ib_number systems and conversions

7/29/2019 IB_Number Systems and Conversions

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Number Systems and Conversions

Abu Ahmed Ferdaus

Assistant ProfessorDepartment of Computer Science & Engineering

Dhaka University


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Number Systems and Conversions

Topics:

Overview of number systems Conversions:

1. Decimal to Binary2. Decimal to Octal3. Decimal to Hexadecimal

4. Binary to Decimal5. Binary to Octal6. Binary to Hexadecimal7. Octal to Decimal

8. Octal to Binary9. Octal to Hexadecimal10. Hexadecimal to Decimal11. Hexadecimal to Binary12. Hexadecimal to Octal

Exercises


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Overview of Number System

Many number systems are in use in digital

technology. The most common are:1. Decimal number system base 10

(0, 1, 2, 3, 4, 5, 6, 7, 8, 9)

2. Binary number system base 2

(0, 1)

3. Octal number system base 8

(0, 1, 2, 3, 4, 5, 6, 7)

4. Hexadecimal number system base 16(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F)

The decimal system is clearly the most familiar to

us because it is a tool that we use every day.


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Decimal System

The decimal system is composed of 10 numerals or

symbols. These 10 symbols are 0, 1, 2, 3, 4, 5, 6, 7, 8,9; using these symbols as digits of a number, we canexpress any quantity. The decimal system, also calledthe base-10 system because it has 10 digits.

103

102

101

100

10-1

10-2

10-3

=1000 =100 =10 =1 . =0.1 =0.01 =0.001

MostSignificant

Digit

Decimalpoint

LeastSignificant

Digit


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Decimal Numbers

decimal means that we have ten digits to use in our

representation (the symbols 0 through 9)

Decimal number 3,546?

it is threethousands plus fivehundreds plus four

tens plus sixones.

Decimal number: 329

329

102 101 100

3x100 + 2x10 + 9x1 = 329


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Binary System

In the binary system, there are only two symbols or

possible digit values, 0 and 1. This base-2 system canbe used to represent any quantity that can berepresented in decimal or other number system.

23

22

21

20

2-1

2-2

2-3

=8 =4 =2 =1 . =1/2 =1/4 =1/8

Most

Significant Bit Binarypoint

Least

Significant Bit


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Binary Numbers

Binary (base two) system:

has two states: 0 and 1

Basic unit of information is the binary digit, or bit.

Binary number 101

101

22 21 20

1x4 + 0x2 + 1x1 = 5

most

significant

least

significant


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Fractional Values

In Decimal

In Binary

0.329

10-1

10-2

10-3

3x0.1 + 2x0.01 + 9x0.001 = 0.329

0.011

2-1 2-2 2-3

0x0.5 + 1x0.25 + 1x0.125 = 0.375


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Octal Number System

The octal number system has a base of eight, meaning

that it has eight possible digits: 0,1,2,3,4,5,6,7.

83

82

81

80

8-1

8-2

8-3

=512 =64 =8 =1 . =1/8 =1/64 =1/512

Most

Significant

Digit

Octal

point

Least

Significant

Digit


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Octal Numbers

Octal means that we have eight digits to use in ourrepresentation (the symbols 0 through 7)

Octal number 327 = ?

3x64 + 2x8 + 7x1 = 215

327

81 8082


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Hexadecimal Number System

The hexadecimal system uses base 16. Thus, it has 16

possible digit symbols. It uses the digits 0 through 9 plusthe letters A, B, C, D, E, and F as the 16 digit symbols.

163 162 161 160 16-1 16-2 16-3

=4096 =256 =16 =1 . =1/16 =1/256 =1/4096

Most

Significant

Digit

Hexadec.

point

Least

Significant

Digit


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Hexadecimal Numbers

Hexadecimal means that we have sixteen digits touse in our representation (the symbols 0 to 9 and

A,B,C,D,E,F)

Hex number 27B = ?

