Basic Digital Electronics - Unit 2

2

NUMBER SYSTEMS AND CODES

At the end of the lesson, students should be able to: 1. 2.

Count in the binary number system. Convert from the decimal form to binary form and from binary form to decimal form including fractions. 3. Add and subtract binary numbers including fractions. 4. Determine l’s and 2’s compliments of a binary number. 5. Express signed numbers in binary form. 6. Carry out arithmetic operations with signed binary numbers. 7. Convert between the binary and octal number systems. 8. Convert between the binary and hexadecimal number system. 9. Express decimal numbers in binary coded decimal (BCD). 10. Add BCD numbers.

1. Understand the Gray and ASCII code is used.

UNIT 2: NUMBER SYSTEMS AND CODES

2.1

COUNT IN BINARY NUMBER SYSTEM

Only have 2 digits Æ 0 and 1 (known as bits)

It is expressed as a base of 2. Weight Binary Number

24

23

22

21

20

2-1

2-2

2-3

2-4

1

0

1

0

0

1

0

1

0

MSB

2.2

LSB

BINARY AND DECIMAL FORM TO BINARY FORM

► CONVERT FROM DECIMAL TO BINARY FORM

It use repeated division-by-2

Example: Convert 1210 to binary form.

Solution: 2

12

Remain 0

2

6

0

2

3

1

2

1

1

2

0

From below

1210 = 1 1 0 0

2

1210 = 11002 Check: Weight

23

22

21

20

Binary Number

1

1

0

0

= 23 + 22 = 8 + 4 = 12

Fatma Syazana Zaini, IKM Besut, 2010

14


► CONVERT DECIMAL TO BINARY FORM INCLUDING FRACTION

Fractional binary numbers are expressed as negative powers of 2.

Multiply the decimal numbers by 2 repeatedly until the fraction become 0 or at the place you want.

Example 1: Convert 0.312510 to binary form.

Solution: Carry 0.3125 × 2

= 0.625

0

0.625

×2

= 1.25

1

0.25

×2

= 0.50

0

0.5

×2

= 1.00

1

=0.0101

Stop until the fraction become 0.

Example 2: Convert 0.634210 to binary form.

Solution: Carry 0.6342

×2

= 1.2684

1

0.2684

×2

= 0.5368

0

0.5368

×2

= 1.0736

1

0.0736

×2

= 0.1472

0

= . 1010

Stop at the place you want


15


► CONVERT FROM BINARY FORM TO DECIMAL POINT

Multiply each bit by positional weight and add the weight of all bits that are 1

Discard the weight of all bits that are 0

Example: Express 11011012 to decimal form.

Solution: Weight

26

25

24

23

22

21

20

Binary Number

1

1

0

1

1

0

1

11011012 = 26 + 25 + 23 + 22 + 21 = 64 + 32 + 8 + 4 + 1 = 109

► CONVERT FROM BINARY FORM TO DECIMAL FORM INCLUDING FRACTION

Example: 1. Convert 0.1011 to decimal point.

Solution: Weight

20

2-1

2-2

2-3

2-4

Binary Number

0.

1

0

1

1

0.10112

= (0 × 20) + (1 × 2-1) + (0 × 2-2) + (1 × 2-3) + (1 × 2-4) = (0 × 0) + (1 × 1/2) + (0 × 1/4) + (1 × 1/8) + (1 × 1/16) = 0.5 + 0.125 + 0.0625

= 0.6875


16


2. Convert 0.11001 to decimal point.

Solution: Weight

20

2-1

2-2

2-3

2-4

2-5

Binary Number

0.

1

1

0

0

1

0.110012

= 2-1 + 2-2 + 2-5 = 0.5 + 0.25 + 0.03125 = 0.78125

2.3

ADD AND SUBTRACT BINARY NUMBER INCLUDING

► ADD BINARY NUMBER

Basic rules: Carry

0+0=0

0

0+1=1

0

1+0=1

0

1+1=0

1

1 + 1 + (carry) = 1

1

Example: 1. Add 1012 to 102.

