complement:

5. Signed Binary Numbers Decimal Signed 2’s compl. Signed 1’s compl.
Signed mag. +7 +6 +5 +4 +3 +2 +1 +0 -0 -1 -2 -3 -4 -5 -6 -7 -8
0111
0111
0111
0110
0110
0110
0101
0101
0101
0100
0100
0100
0011
0011
0011
0010
0010
0010
0001
0001
0001
0000
0000
0000 -
1111
1000
1111
1110
1001
1110
1101
1010
1101
1100
1011
1100
1011
1100
1011
1010
1101
1010
1001
1110
1001
1000
1111
1000 - -

6. Addition of Signed Numbers
Signed magnitude
Just like regular arithmetic
Same signs: add magnitudes
Different signs: subtract smaller magnitude from larger, keep the sign of the larger magnitude
Requires comparison and subtraction
Complex circuit design
Better to use signed complement representation
= + 68
(-44) = - 20

7. Addition of Signed Numbers
Use 2’s complement:
Represent negative numbers in 2’s complement
Perform addition as usual
MUST USE THE SAME NUMBER OF BITS FOR BOTH NUMBERS AND THE SUM

8. Addition of Signed Numbers
Signed 2’s complement: 3+4, -2+(-6), 6+(-3), 4+(-7) +3 + -2 +
+ +4
+ -6 +7 -8 1
Discard carry +6 + +4 +
+ -3
+ -7 +3 -3
Discard carry

9. Overflow
Addition can result in overflow if there isn’t enough space available to represent the result
Can occur only if operands have the same sign
Overflow occurs when the carry bits into and out of the sign position are different
E.g.: + +7
+ +7
+14
= -2 !! 1
carry into
sign bit
carry out of + -3
+ -6 -9 1
carry into
sign bit
carry out of
= 7 !!

10. Subtraction with Signed 2’s Complement
Take the 2’s complement of the subtrahend (including the sign bit)
Add it to the minuend (including the sign bit)
Ignore any carry out of the sign-bit
Make sure to check for overflow
A – (+B) = A + (-B)
A – (-B) = A + (+B)
Equivalent to taking
the 2’s complement

11. Examples: Subtraction Signed 2’s Complement
4 – 3 = 0100 – 0011 = = 10001
3 – 4 = 0011 – 0100 = = 1111
1111 is equal to -1 in 2’s complement
3 – (-4) = 0011 – (-(0100))in 2’s complement =
= 0011 – 1100 = = 0111 = 710
-3 – (-4) = 1101 – 1100 = = 10001
= 110
Discard
the carry
Instead add 2’s complement
Discard
the carry

12. Binary Codes Is binary the only way to encode information?
Same information can be coded using a different code
The encoding still uses 0’s and 1’s
E.g.: represent the decimal values 1-9:
Binary: 0001  1001
One out of 9: , , ,
… ,
A code is a set of n-bit strings, which assigns a unique string to any symbol

13. Binary Coded Decimal (BCD) Code
In BCD, each decimal digit is represented with its binary encoding
Easier for people to understand than binary
How many bits do we need to represent numbers 0 through 9 in BCD?
4 bits
E.g.: (527)10 = ( )BCD
A number with k digits in decimal requires
4k bits in BCD
Do not confuse with binary representation! 5 2 7

14. Addition using BCD code
Similar to adding 4-bit unsigned binary numbers
A correction must be made if the result exceeds 1001 (9)
Addition result range is 0 to 19 (9+9+1carry)
Binary: 0000 to 10011
BCD: to
The difference in result between the BCD and binary values is 6 (110)
If the result is  1010 (10 decimal), adding 6 (110) to the result will correct the problem

15. BCD Addition Example 5 + 9 , 8 + 8, 9 + 9 Results:
5 + 9 = = 1110 (need correction) add 110: = (1410)
8 + 8 = = (need correction)  add 110: = (1610)
9 + 9 = = (need correction)  add 110: = (1810)

16. Readings
Chapter 2
Sections 1.7