2x256 + 7x16 + 11x1 = 635

27B

161 160162


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Comparison of Four Number Systems

Binary

00000001

0010

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

1101

1110

1111

Base-2

Octal

01

2

3

4

5

6

7

10

11

12

13

14

15

16

17

Base-8

Decimal

01

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Base-10

Hexadecimal

01

2

3

4

5

6

7

8

9

A

B

C

D

E

F

Base-16


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Number Conversion

Number Conversion can be viewed as:

Decimal

HexadecimalOctal

Binary


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1. Decimal to Binary Conversion

Repeated Division: repeatedly divide the decimal

number by 2, the base of the binary system. Divisionby 2 will either give a remainder of 1 (dividing an oddnumber) or no remainder (dividing an even number).Collecting the remainders from the repeated divisionswill give the binary answer. Example: convert 25

10to

binary.

25/ 2 = 12+ remainder of 1 1 (Least Significant Bit)

12/ 2 = 6 + remainder of 0 06 / 2 = 3 + remainder of 0 0

3 / 2 = 1 + remainder of 1 1

1 / 2 = 0 + remainder of 1 1(Most Significant Bit)Result 2510 = 1 1 0 0 12


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1. Decimal to Binary Conversion (Cont.)

Reading from bottom to top, the final answer is 11001.

Remember that the first division gives us the leastsignificant digit of our answer, and the final divisiongives us the most significant digit of our answer. Also,the result of the final division is always 0.

Converting Fraction: When converting a fractionaldecimal value to binary, we need to use a slightlydifferent approach. Instead of dividing by 2, werepeatedly multiply the decimal fraction by 2. If theresult is greater than or equal to 1, we add a 1 to ouranswer. If the result is less than 1, we add a 0 to ouranswer. The 0 fractional parts indicates the end of themultiplication.


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1. Decimal to Binary Conversion (Cont.)

Convert 0.37510 to binary.

Result

0.375 * 2 = 0.75 0 (Leftmost digit)

0.75 * 2 = 1.5 1

0.5 * 2 = 1.0 1 (Rightmost digit)

So, 0.37510 = 0.0112

So, a decimal number with fraction can be convertedto binary using the above methods.

25.37510 = 25 + 0.375 = 11001.0112


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2. Decimal to Octal Conversion


number by 8, the base of the octal system. Division by8 will give a remainder of 0..7. Also, the result of thefinal division is always 0. Collecting the remaindersfrom the repeated divisions will give the octal number.

Example: convert 17710 to octal and binary:

177/8 = 22+ remainder of 1 1 (Least Significant Bit)

22/ 8 = 2 + remainder of 6 6

2 / 8 = 0 + remainder of 2 2 (Most Significant Bit)Result17710 = 2618

Convert to Binary = 0101100012


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3. Decimal to Hexadecimal Conversion


number by 16, the base of the hexadecimal system.Division by 16 will give a remainder of 0..F(15). Also,the result of the final division is always 0. Collectingthe remainders from the repeated divisions will give

the hexadecimal number. Example: convert 37810 tohexadecimal and binary:

378/16 = 23+ remainder of 10A (Least Significant Bit)

23/ 16 = 1 + remainder of 7 7

1 / 16 = 0 + remainder of 1 1 (Most Significant Bit)

Result37810 = 17A8

Convert to Binary= 0001 0111 10102= 0000 0001 0111 1010

(16 bits)


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4. Binary to Decimal Conversion

Any binary number can be converted to its decimal

equivalent simply by summing together the weights of thevarious positions in the binary number which contain a 1.The method is find the weights (i.e., powers of 2) for eachbit position that contains a 1, and then to add them up.

1 1 0 1 12

(binary)

24+23+0+21+20 = 16+8+0+2+1

= 2710 (decimal)

1 0 1 1 0 1 0 12

(binary)

27+0+25+24+0+22+0+20 = 128+0+32+16+0+4+0+1

= 18110 (decimal)


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4. Binary to Decimal Conversion (Cont.)

Convert 1101012 to decimal number.