Check: 1 + 1

0

1

1

0

1

1

5 +

2 7


17


2. Add 10102 to 112.

1

0

Check: 10

1

1

+

0

1

1

1

0

+ 1

1

3 13

3. Add 11012 to 1102.

Check: 13

1

1 + 1

0

1

0

1

1

1

0

0

1

1

+

6 19

► SUBTRACT BINARY NUMBER

Case 1: 1 1

1

0

1

1

0

0

0

0

1

Check: 13 -

4 9

Case 2:

Check: 1 -

10

0

1

1

0

0

1

0

1

9 -

4 5

When borrowing 1 from left, it becomes 10. 10 – 1 = 1 (2 - 1 = 1)


18


Case 3:

1 is borrowed from this column leaving 0 When 1 is borrowed from left, it becomes 1 When 1 is borrowed from left, it becomes 102. 102 – 12 = 12 1 – 0 = 0 without borrow

0

1

1

1

1

0

0

0

10

1

Check: 33

0

1

0

1

0

-

1

0

1

1

1

-

2.4

10 23

1’S AND 2’S COMPLEMENTS OF BINARY NUMBER

1’s and 2’s compliment of binary numbers are important because they permit the representation of negative (-ve) numbers. ► 1’S COMPLEMENTS

The 1’s compliment of a binary is found by changing all 1s to 0s and all 0s to 1s. In other words, change each bits in the number to its complement. Example: Find the 1’s complement of 11012. Solution: 1

1

0

1

0 0 1 0 11012 = 00102


19


► 2’S COMPLEMENTS

The 2’s complement of a binary number is formed by taking the 1’s complement of the number and adding 1 to the least significant bit (LSB) position.

Example 1: Find the 2’s complement of 11012.

Solution: 1

1

0

1

Binary number

0

0

1

0

1’s complement

+

1 0

0

1

2’s complement

1

Example 2: Find the 2’s complement of 10011012.

Solution: 1

0

0

1

1

0

1

Binary number

0

1

1

0

0

1

0

1’s complement

1

Add 1

1

2’s complement

+ 0

1

1

0

0

1


20


2.5

EXPRESS SIGNED NUMBERS IN BINARY FORM

► THE SIGNED BITS

The left – most bit in a signed binary number is the sign bit, which tells you whether the number is positive or negative.

A 0 sign bit indicates a positive number, and a 1 sign bit indicates a negative number.

► SIGN – MAGNITUDE FORM

When a signed binary number is represented in sign – magnitude, the left – most bits is the sign bit and the remaining bits are the magnitude bits.

The magnitude bits are in true (un-complemented) binary for both positive and negative numbers.

For example, the decimal number +25 is expressed as an 8 – bit signed number using the sign – magnitude form as: 0 0 0 1 1 0 0 12 Sign bit

Magnitude bit

The decimal number – 25 is expressed as 1 0 0 1 1 0 0 12

REMEMBER: The MSB is the sign bit, which tells us weather the number is positive or negative. Sign bit 0

=

positive

Sign bit 1

=

negative


21


► 1’S COMPLEMENT

Example: Using 8 bits, express decimal number +25 in 1’s compliment. +25

: 0001 1001

binary number

1110 0110

1’s complement

In 1’s complement form, a negative number is the 1’s complement of the corresponding positive number. (dalam bentuk 1’s C, nombor negative adalah 1’s C untuk nombor positif)

► 2’S COMPLEMENT

Negative number is the 2’s complement of the corresponding positive number.

Example: Express decimal number -25 as 2’s complement.

0

0

0

1

1

0

0

1

Binary number of +25

1

1

1

0

0

1

1

0

1’s complement

1

Add 1

1

2’s complement

+ 1

1

1

0

0

1

1

In the 2’s complement form, a negative number is the 2’s complement of the corresponding positive number.


