Digital Arithmetic: Operations and Circuits discusses binary addition, subtraction, multiplication, and division. It also covers different systems for representing signed numbers, including sign-magnitude, 1's complement, and 2's complement. Key topics include performing arithmetic using the 2's complement system, detecting overflow, and representing decimal values in binary coded decimal. The document provides examples and review questions to illustrate binary arithmetic concepts.

1. Digital Arithmetic:
Operations and
Circuits

2. Objectives
• Addition, subtraction, multiplication, and division
of two binary numbers.
• Addition and subtraction of hexadecimal numbers
• Difference between binary addition and OR addition.
• Comparison among three different systems for
representing signed numbers.
• Manipulate signed binary numbers using the 2’s –
complement system.
• BCD adder circuit and addition process.
• Basic operation of an arithmetic/logic unit

3. Binary Addition
• The addition of two binary numbers is performed in
exactly the same manner as the addition of decimal
numbers.
• Least-significant-digit first.
• “Carry” of 1 into the next position may be needed.
• 4 different cases for binary addition
position
next
into
1
of
carry
1
11
1
1
1
position
next
into
1
of
carry
0
10
1
1
1
0
1
0
0
0













•The operations of subtraction, multiplication, and division
actually use only addition as their basic operation

4. Example
1001(9)
110(6)
(3)
011

)
24
(
11000
1111(15)
1001(9)


5. Review Question
• Add the following pairs of binary numbers.
– 10110+00111
– 10001111+00000001

6. Representing signed numbers
Considerations: representing both positive
and negative numbers; efficient computation.
Sign-magnitude system

7. Sign-magnitude system: represents + and –
numbers, but we need to design special circuits
to add positive and negative numbers.
ex: 5 + (-5) = 0
use 8 bit signed-binary numbers and add using
full-adder circuits :
0 0 0 0 0 1 0 1 ( 5)
+ 1 0 0 0 0 1 0 1 (-5)
_____________________
1 0 0 0 1 0 1 0 (-10)

8. 1’s complement system
– Change each 0 to 1, and each 1 to 0.
– Example
(45) 1 0 1 1 0 1 original binary number
     
(-45) 0 1 0 0 1 0 complement each bit
1 0 1 1 0 1 (45)
+ 0 1 0 0 1 0 (-45)
____________________
1 1 1 1 1 1
Add one to this result, get zero.

9. 2’s complement of a binary number:
– Take the 1’s complement of the number
– Add 1 to the least-significant-bit position
number
binary
original
of
complement
s
2'
010011
complement
s
2'
form
to
1
add
1
complement
s
1'
form
bit to
each
complement
010010
45
of
equivalent
binary
101101


10. Representing signed numbers
using 2’s complement form
• If the number is positive, the magnitude is
represented in its positional-weighted binary form,
and a sign bit of 0 is placed in front of the MSB.
• If the number is negative, the magnitude is
represented in its 2’s complement form, and a sign
bit of 1 is placed in front of the MSB.

11. example

12. Example
• Represent each of the following signed decimal
numbers as a signed binary number in the 2’s-
complement system. Use a total of five bits including
the sign bit.
(a) +13 (b) –9 (c) +3 (d) –2 (e) -8

13. Negation
• Negation is the operation of converting a postive
number to its negative equivalent or a negative
number to its positive equivalent.
• We negate a signed binary number by 2’s-complementing
it.
• Example
– Each of the following numbers is a signed binary
number in the 2’s-complement system. Determine the
decimal value in each case:
(a)01100 (b) 11010 (c)10001

14. Special case in 2’s-complement
representation
• Whenever a signed number has a 1 in the sign bit and
all 0s for the magnitude bits, its decimal equivalent
is –2N, where N is the number of bits in the
magnitude.
• The complete range of values that can be represented
in the 2’s-complement system having N magnitude bits
is –2N to +(2N - 1).
• What is the range of unsigned decimal values that can
be represented in a byte?
• What is the range of signed decimal values that can
be represented in a byte?

15. Review Questions
• Represent each of the following values as an eight-bit
signed number in the 2’s-complement system
(a)+13 (b) –7 (c)-128
• Each of the following is a signed binary number in the 2’s-
complement system. Determine the decimal equivalent for
each.
(a)100011 (b)1000000 (c)01111110
• What range of signed decimal values can be represented in
12 bits(including the sign bit)?
• How many bits are required to represent decimal values
ranging from –50 to +50?
• What is the largest negative decimal value that can be
represented by a two-byte number?
• Perform the 2’s-complement operation on each of the
following.
(a)10000 (b)10000000 ©1000
• Define the negation operation

16. Addition in the 2’s-complement
system
• Case I: Two Postive Numbers.
+9  0 1001 (augend)
+4  0 0100 (addend)
0 1101 (sum = +13)
Sign bits

17. Addition, cont.
• Case II: Positive Number and Smaller Negative Number
+9  0 1001 (augend)
-4  1 1100 (addend)
1 0 0101
Sign bits
This carry is disregarded; the result
is 01001(sum=+5)

18. Addition, cont.
• Case III: Positive Number and Larger Negative Number
-9  10111
+4  00100
11011 (sum = -5)
Negative sign bit

