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Eng. Mohammed Timraz Electronics & Communication Engineer

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Faculty of Engineering and Urban planning

Software Engineering Department

Digital Logic Design ESGD2201

Lecture 2

Number Systems

Eng. Mohammed Timraz

Electronics & Communication Engineer

Wednesday, 10th September 2008

1. Decimal Number

2. Binary Numbers

2.1 Binary to Decimal Conversion

2.2 Decimal to Binary Conversion

3. Octal Numbers

3.1 Octal to Decimal Conversion

3.2 Decimal to Octal Conversion

3.3 Binary to Octal Conversion

3.4 Octal to Binary Conversion

4. Hexa Decimal Numbers

4.1 Binary to Hexa Decimal Conversion

4.2 Hexa Decimal to Binary Conversion

4.3 Hexa Decimal to Decimal Conversion

4.4 Decimal to Hexa Decimal Conversion

1- Decimal Number:

[Base10],

*Integer Number:[0,1,2,3,………..9]

Example:-

[23]10=3×100 +2×101

=3×1+2×10

=3+20= 23

Decimal Number

*Float Number:

[16.15]

Integer part float part

Decimal Point

Example:-[16.15]

16.15=6*100+1*101+1*10-1+5*10-2

=6 + 10+ 0.1 + 0.05

=16.15

…..104 103 102 101 . 10-1 10-2 10-3 10-4 …..Decimal Point

2-1 Binary to Decimal Conversion:

Example:- [1101101]2

1 1 0 1 1 0 1

MSB LSB

Most significant BIT Least Significant Bit

2-1 Binary to Decimal Conversion:

Example:-

For integer numbers,

[1101101]2 = [ ? ]10

1 1 0 1 1 0 1

26 25 24 23 22 21 20

[1101101]2 =1×20+0×21+1×22+1×23+0×24+1×25+1×26

=[109]10

MSB LSB

2-1 Binary to Decimal Conversion:

Example:-

For float numbers,

[10.111]2 = [ ? ]10

1 0 . 1 1 1

21 20 2-1 2-2 2-3

[10.111]2 =0×20+1×21+1×2-1+1×2-2+1×2-3

=[2.875]10

Note: MSB & LSB just for integer part.

Binary

point

Number Systems

2-1 Binary to Decimal Conversion:

Example:-

[1011]2 = [ ? ]10

1 0 1 1

23 22 21 20

[1011]2 =1×20+1×21+0×22+1×23

=[11]10

Number Systems

- 2-2 Decimal to Binary Conversion:
- There are two ways to convert the decimal numbers to binary numbers.
- By using the Truth Table: we can use the truth table for small decimal numbers, but for large decimal numbers it will be more difficult.
- So, we will use the division: which we can use it for small and large decimal numbers

Number Systems

Decimal to Binary Conversion:

The Division:

[198]10 = [ ? ]2

By using the division system:

Carrier

0

1

1

0

0

0

1

1

1

1

0

1

1

0

0

0

Number Systems

Decimal to Binary Conversion:

The Division:

For float numbers,

[0.3125]10 = [ ? ]2

By using

the multiplication system:

MSB LSB

. 0

0

1

1

Carrier

0

0.3125 × 2 = 0.625

1

0.625 × 2 = 1.25

0

1.25 × 2 = 0.5

0.5 × 2 = 1.00

1

Continuo to the desired number of decimal places or stop when fractional part is all zero

3. Octal Numbers:

[Base 8],[ 0,1,3,4,5,6,7]

3-1 Octal to Decimal Conversion:

Example:-

[2374]8 = [ ? ]10

=4×80+7×81+3×82+2×83

=[1276]10

Number Systems

3-2 Decimal to Octal Conversion:

The Division:

[359]10 = [ ? ]8

By using the division system:

Carrier

7

4

5

4

7

5

3-3 Binary to Octal Conversion:

Example:-

[110101]2 = [ ? ]8

Here we will take 3 bits and convert it from binary to decimal by using the decimal to binary truth table:

BinaryDecimal

1 1 0 1 0 1 = (65)8

{

{

6

5

3-4 Octal to Binary Conversion:

Example:-

[13]8 = [ ? ]2

Here we will convert each decimal digit from decimal to binary (3 bits) using the decimal to binary truth table:

BinaryDecimal

(13)8 = (001011)2

4. Hexa Decimal Numbers:

[Base 16],[ 0,1,3,4,5,6,7,8,9,A,B,C,D,E,F]

4-1 Binary to Hexa Decimal Conversion:

Example:-

[1100101001010111]2 = [ ? ]16

Here we will take 4 bits and convert it from binary to decimal by using the decimal to binary truth table:

1100 1010 0101 0111

{

{

{

{

C

A

5

7

Then, [1100101001010111]2 = [CA57]16

1110

0010

0111

Number Systems

4-2 Hexa Decimal to Binary Conversion:

Example:-

[DE27]16 = [ ? ]2

Here we will convert each Hexa decimal digit from decimal to binary (4 bits) using the decimal to binary truth table:

D

E

2

7

Then, [DE27]16 = [1101111000100111]2

4-3 Hexa Decimal to Decimal Conversion:

Example:-

[B2F8]16 = [ ? ]10

=8×160+F×161+2×162+B×163

=[45816]10

Where, B=11, and F=15

4-4 Decimal to Hexa Decimal Conversion:

Example:-

[650]10 = [ ? ]16

By using the division system:

Carrier

10

8

2

8

A

2

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