22


Example: Express the decimal number -39 as an 8 bit number in the sign-magnitude, 1’s complement and 2’s complement forms. + 39 = 0010 01112 In the sign-magnitude form -39 = 1010 01112 In 1’s complement form -39 :0 0 1 0

0 1 1 1

Binary number (+39)

1 1 0 1

1 0 0 0

1’s complement

In 2’s complementform -39 :0 0 1 0

0 1 1 1

Binary number (+39)

1 1 0 1

1 0 0 0

1’s complement

+ 1 1 0 1

2.6

Add 1

1 1 0 0 1

2’s complement (-39)

ARITHMETIC OPERATIONS WITH SIGNED NUMBERS

► ADDITION IN 2’S COMPLEMENT SYSTEM o

Case 1 :

Two positive numbers The addition of +7 and +4 0 0 0 0 0 1 1 1

(+ 7)

+ 0 0 0 0 0 1 0 0

(+ 4)

0 0 0 0 1 0 1 1

(+ 11)

If the sum is positive, therefore, in true binary


23


o

Case 2 :

Negative number < Positive number To get -6, change +6 to 2’s

The addition of +15 and -6

Complement

0 0 0 0

0 1 1 0

Binary number (+6)

1 1 1 1

1 0 0 1

1’s complement

+

1 1 1 1 1

1

1

1

1

1 0 1 0

1

Add 1 2’s complement (-6)

1

0 0 0 0 1 1 1 1

(+ 15)

+ 1 1 1 1 1 0 1 0

(- 6)

1 0 0 0 0 1 0 0 1

(+ 9)

If there is a final carry bit, then discard. The sum is positive.

o

Case 3 :

Positive number < Negative number The addition of +16 and -24 0 0 0 1

1 0 0 0

Binary number (+24)

1 1 1 0

0 1 1 1

1’s complement

+

1 1 1 1 0

1 0 0 0

Add 1 2’s complement (-24)

0 0 0 1 0 0 0 0

(+ 16)

+ 1 1 1 0 1 0 0 0

(- 24)

1 1 1 1 1 0 0 0

(- 8)


24


o

Case 4 :

Two negative number The addition of -5 and -9 -9 :-

- 5 :0 0 0 0

0 1 0 1 (+5)

0 0 0 0

1 0 0 1 (+9)

1 1 1 1

1 0 1 0

1 1 1 1

0 1 1 0

+

1 1 0 1 1 (-5)

1 1 1 1

1

1

1

1

+

1

1

1 1 1 1 1

0 1 1 1 (-9)

1

1 1 1 1 1 0 1 1

(- 5)

+ 1 1 1 1 0 1 1 1

(- 9)

1 1 1 1 1 0 0 1 0

( - 14)

The final carry bit is discarded. The sum is negative.

o

Case 5 :- Equal and opposite number The addition of -9 and +9 1

1

1

1

1

1

1

1 1 1 0 1 1 1 1

- 9 (in 2’s complement form)

+ 0 0 0 1 0 0 0 1

+ 9

1 0 0 0 0 0 0 0 0

0

The final carry bit is discarded.


25


► SUBTRACTION IN 2’S COMPLEMENT SYSTEM

Procedure: i.

Negate the subtrahend – will change the subtrahend to its equivalent value of opposite sign.

ii.

Add the negation to the minuend – the result will represent the difference between the subtrahend and minuend.

iii.

Discarded any final carry bit.

Example: Perform each of the following subtractions of the signed numbers: a) +8 – (+3) = 5 b) 12 – (-9) = 21 c) -25 – (+19) = -44 d) -120 – (-30) = -90

Solutions: a) +8 – (+3) = 5 1

1

1

b) 12 – (-9) = 21

1

1

0 0 0 0 1 0 0 0

0 0 0 0 1 1 0 0

+ 1 1 1 1 1 1 0 1

+ 0 0 0 0 1 0 0 1

1 0 0 0 0 0 1 0 1

0 0 0 1 0 1 0 1

c) -25 – (+19) = -44 1

1

1

1

d) -120 – (-30) = -90 1

1

1

1

1 1 1 0 0 1 1 1

1 0 0 0 1 0 0 0

+ 1 1 1 0 1 1 0 1

+ 0 0 0 1 1 1 1 0

1 1 1 0 1 0 1 0 0

1 0 1 0 0 1 1 0


26


2.7

OCTAL NUMBERING SYSTEM

Octal number system provides a convenient way to express binary numbers and code. However, it is used less frequently than hexadecimal in conjunction with computers and microprocessor to express binary quantities for input and output purposes.

