1. Lecture 1.2 (Chapter 1) Prepared by Dr. Lamiaa Elshenawy
Logic Design (CE1111)
Lecture 1.2 (Chapter 1)
Prepared by Dr. Lamiaa Elshenawy

2. Outlines Digital Systems Binary Numbers Number-Base Conversions
Octal and Hexadecimal Numbers
Complements of Numbers
Signed Binary Numbers
Binary Codes
Binary Storage and Registers
Binary Logic

3. Complements of Numbers
Complements are used in digital computers to simplify the subtraction operation and for logical manipulation
Simplifying operations leads to simpler, less expensive circuits to implement the operations

4. Complements of Numbers
Types of complements
Radix complement (r’s complement)
Diminished radix complement ((r-1)’s complement)
Decimal Numbers
10’s complement and 9’s complement
Binary Numbers
2’s complement and 1’s complement

5. Complements of Numbers
How can we calculate r’s complement and (r-1)’s complement ?
Radix complement
𝑟 𝑛 −𝑁
Diminished radix complement
𝑟 𝑛 −1 −𝑁
Where n is number of digits

6. Complements of Numbers
Calculate the 10’s complement and 9’s complement of
Solution
10’s complement:
=453300
9’s complement:
=453299

7. Complements of Numbers
Calculate the 2’s complement and 1’s complement of
Hint
Solution
2’s complement:
1’s complement:
2’s complement: leave all least significant 0’s and first 1 unchanged, change from 0 to 1 or from 1 to 0 for all other higher significant digits
1’s complement: change from 0 to 1 or from 1 to 0

8. Signed Binary Numbers Leftmost position of the number used for sign:
Bit Positive number
Bit Negative number
Example
(+9)10 =(01001) signed-magnitude representation
(-9)10 =(11001) signed-magnitude representation
(-9)10 =(10110) signed-1’s complement representation
(-9)10 =(10111) signed-2’s complement representation

10. Arithmetic Addition Unsigned numbers: As ordinary arithmetic Example
25+12=37
13+ −30 =− 30−13 =−17
Signed numbers:
Positive numbers as ordinary arithmetic
=+37
Negative numbers
Write negative number in signed-2’s complement
Discard the end carry

11. Arithmetic Addition
Example

12. Arithmetic Subtraction
Unsigned numbers:
As ordinary arithmetic
Example
27−14=13
15− −30 = =45
Signed numbers:

13. Arithmetic Subtraction
Example
−6 − −13 =−6+13=+7
−6 10=( )2
=( ) (signed-2’s complement)
+13 10=( )2
𝟏 𝟎𝟏𝟏𝟏 =+7
Discard

14. Binary Codes What is a binary code?
An n‐bit binary code is a group of n bits that assumes up to 2 𝑛 distinct combinations of 1’s and 0’s to represent one element that is being coded.
Why we use binary codes?
Because digital systems understands only 1 or 0

15. Binary Codes What are the most common binary codes?
Binary-Coded Decimal (BCD)
2421-Code
Excess-3 Code
8,4,-2,-1 Code
Gray Code
ASCII Character Code

16. Binary-Coded Decimal Converts Decimal Numbers Binary Numbers How?
Write each decimal digits in 4bits
Example

17. Binary-Coded Decimal

18. Signed-BCD Addition Write each decimal number in BCD
Write sign in the leftmost position (0 for +, 1 for -)
Add (6) 10=(0110) 2 to the binary sum If
𝑠𝑢𝑚≥ (1010) 2
Example

21. ASCII Character Code It codes 128 Characters in7-bits
26 capital letters
26 small letters
10 decimal digits
Special characters
ASCII: American Standard Code for Information Interchange

24. Registers What is a register?
A register is a group of binary cells. A register with n cells can store any discrete quantity of information that contains n bits

25. Register Transfer

26. Information Processing

27. Binary Logic What is a binary logic?
Binary logic consists of binary variables and a set of logical operations
Variables are designated by letters of the alphabet, such as A, B, C, x, y, z, etc., with each variable having two and only two distinct possible values: 1 and 0

28. Binary Logic What are the most common logic gates? AND: 𝑥.𝑦=𝑥 𝐴𝑁𝐷 𝑦=𝑧
OR: 𝑥+𝑦=𝑥 𝑂𝑅 𝑦=𝑧
NOT: 𝑥 ′ = 𝑥 =𝑧

29. Binary Logic

30. Thank you for your attention