1. Introduction
The term digital is derived from the way computers perform operation, by counting digits.
Application of digital technology: television, communication systems, radar, navigation and guidance system, military system, medical instrument, industrial process control and etc.

2. Digital and Analog Quantities
Analog quantities have continuous values
Digital quantities have discrete sets of values
Analog : a quantity represented by voltage, current or meter movement that is proportional to the value of that quantity
Digital : the quantities are represented not by proportional quantities but by symbols called digits

3. Digital and Analog Quantities
Digital quantities have discrete sets of values
Analog quantities have continuous values

4. Advantages of Digital Systems Over Analog Systems
More reliable than analog systems due to better immunity to noise & better accuracy.
Ease of design: No special math skills needed to visualize the behavior of small digital (logic) circuits.
Programmability.
Speed: A digital logic element can produce an output in less than 10 nanoseconds (10-8 seconds).
Economy: Due to the integration of millions of digital logic elements on a single miniature chip forming low cost integrated circuit (ICs).

5. Binary Digits
The conventional numbering system uses ten digits: 0,1,2,3,4,5,6,7,8, and 9.
The binary numbering system uses just two digits: 0 and 1.
The two binary digits are designated 0 and 1
They can also be called LOW and HIGH, where LOW = 0 and HIGH = 1

6. Logic Level

7. Number Systems, Operations and Codes

8. Introduction
The binary system and digital codes are fundamental to computers and to digital electronic in general.
The binary number system such as decimal, hexadecimal and octal is presented.
Arithmetic operations with binary numbers are covered to provide a basis for understanding how computers and many other types of digital systems work.

9. Number System Decimal Binary Octal Hexadecimal 0 ~ 9 0 ~ 1 0 ~ 7 0 ~ F

10. Decimal Numbers
In decimal number system each of the ten digits, 0 through 9
Example:
Express the decimal number as a sum of the values of each digit

11. Binary Numbers
The binary number system is another way to represent quantities. There are 1 (HIGH) and 0 (LOW)
The binary numbering system has a base of 2 with each position weighted by a factor of 2:

12. Example Convert the binary whole number 1101101 to decimal Try This:
Solution:
Weight:
Binary : =
= = 109

13. Decimal Number
Binary Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

14. Illustration of a Simple Binary Counting Application
An Application
Illustration of a Simple Binary Counting Application

15. Decimal-to-Binary Conversion
Sum-of-Weight Method
Repeated Division by 2 Method
Converting Decimal Fractions to Binary

16. Sum-of-Weight Method 1 0 0 1 Example:
Example:
Convert the following decimal numbers to binary:
a) 12 b) 25 c) 58 d) 82
The decimal number 9, for example, can be expressed as the sum of binary weight of:

17. Repeated Division by 2 Method
A systematic method of converting whole numbers from decimal to binary is the repeated division-by-2 process.
Remainder
Example
Convert the decimal number 12 to binary
MSB
LSB
Stop when the whole-number quotient is 0
Convert decimal number 39 to binary?

18. Converting Decimal Fractions to Binary
Sum-of-Weight
0.625 = = = 0.101
Repeated Multiplication by 2
MSB
LSB
Carry
0.625 x 2 = 1.25
0.25 x 2 = 0.50
0.50 x 2 = 1.00 1
Stop when the fractional part is all zeros

19. Binary Arithmetic
Binary arithmetic is essential in all digital computers and in many other types of digital systems.
Addition, Subtraction, Multiplication, and Division

20. Binary Addition Example:
The four basic rules for adding binary digits (bits) are as follows:
0 + 0 = 0 sum of 0 with a carry of 0
0 + 1 = 1 sum of 1 with a carry 0f 0
1 + 0 = 1 sum of 1 with a carry of 0
1+ 1 = 10 sum of 0 with a carry 0f 1
1 1
0 1 1
1 0 0
Carry
Example:
Try This:
= ??

21. Binary Subtraction Example:
The four basic rules for subtracting bits are as follows:
0 – 0 = 0
1 – 1 = 0
1 – 0 = 1
10 – 1 = – 1 with a borrow of 1
1 1 – 0 1 = ??
1 1
0 1
1 0
Example:
Try This:
1 0 1 – = ???