► OCTAL – TO – BINARY CONVERSION Octal digit is represented by a 3-bit binary number. OCTAL DIGIT

BINARY

0

000

1

001

2

010

3

011

4

100

5

101

6

110

7

111 Table 1

Example: Convert each of the following octal numbers to binary. b)

a) 138

258

c)

1408

d) 72568

Solution: b)

a) 1 0 0 1

2 5

3 0 1 1

138 = 001 0112

0 1 0

1 0 1

258 = 010 1012


27


c) 1 0 0 1

4

0

1 0 0

0 0 0

1408 = 001 100 0002 d) 7 1 1 1

2

5

0 1 0

6

1 0 1

1 1 0

72568 = 111 010 101 1102

► BINARY – TO – OCTAL CONVERSION Example: Convert each of the following binary numbers to octal. a) 110 101

b) 101 111 001

c) 100 110 011 101

d) 011 010 000 100

Solution: a) 110 101

6

5

b) 101 111 001

5

8

6

3

5

1

8

d) 0 1 1 010 000 100

c) 100 110 011 101

4

7

8

3

2

0

4

8


28


2.8

HEXADECIMAL NUMBERING SYSTEM

► The hexadecimal number system has sixteen (16) digits. It is composed of 16 digits and alphabets characters. ► Each hexadecimal digit represents a 4-bit binary number as listed in table 2. ► Hexadecimal is widely used in computer and microprocessor application. DECIMAL

BINARY

HEXADECIMAL

0

0000

0

1

0001

1

2

0010

2

3

0011

3

4

0100

4

5

0101

5

6

0110

6

7

0111

7

8

1000

8

9

1001

9

10

1010

A

11

1011

B

12

1100

C

13

1101

D

14

1110

E

15

1111

F

Table 2


29


► BINARY TO – HEXADECIMAL CONVERSION

Example 1: Convert the following binary numbers to hexadecimal.

Solutions; a) 1100 1010 0101 0111

C

A

5

b)

7

11 1111 0001 0110 1001

3

1100 1010 0101 01112 = CA5716

F

1

6

9

11 1111 0001 0110 10012 = 3F16916

► HEXADECIMAL – TO – BINARY CONVERSION

Example 1: Determine the binary numbers for the following hexadecimal numbers.

a)

10A416 1 0 0 0 1

0 0 0 0

0

A

4 1 0 1 0

0 1 0 0

10A416 = 0001 0000 1010 01002


30


b)

CF8E16 C 1 1 0 0

F

8

1 1 1 1

E 1 0 0 0

1 1 1 0

CF8E16 = 1100 1111 1000 11102

c)

974216 9 1 0 0 1

0 1 1 1

7

4

2 0 1 0 0

0 0 1 0

974216 = 1001 0111 0100 00102

►

DECIMAL TO HEXADECIMAL CONVERSION •

To convert decimal numbers to hexadecimal, the decimal numbers are divided by 16 using the same repeated-division-method.

•

Example 1 : Convert 4428910 to hexadecimal numbers.

Solution: Remain

16

44289

1

16

2768

0

16

173

13 = D

16

10

10 = A

4428910 = A D 0 1

0 4428910 = AD0116


31


•

For fractional decimal number, repeated-multiplication-method is used.

•

Example 1: Convert 0.678510 to hexadecimal numbers.

Carry 0.6785

× 16

= 0.856

10

0.856

× 16

= 0.696

13

0.696

× 16

= 0.136

11

0.136

× 16

= 0.176

2

=

. A D B2

0.678510 = 0.ADB2

►

HEXADECIMAL TO DECIMAL CONVERSION •

Example : Convert A5C1.0116 to decimal numbers.

Solutions: Weight

163

162

161

160

16-1

16-2

Hex Number

A

5

C

1.

