Decimal Number System

(base₁₀)

This is the numbering system that we have grown up with.

To understand the other numbering systems (binary, octal, hexadecimal) we will start by recalling the 'rules' associated with the decimal (or base₁₀) system.

With the decimal numbering system, we use a base of 10.

This means that we go up in increments of 10:

1

10

100

1000

etc…..

Example

Consider the following decimal number:

136.25.

What does this actually mean?

This number means:

102  * 1  =  100.0
101  * 3  =   30.0
100  * 6  =    6.0
10-1 * 2  =    0.2
10-2 * 5  =    0.05
 
total      =  136.25

Binary Number System

(base₂)

With the binary numbering system, we use a base of 2 instead of a base of 10.

This means that we go up in increments of 2:

1

2

4

8

16

32

64

128

256

512

etc…..

Example:

Consider the binary number:

1101.01.

Again in its full form this means:

23  * 1  =  1000.0  (8 in decimal)
22  * 1  =   100.0    (4 in decimal)
21  * 0  =    00.0    (0 in decimal)
20  * 1  =     1.0    (1 in decimal)
2-1 * 0  =     0.0    (0.0 in decimal)
2-2 * 1  =     0.01   (0.25 in decimal) 
 
total     =  1101.01 (13.25 in decimal)

The following shows the first integers and their binary equivalents (and how they are derived):

decimal  binary
   0      0000    (0 * 23 + 0 * 22 + 0 * 21 + 0 * 20)
   1      0001    (0 * 23 + 0 * 22 + 0 * 21 + 1 * 20)
   2      0010    (0 * 23 + 0 * 22 + 1 * 21 + 0 * 20)
   3      0011    (0 * 23 + 0 * 22 + 1 * 21 + 1 * 20)
   4      0100    (0 * 23 + 1 * 22 + 0 * 21 + 0 * 20)
   5      0101    (0 * 23 + 1 * 22 + 0 * 21 + 1 * 20)
   6      0110    (0 * 23 + 1 * 22 + 1 * 21 + 0 * 20)
   7      0111    (0 * 23 + 1 * 22 + 1 * 21 + 1 * 20)
   8      1000    (1 * 23 + 0 * 22 + 0 * 21 + 0 * 20)
   9      1001    (1 * 23 + 0 * 22 + 0 * 21 + 1 * 20)

Octal Number System

(base₈)

With the octal numbering system, we use a base of 8 instead of a base of 10 or 2.

As it turns out, each octal "digit" will represent a group of 3 bits or binary digits.

Example:

Consider the binary number

110101010

This 9 bit binary number can be grouped into 3 groups of 3 bits:

110 101 010

6 5 2

This gives us the octal number:

652₈

 
This octal number (and associated binary number) is a representation of the following decimal number:
 
42610
 
 

Hexadecimal Number System

(base₁₆ or base_H)

Although it is very convenient for computers to use 0 and 1 it is difficult for humans to readily identify binary numbers.

Hexadecimal (Hex) acts as an interim number system between binary and ourselves as it is more recognizable yet easy to convert to and from binary.

With the hexadecimal numbering system, we use a base of 16 instead of a base of 10, 8, or 2.

As it turns out, each hexadecimal "digit" will represent a group of 4 bits or binary digits (1 byte can be expressed by two hexadecimal digits).

The Hexadecimal system uses the following "digits"

Hexadecimal Digit Decimal Digit

1 1

2 2

3 3

4 4

5 5

6 6

7 7

8 8

9 9

A 10

B 11

C 12

D 13

E 14

F 15

Example:

Consider the binary number

100101110100

This 12 bit binary number can be grouped into 3 groups of 4 bits:

1001 0111 0100

9 7 4

This gives us the hexadecimal number:

974_{16 =} 974_H

A table providing decimal, binary, and hex equivalents for the first 16 integers is:

decimal  binary  Hex
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

A few further examples of binary and hexadecimal numbers follow:

 
 0100 1100 0001 1111 = 4C1F16  = 4C1FH
  1101 0110 1100 0000 = D6C016 = D6C0H