22. Binary Multiplication
The four basic rules for multiplying bits are as follows:
0 X 0 = 0
0 X 1 = 0
1 X 0 = 0
1 X 1 = 1
1 1 X 1 1 = ??
1 1
X 1 1
+1 1
Try This:
1 1 1 X = ??
Example:

23. Binary Division Example:
Division in binary follows the same procedure as division in decimal.
1 1 0 ÷ 11 = ??
1 0
1 1
0 0 0
Example:
Try This:
1 1 0 ÷ 10 = ??

24. 1’s and 2’s Complements of Binary Numbers
The 1’s and 2’s Complements of Binary Numbers are very important because they permit the representation of negative numbers.
The method of 2’s compliment arithmetic is commonly used in computers to handle negative numbers

25. Finding the 1’s Complement
The 1’s complement of a binary number is found by changing all 1s to 0s and all 0s to 1s.
Example:
(Binary Number)
(1’s Complement)
NOT Gate

26. Finding the 2’s Complement
The 2’s complement of a binary number is found by adding 1 to the LSB of the 1’s complement
2's Complement = (1's Complement) + 1
Find the 2’s complement of
(Binary number)
(1’s complement)
1 (Add 1)
Example:

27. Alternative Method to find 2’s Complement
Start at the right with the LSB and write the bits as they are up and including the first 1
Take the 1’s complements of the remaining bits
(Binary Number)
(2’s Complement)
Try This:
Example:
These bits stay the same
1’s Complements of original bits

28. Signed Numbers
Digital systems, such as the computer, must be able to handle both positive and negative numbers. A signed binary number consists of both sign and magnitude information. The sign indicates whether a number is positive or negative and the magnitude is the value of the number. There three forms in which signed integer (whole) numbers can be represented in binary:
Sign-Magnitude
1’s Complement
2’s Complement

29. 0 = Positive Number and 1 = Negative Number
The Sign Bit
The left-most bit in a signed binary number is the sign bit, which tells you whether the number is positive or negative.
0 = Positive Number and 1 = Negative Number
Sign-Magnitude Form
When a signed binary number is represented in sign-magnitude, the left-most bit is the sign bit and the remaining bits are the magnitude bits. The magnitude bits are in true (uncomplemented) binary for both positive and negative numbers.
Decimal number, +25 is expressed as an 8-bit signed binary number using sign-magnitude form as:
Example:
Magnitude Bit
Sign Bit

30. 1’s Complement Form
Positive numbers in 1’s complement form are represented the same way as the positive sign-magnitude numbers. Negative numbers, however, are the 1’s complements of the corresponding positive numbers. Example: The decimal number -25 is expressed as the 1’s complement of +25 ( ) as ( )
2’s Complement Form
In the 2’s complement form, a negative number is the 2’s complement of the corresponding positive number

31. Example: Sign-Magnitude: 1's Complement: 2's Complement: 00100111
Express the decimal number -39 in sign-magnitude, 1’s complement and 2’s complement
Sign-Magnitude:
>>>
1's Complement:
>>>
2's Complement:
>>>

32. The Decimal Value of Signed Numbers
Decimal Value of positive and negative numbers in the sign-magnitude form are determined by summing the weights in all the magnitude bit positions where there are 1s and ignoring those positions where there are zeros.
Sign-Magnitude:
Determine the decimal value of this signed binary number expressed in sign magnitude:
Example:
>> = 21
The sign bit is 1: Therefore, the decimal number is -21

33. The Decimal Value of Signed Numbers
1’s Complement:
Decimal values of negative numbers are determined by assigning a negative value to the weight of the sign bit, summing all the weight where there are 1s and adding 1 to the result
Determine the decimal values of this signed binary numbers expressed in 1’s complement
Example:
= +23
= = -23

34. The Decimal Value of Signed Numbers
The weight of the sign bit in a negative number is given a negative value
2’s Complement:
Determine the decimal values of this signed binary numbers expressed in 1’s complement
Example:
= +86
= -86

35. Arithmetic Operations with Signed Number
In this section we will learn how signed numbers are added, subtracted, multiplied and divided. This section will cover only on the 2’s complement arithmetic, because, it widely used in computers and microprocessor-based system .