0

1

A5C1.0116 = 10 × 163 + 5 × 162 + 12 × 161 + 1 × 160 + 1 × 16-2 = 42433.0039

REMEMBER: For hexadecimal system, if remainder larger than 9, replace as follow: 10 = A

13 = D

11 = B

14 = E

12= C

15 = F


32


►

OCTAL TO HEXADECIMAL CONVERSION To convert from octal to hexadecimal number, you can use oct-dec-oct method or oct-bin-hex method.

Example: Convert 74218 to hexadecimal number.

Solution:

7

Octal Number

Step 1: Expand each octal digits to three binary bit Step 2: binary

Result

in

Step 3: Group in four

1

1

1

1

1

1

1

1

1

1

1

1

0

4

2

0

1

0

1

0

0

0

1

0

0

0

1

Step 4: Convert

F

1

0

0

0

0

0

0

0

0

0

1

1

1

1

74218 = F1116


33


►

HEXADECIMAL TO OCTAL CONVERSION Same with octal to hexadecimal conversion, you can either use hex-dec-oct method or hex-bin-oct method. Example: Convert E64B16 to octal number.

Solution:

E

Hex Number

Step 1: Expand each octal digits to four binary bit

1 1 1 0

Step 2: Result in binary

Step 3: Group in three

Step Convert

4:

6

4

0 1 1 0

B

0 1 0 0

1 0 1 1

1 1 1 0 0 1 1 0 0 1 0 0 1 0 1 1

1

1 1 0

0 1 1

0 0 1

0 0 1

0 1 1

1 6 3 1 1 3

E64B16 = 1631138


34


2.9

DECIMAL NUMBERS IN BINARY CODED DECIMAL (BCD) FORM

► BCD is a way to express each of the decimal digits with a binary code. ► BCD means that each decimal digit 0 through 9 is represented by a binary code of four bits. ► There are only ten code groups in the DCB system. ► To express any decimal number in BCD, replace each decimal digit with the appropriate 4-bit code. DECIMAL DIGIT 0 1 2 3

0000 0001 0010 0011

4 5 6 7 8 9

0100 0101 0110 0111 1000 1001

BCD

Table 3

► Example : Convert each of the following numbers to BCD. a) 35

b)

98

c)

170

d)

2469

1

0

Solutions: a) 3 0

0

1

1

5 0

1

35 = 0011 0101BCD Fatma Syazana Zaini, IKM Besut, 2010

35


b) 9 1

0

0

8

1

1

0

0

0

98 = 1001 1000BCD

c) 1 0

0

0

1

0

7 1

0 1

1

6

9

0

0

0

0

170 = 0001 0111 0000BCD

d) 2 0

0

1

0

0

1

0

4 0

0

1

1

0

1

0

0

1

2469 = 0010 0100 1001 1001BCD

► Example : Convert each of the following BCD to decimal. a) 1000 0110

b)

0011 0101 0001

Solutions: a) 1000 0110

8

6

1000 0110BCD = 8610

b) 0011 0101 0001

3

5

1

0011 0101 0001BCD = 35110


36


2.10

ADD BCD NUMBERS

BCD is a numerical code and can be used in arithmetic operation. Addition is the most important operation because the other three operations (subtraction, multiplication and division) can be accomplished by the use of addition.

► Step Procedure To Add BCD Number Step 1:

Add the two BCD number using rules for binary addition.

Step 2:

if a 4-bit sum is equal or less than 9, it is a valid BCD number.

Step 3:

if a 4-bit sum is greater than 9, or if a carry out of the 4-bit group is generated, it is an invalid result. Add 6 (0110) to the 4-bit sum in order to skip the six invalid states and return the code to BCD. If a carry result when 6 is added, simply add the carry to the next 4-bit group.

