Numbers for Digital System

This article describes the number system and the conversion process between number systems. There are four number systems that can be used : decimal, binary, octal, and hexadecimal. The decimal system is certainly familiar because it is always used every day. For digital techniques used binary and hexadecimal number systems.

Decimal Numbers

Decimal numbers consist of 10 numeric symbols :  0, 1, 2, 3, 4, 5, 6, 7, 8, 9; The decimal system is often called the base 10 number. Because it consists of 10 digits. In the application, the value of a decimal number is determined by its position in the data as given in table 1 below

Table 1 : The value of the decimal number in its position

103 102 101 100 10-1 10-2 10-3
= 1000 = 100 = 10 = 1 . = 0.1 = 0.01 = 0.001
MSD Decimal pont LSD

MSD = Most Significant Digit
LSD  = Least Significant Digit

Examples:

123

1 = 100

2 = 20

3 = 10

 

Binary Numbers

In a binary system, there are two digit values 0 and 1, so it is often called the base number two. With these two values ​​it can be used to represent all quantifiable quantities that can be expressed in decimal form or other number systems. The bit value is determined by the bit position in a binary data series, as given in Table 2 and the 4-bit Binary System is given in Table 3 below

Table 2 : The Value of bit positions in binary

23 22 21 20 2-1 2-2 2-3
= 8 = 4 = 2 = 1 . = 1/2 = 1/4 = 1/8
MSB Binary Point LSB

MSB = Most Significant Bit
LSB  = Least Significant Bit

Table 3  : 4 bit binary system

23 22 21 20 Decimal
0 0 0 0 0
0 0 0 1 1
0 0 1 0 2
0 0 1 1 3
0 1 0 0 4
0 1 0 1 5
0 1 1 0 6
0 1 1 1 7
1 0 0 0 8
1 0 0 1 9
1 0 1 0 10 (A)
1 0 1 1 11 (B)
1 1 0 0 12 (C)
1 1 0 1 13 (D)
1 1 1 0 14 (E)
1 1 1 1 15 (F)

 

Converting Binary to Decimal

Binary numbers can be converted to decimal form by summing the 1 bit values ​​based on their position as given in Fig.

Examples:

1 1 0 1 1 2 (binary)

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

= 2710 (decimal)

and

1 0 1 1 0 1 0 1 2 (binary)

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

= 18110 (decimal)

 

Decimal to Binary Conversion

There are 2 methods:

(A) Reverse of the Binary to Decimal method

45 10 = 32 + 0 + 8 + 4 +0 + 1

= 25+ 0 + 23+22+ 0 + 20

= 1 0 1 1 0 12

(B) repeated division

This method uses division by 2 repeatedly, the result is read from the result of the last division.

Example: conversion 2510 to binary 25/2

= 12 remainder 1 1 (LSB) 12/2

= 6 remainder 0 0

6/2 = 3 remainder 0 0

3/2 = 1 remainder 1 1

2.4

1/2 = 0 remainder 1 1 (MSB)

Result 2510 = 110012

 

Octal Numbers

Octal numbers are often called base 8 numbers because they have 8 digits, namely: 0,1,2,3,4,5,6,7 with values ​​based on digit positions as given in Table.4.

Table 4 Value of octal number

83 82 81 80 8-1 8-2 8-3
= 512 = 64 = 8 = 1 . = 1/8 = 1/64 = 1/1212
MSD Octal Point LSD

 

Convert Octal to Decimal

Example: 24.68 = (……..)10

= 2 x (81) + 4 x (80) + 6 x (8-1)

= 20.7510

Converting Binary to Octal

Octal Digits 0 1 2 3 4 5 6 7
Binary Equivalent 000 001 010 011 100 101 110 111

To convert a binary number to octal, each octal digit is given as a 3 bit binary number.

Example: 100 111 0102 = ……….8

= (100) (111) (010)

= 4 7 28

2.5

Octal to Binary Conversion

The method is to use repeated division. This method divides the decimal number by 8 repeatedly and the result is read from the last division.

Example: Conversion says a 17710 to Octal and binary form

Convert to octal:

177/8 = 22 remainder 1 1 (LSB) 22/8 =

2 remainder 6 6

= 0 remainder 2 2 (MSB)

2/8Result 17710 = 2618

Convert to Binary = 010 110 0012

Hexadecimal Numbers

The hexadecimal number system is a number system with 15 numeric symbols so it is often called the base 16 number system, namely: 0 to plus the letters A, B, C, D, E, and F.The digit position values ​​are given in Table 6. .

Table 6. The position value of the hexadecimal number

163 162 161 160 16-1 16-2 16-3
= 4096 = 256 = 16 = 1 . = 1/16 = 1/256 = 1/4096
MSD Hex Point LSD

Convert Hexadecimal to Decimal

Example:

2AF16 = ……10.

2.6

= 2 x (162) + 10 x (161) + 15 x (160)

= 68710

Repeating divisor: Convert decimal to hexadecimal

The method is the same in the decimal system but the divisor is 16. Example:

Convert 37810 to hexadecimal and binary:

378/16 = 23+ remainder 10 A (LSB) 23/16

= 1 + remainder 7 7

1/16 = 0 + remainder 1 1 (MSB)

Result 37810 = 17A8

Conversion to binary = 0001 0111 1010

Converting Hexadecimal to Binary

Each hexadecimal digit consists of 4 bits of binary digits as given in Table 7.

Table 7. Binary equivalent of the hexadecimal number

Hexadecimal 0 1 2 3 4 5 6 7
Binary equivalent 0000 0001 0010 0011 0100 0101 0110 0111
Hexadecimal 8 9 A (10) B (11) C (12) D (13) E (14) F (15) )
Binary Equivalent 1000 1001 1010 1011 1100 1101 1110 1111

Converting Binary to Hexadecimal

Example:

1011 0010 11112 = …. 16

2.7

= (1011) (0010) (1111)2

= B 2 F16

Convert Hexadecimal to Binary

Example:

123E16 = …… 2

 = 10001 0010 0011 11102

Convert Hexadecimal to Octal

Steps:

1) Convert Hexadecimal to Binary form.

2) Arrange binary numbers in 3 bit form starting from LSB.

Example.

Convert 5A816 to Octal

Convert to the binary form

45616 = 0100 0101 0110 (binary)

Create in a 3 bit group.

Obtained = 010 001 010 110

= 2 1 2 38

Conclusion

  1. There are 4 number systems : binary, octal, decimal, and hexa decimal.
  2. The four numbers can be converted to one another.
  3. Digital techniques using binary numbers