36. Addition 0 0 0 0 0 1 1 1 +0 0 0 0 0 1 0 0 Both Number Positive:
7 + 4
The Sum is Positive and is therefore in true binary
Positive Number with
Magnitude Larger than
Negative Number:
Discard
Carry
15 + (-6) 1
The Final Carry is Discarded.
The Sum is Positive and is therefore in true binary

37. Addition 0 0 0 1 0 0 0 0 +1 1 1 0 1 0 0 0 Negative Number with
Negative Number with
Magnitude Larger than
Positive Number:
16 + (-24)
The Sum is Negative and is therefore in
2’s complement form
Discard
Carry
Both Number Negative:
-5 + (-9) 1
The Final Carry is Discarded.
The Sum is Negative and is therefore in
2’s complement form

38. Subtraction Example: Solution:
To subtract two signed numbers, take the 2’s Complement of the subtrahend and ADD. Discard any final carry bit
Example:
8 – 3 = 8 + (-3) = 5
Solution: +
2’s Complement 1
Discard Cary
Difference

39. Multiplication Direct Addition: Partial Product:
The numbers in a multiplication are the multiplicand, the multiplier and the product. Direct Addition and Partial Products are two basic methods for performing multiplication using addition.
Direct Addition:
8 X 3 = 24 +
= 24
(Decimal) +
Partial Product:
Standard Procedure

40. Division
The division operation in computers is accomplished using subtraction. Since subtraction is done with an adder, division can also be accomplished with an adder. The result of a division is called the quotient.
Step 1:
Determine the SIGN BIT for both DIVIDEND and DIVISOR
Step 2:
Subtract the DIVISOR from the DIVIDEND using 2’s Complement addition to get the first partial remainder and ADD 1 to quotient. If ZERO or NEGATIVE the division is complete.
Step 3:
Subtract the divisor from the partial remainder and ADD 1 to the quotient. If the result is POSITIVE repeat Step 2 or If ZERO or NEGATIVE the division is complete.

41. Hexadecimal Numbers
Most digital systems deal with groups of bits in even powers of 2 such as 8, 16, 32, and 64 bits.
Hexadecimal uses groups of 4 bits.
Base 16
16 possible symbols
0-9 and A-F
Allows for convenient handling of long binary strings.

42. Hexadecimal Numbers
Convert from hex to decimal by multiplying each hex digit by its positional weight.
Example:

43. Hexadecimal Numbers
Convert from decimal to hex by using the repeated division method used for decimal to binary and decimal to octal conversion.
Divide the decimal number by 16
The first remainder is the LSB and the last is the MSB.
Note, when done on a calculator a decimal remainder can be multiplied by 16 to get the result. If the remainder is greater than 9, the letters A through F are used.

44. Hexadecimal Numbers Example of hex to binary conversion:

45. Hexadecimal Numbers

46. Hexadecimal Numbers
Hexadecimal is useful for representing long strings of bits.
Understanding the conversion process and memorizing the 4 bit patterns for each hexadecimal digit will prove valuable later.

47. BCD
Binary Coded Decimal (BCD) is another way to present decimal numbers in binary form.
BCD is widely used and combines features of both decimal and binary systems.
Each digit is converted to a binary equivalent.

48. BCD To convert the number 87410 to BCD: 8 7 4
8 7 4
= BCD
Each decimal digit is represented using 4 bits.
Each 4-bit group can never be greater than 9.
Reverse the process to convert BCD to decimal.

49. BCD BCD is not a number system.
BCD is a decimal number with each digit encoded to its binary equivalent.
A BCD number is not the same as a straight binary number.
The primary advantage of BCD is the relative ease of converting to and from decimal.

50. Alphanumeric Codes
Represents characters and functions found on a computer keyboard.
ASCII – American Standard Code for Information Interchange.
Seven bit code: 27 = 128 possible code groups
Table 2-4 lists the standard ASCII codes
Examples of use are: to transfer information between computers, between computers and printers, and for internal storage.

51. Thank You
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