Example 1 : Add the following BCD numbers: a. 0011 + 0100

+

0 0 1 1 0 1 0 0 0 1 1 1

b. 0010 0011 + 0001 0101

+

0 0 1 0 0 0 0 1 0 0 1 1

1

1

1

0 0 1 1 0 1 0 1 1 0 0 0


37


c. 1000 0110 + 0001 0011

+

1

1 0 0 0 0 0 0 1 1 0 0 1

1

0 1 1 0 0 0 1 1 1 0 0 1

d. 0100 0101 0000 + 0100 0001 0111 1

+

1

0 1 0 0 0 1 0 0 1 0 0 0

0 1 0 1 0 0 0 1 0 1 1 0

0 0 0 0 0 1 1 1 0 1 1 1

► Example 2: Add the following BCD numbers: a. 1001 + 0100

+ + 0

0

0

1

1 0 1 0 0

0 1 1 1 0

1

0 0 0 1 1

1 0 1 0 1

Invalid BCD number (>9) Add 0110 (6)

3

b. 1001 + 1001

+ 1 + 0 0 0 1 1

1 1 0 0 1

1

0 0 0 1 0

0 0 1 1 0

1 1 0 0 0

Invalid because of carry Add 0110 (6)

8


38


c. 0001 0110 + 0001 0101 1

1

0 0 0 1 + 0 0 0 1

0 1 1 0 0 1 0 1

0 0 1 0

1 0 1 1

0 0 1 1

0 1 1 0 0 0 0 1

3

1

+

Right group is invalid (>9), left group is valid Add 0110 (6)

d. 0110 0111 + 0101 0011 1

1

1

1

0 1 1 0 0 1 0 1

0 1 1 1 0 0 1 1

1 0 1 1

1 0 1 0

+

0 1 1 0

0 1 1 0

0 0 0 1

0 0 1 0

0 0 0 0

1

2

0

+

Both groups are invalid (>9) Add 0110 (6) to both group


39


2.11 ►

GRAY AND ASCII CODES

THE GRAY CODES

•

There are many specialized codes used in digital systems. Some codes are strictly numeric like BCD and others are alphanumeric that is they used to represent numbers, letters, symbols and instructions. The code introduced in this topic is the Gray code.

•

The Gray Code is un-weighted and is not an arithmetic code: that is there are no specific weights assigned to the bit positions. The important feature of the Gray code is that it exhibits only a single bit change from one code word to the next in sequence.

•

This property is important in many applications, such as shaft position encoders, where error susceptibility increases with the number of bit changes between adjacent numbers in a sequence.

•

Table 4 is a listing of the 4-bit Gray code for decimal numbers 0 through 15. Binary numbers are shown in the table 4 for reference.

DECIMAL 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

BINARY 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

GRAY CODE 0000 0001 0011 0010 0110 0111 0101 0100 1100 1101 1111 1110 1010 1011 1001 1000

Table 4 Fatma Syazana Zaini, IKM Besut, 2010

40


►

BINARY – TO – GRAY CODE CONVERSION Procedure: i.

The most significant bit (left-most) is the Gray code is the same as the corresponding MSB in the binary number.

ii.

Going from left to right, add each adjacent pair of binary code bits to get the next Gray code bit. Discard carries.

Example 1: Convert the binary number 10110 to Gray code.

1

+

0

1

+

1

1

+

1

1

+

0

0

Binary

1

Gray

Example 2: Convert the binary number 1100 0110 to Gray code

1 + 1 + 0 + 0 + 0 + 1 + 1 + 0 Binary 1

►

0

1

0

0

1

0

1 Gray

GRAY – TO – BINARY CONVERSION Procedure: i.

The most significant bit (left-most) in the binary code is the same as the corresponding bit in the Gray code.

ii.

Add each binary code bit generated to the Gray code bit in the next adjacent position. Discard carries.


41


Example1: Convert the Gray code below to binary. a. 11011

b. 1011 1111

► ASCII CODES

•

ASCII is the abbreviation for American Standard Code for Information Interchange. Pronounce “askee”. ASCII is a universally accepted alphanumeric code used in most computers and other electronic equipment. Most computer keyboards are standardized with the ASCII. When we enter a letter, a number or control command, the corresponding ASCII code goes into the computer.

•

ASCII has 128 characters and symbols represented by a 7-bit binary code. Actually ASCII can be considered an 8-bit code with the MSB always 0. This 8-bit code is 00 through 7F in hexadecimal.