19. Addition, cont.
• Case IV: two negative Numbers
-9  10111
-4  11100
1 10011
Sign bit
This carry is disregarded; the result is
10011(sum =-13)

20. Addition, cont.
• Case V: Equal and Opposite Numbers
-9  10111
+9  01001
0 100000
Disregard; the result is
00000(sum = +0)

21. Review Questions
• Assume the 2’s complement system for both questions
– True or False: Whenever the sum of two signed
binary numbers has a sign bit of 1, the magnitude
of the sum is in 2’s complement form.
– Add the following pairs of signed numbers. Express
the sum as a signed binary number and as a decimal
number
• (a) 100111+111011
• (b) 100111+011001

22. Subtraction in the 2’s-
complement System
• The procedure for subtracting one binary number(the
subtrahend) from another binary number(the minuend)
– Negate the subtrahend. This will change the
subtrahend to its equivalent value of opposite
sign.
– Add this to the minuend. The result of this
addition will represent the difference between the
subtrahend and the minuend.

23. Arithmetic Overflow
• When two positive or two negative numbers are being
added, an overflow could occur if there is a carry
happening to the sign-bit position.
• Overflow can occur when the minuend and subtrahend
have different signs.

24. Review Questions
• Perform the subtraction on the following pairs of
signed numbers using the 2’s-complement system.
Express the results as signed binary numbers and as
decimal values.
(a)01001-11010 (b)10010-10011
• How can arithmetic overflow be detected when signed
numbers are being added? Subtracted?

25. Multiplication of Binary
numbers
• The same manner as the multiplication of decimal
numbers.
1001  multiplicand = 910
1011  multiplier=1110
1001
1001
0000
1001
1100011 final product = 9910

26. Multiplication in the 2’s-
Complement System
• If the two numbers to be multiplied are positive,
they are already in true binary form and are
multiplied as they are.
• When the two numbers are negative, they will be in
2’s-complement form. Each is converted to a positive
number, and then the two numbers are multiplied. The
product is kept as a positive number and is given a
sign bit of 0.
• When one of the number is positive and the other is
negative, the negative number is first converted to a
positive magnitude by taking its 2’s complement. The
product will be in true-magnitude form, should be
changed to 2’s complement form and given a sign bit
of 1.

27. Review Question
• Multiply the unsigned numbers 0111 and 1110

28. Binary Division
• The same as for decimal numbers---long division
0
11
0011
011
1001
0010
11
0
100
100
100
1
.
0010
0
.
1010
100
The division of signed numbers is handled
in the same way as multiplication.

29. 1.7 Binary Codes
• BCD Code
– A number with k decimal
digits will require 4k bits
in BCD.
– Decimal 396 is represented
in BCD with 12bits as 0011
1001 0110, with each group
of 4 bits representing one
decimal digit.
– A decimal number in BCD is
the same as its equivalent
binary number only when the
number is between 0 and 9.
– The binary combinations 1010
through 1111 are not used
and have no meaning in BCD.

30. Binary Code
• Example:
– Consider decimal 185 and its corresponding value
in BCD and binary:
• BCD addition

31. Binary Code
• Example:
– Consider the addition of 184 + 576 = 760 in BCD:
• Decimal Arithmetic: (+375) + (-240) = +135
Hint 6: using 10’s of BCD

32. Binary Codes
• Other Decimal Codes

33. Binary Codes)
• Gray Code
– The advantage is that only
bit in the code group
changes in going from one
number to the next.
• Error detection.
• Representation of analog
data.
• Low power design.
000 001
010
100
110 111
101
011
1-1 and onto!!

34. Binary Codes
• American Standard Code for Information
Interchange (ASCII) Character Code

35. Binary Codes
• ASCII Character Code

36. ASCII Character Codes
• American Standard Code for Information Interchange
(Refer to Table 1.7)
• A popular code used to represent information sent as
character-based data.
• It uses 7-bits to represent:
– 94 Graphic printing characters.
– 34 Non-printing characters.
• Some non-printing characters are used for text format
(e.g. BS = Backspace, CR = carriage return).
• Other non-printing characters are used for record
marking and flow control (e.g. STX and ETX start and
end text areas).

37. ASCII Properties
• ASCII has some interesting properties:
– Digits 0 to 9 span Hexadecimal values 3016 to 3916
– Upper case A-Z span 4116 to 5A16
– Lower case a-z span 6116 to 7A16
• Lower to upper case translation (and vice versa)
occurs by flipping bit 6.

38. Binary Codes
• Error-Detecting Code
– To detect errors in data communication and
processing, an eighth bit is sometimes added to
the ASCII character to indicate its parity.
– A parity bit is an extra bit included with a
message to make the total number of 1's either
even or odd.
• Example:
– Consider the following two characters and their
even and odd parity:

39. Binary Codes
• Error-Detecting Code
– Redundancy (e.g. extra information), in the form of
extra bits, can be incorporated into binary code
words to detect and correct errors.
– A simple form of redundancy is parity, an extra bit
appended onto the code word to make the number of
1’s odd or even. Parity can detect all single-bit
errors and some multiple-bit errors.
– A code word has even parity if the number of 1’s in
the code word is even.
– A code word has odd parity if the number of 1’s in
the code word is odd.
– Example:
10001001
10001001
1
0 (odd parity)
Message B:
Message A: (even parity)