Binary

Weights

25 24 23 22 21 20

Weight

Value

32 16 8 4 2 1

BinaryNumber

1 1 0 1 0 1

Decimal

Value

32 16 0 4 0 1 Total

(53)10


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5. Binary to Octal Conversion

Each Octal digit is represented by three bits of binary

digit.

Partition each binary number into 3-bit group to theleft and right of the fractional point and then replacingeach 3-bit group by its equivalent octal number.

Octal Digit 0 1 2 3 4 5 6 7

Binary Equivalent 000 001 010 011 100 101 110 111

1001110102 = (100) (111) (010)2 = 4728

10011.10112

=(010) (011).(101)(100)2

= 23.548


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6. Binary to Hexadecimal

Each Hexadecimal digit is represented by four bits of

binary digit.

Partition each binary number into 4-bit group to the

left and right of the fractional point and then replacingeach 4-bit group by its equivalent hexadecimalnumber.

1011001011112 = (1011) (0010) (1111)2 = B2F16

11011.1012 = (0001)(1011).(1010)2 = 1B.A16

Hexadecimal Digit 0 1 2 3 4 5 6 7


Hexadecimal Digit 8 9 A B C D E F



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7. Octal to Decimal Conversion

To express the value of a given octal number as its

decimal equivalent we just need to sum the digits aftereach has been multiplied by its associated weight.

Convert (237.04)8 to decimal

Again, 24.68= 2 x (81

) + 4 x (80

) + 6 x (8-1

) = 20.7510

Weights 82 81 80 8-1 8-2

Weight

Value 64 8 1 0.125 0.015625

OctalNumber 2 3 7 0 4

Decimal

Value 128 24 7 0 0.0625 Total(159.0625)10


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8. Octal to Binary Conversion


digit.

For Octal to Binary conversion, convert each octal digit

to its corresponding binary digits.

Octal Digit 0 1 2 3 4 5 6 7


4728 = (100) (111) (010)2 = 1001110102

53.68 = (101)(011).(110)2 = 101011.112


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9. Octal to Hexadecimal Conversion


digit.

1) Convert Octal to Binary first.2) Partition each binary number into 4-bit group to theleft and right of the fractional point and then replacingeach 4-bit group by its equivalent hexadecimal

number.

Octal Digit 0 1 2 3 4 5 6 7


4728 = (100) (111) (010)2 = (0001) (0011) (1010)2 = 13A16

67.528 = (110)(111).(101)(010)2 =(0011)(0111).(1010)(1000)

2= 37.A8

16


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10. Hexadecimal to Decimal Conversion

To express the value of a given hexadecimal number

as its decimal equivalent, we just need to sum thedigits after each has been multiplied by its associatedweight.

The hexadecimal number is converted to decimal as

follow:2AF16 = 2 x (16

2) + 10 x (161) + 15 x (160)

= 2 x 256 + 10 x 16 + 15 x 1 = 512+160+15

= 68710

Weights 162 161 160 . 16-1 16-2


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11. Hexadecimal to Binary Conversion

Each Hexadecimal digit is represented byfourbits of

binary digit.

For Hexadecimal to Binary conversion, convert each

hexadecimal digit to its corresponding 4-bit binarydigit.

B2F16 = (1011) (0010) (1111)2 = 1011001011112

7.A3C16

= (0111).(1010)(0011)(1100) =111.10100011112



Hexadecimal Digit 8 9 A B C D E FBinary Equivalent 1000 1001 1010 1011 1100 1101 1110 1111


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12. Hexadecimal to Octal Conversion

Each Hexadecimal digit is represented by fourbits of

binary digit.

1) Convert Hexadecimal to Binary first.2) Partition each binary number into 3-bit group to the

left and right of the fractional point and then replacingeach 3-bit group by its equivalent hexadecimalnumber.

B2F16 = (1011) (0010) (1111)2 = (101) (100) (101) (111)2

= 54578



Hexadecimal Digit 8 9 A B C D E FBinary Equivalent 1000 1001 1010 1011 1100 1101 1110 1111


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THANK YOU ALL

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