•

The first thirty-two ASCII characters are non-graphic commands that are never printed or displayed and are used only for control purposes.

•

Examples of the control characters are “null”, “line feed”, “start of text” and “escape”. The other characters are graphic symbols that can be printed or displayed and include the letters of the alphabet (lowercase and uppercase), the ten decimal digits, punctuation signs and other commonly used symbols.


42


•

Table 5 is a listing of the ASCII code showing the decimal, hexadecimal and binary representations for each character and symbol. The left section of the table lists the names of the 32-control character (00 through 1F hexadecimal).

Table 5

•

The first thirty-two codes in the ASCII table represent the control characters. These are used to allow devices such as a computer and printer to communicate with each other when passing information and data.

•

Table 2 lists the control characters and the control key function that allows them to be entered directly from an ASCII keyboard by pressing the control key (CTRL) and the corresponding symbol.


43


•

The extended ASCII contains characters in the following general categories: a. b. c. d. e. f. g.

Foreign (non-English) alphabetic characters Foreign currency symbols Greek Letter Mathematical symbols Drawing characters Bar graphing characters Shading character

Example 1: Determine the binary ASCII codes that are entered from the computer’s keyboard when the following BASIC program statement is typed in also express each code in hexadecimal. 20 PRINT “A=” ; X

Solution; The ASCII code for each symbol is found in table 1.

Symbol 2 0 Space P R I N T Space “ A = “ ; X

Binary 011 0010 011 0000 010 0000 101 0000 101 0010 100 1001 100 1110 101 0100 010 0000 010 0010 100 0001 011 1101 010 0010 011 1011 101 1000

Hexadecimal 3216 3016 2016 5016 5216 4916 4E16 5416 2016 2216 4116 3D16 2216 3B16 5816


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SUMMARY

In this topic, students have learned the numeric quantities occur naturally in analogue but must be converted to digital form to be used by computers or digital circuitry. Student must understand that the binary numbering system is used in digital systems because the 1s and 0s are easily represented by ON or OFF transistors, which output 0volt for zero(0) and 5volt for one(1). Students also must know that the ASCII is used by computers to represent all letters, numbers and symbols in digital form.

EXERCISE 1. Convert each decimal number to binary. a. 23 b. 57 c. 45 2. Convert the following binary to decimal a. 101102 b. 1101 10112 c. 110 11112 d. 1001 11012 3. Determine the 1’s complement of each binary number: a. 0001 1010 b. 1111 0111 c. 1000 1101 4. Determine the 2’s complement of each binary number: a. 0001 0110 b. 1111 1100 c. 1001 0001 Fatma Syazana Zaini, IKM Besut, 2010

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5. Using a. b. c.

the rules for binary addition, evaluate the following 11112 + 1002 01112 + 10102 11002 + 0102

6. Using a. b. c.

the rules for binary subtraction evaluate the following. 10002 – 00112 01012 – 00102 11112 - 11002

7. Multiply 0111 1111 by 0000 0101. 8. Convert the following decimal numbers to binary: a. 468 b. 7238 c. 56248 9. Convert the following binary numbers to octal: a. 1 1010 11112 b. 10 0110 00102 c. 101 1111 10012 10. Convert the following decimal numbers to BCD: a. 6 b. 15 c. 273 d. 849 11. What a. b. c.

decimal numbers are represented by each BCD code? 1000 10012 0010 0111 10002 0001 0101 01112

12. Convert the following binary numbers to the Gray code: a. 11002 b. 10102 c. 110102


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13. Convert the following Gray codes to binary: a. 10002 b. 10102 c. 111012

14. What is the ASCII representation for each of the following characters? Express each as a bit and in hexadecimal notation. a. K b. r c. $ d. +

REFERENCES

1. Digital System – Principle And Applications, Tocci, R.J, Prentice Hall international 2. Digital Fundamentals, Floyd T.L, Merrill Publishing 3. BPL(K) Module TFV 2033 Digital Electronics 1 4. Digital Electronics: Teaching Module, KUITHO


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Basic Digital Electronics - Unit 2

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