# 2.5 – Numbering Systems

by on August 15, 2017

### Numbering systems:

Before going through the constants we will first discourse the numbering systems. Numbering system is a way of counting, labeling and measuring using a series of symbols or digits. These symbols represent different values depending on the position they occupy in the number. Now we will see the four most common number systems used.

• Decimal numbering system
• Octal numbering system
• Hexadecimal numbering system
• Binary numbering system

### Decimal numbering system:

It is the most commonly used numbering system used in day-to-day life to count, measure and label. Combination of ten digits from 0 to 9 are used to represent any number. In this system the next position to the left from the decimal point represents units, tens, hundreds, thousands etc. and the next position to the right after the decimal point represents (1/10)’s, (1/100)’s, (1/1000)’s etc.

Example: ### Octal numbering system:

In octal numbering system combination of eight digits from 0 to 7 are used to represent a number. It will go from 0…7, a two digit sequence is from 10..77 and a three digit sequence is from 100…777 and so on.

0    1    2    3    4    5    6    7    10   11    12    13    14     15    16    17     20……..

#### Converting from octal to decimal:

Example:

3452(8) =  3×83     + 4×82  + 5×81 + 2×80

=  3×512 + 4×64 + 5×8  +  2×1

=  1536   +  256  +  40  +  2

=  1834(10)

### Hexadecimal numbering system:

In hexadecimal numbering system combination of 16 digits from 0 to 9 and A to F are used to represent a number. It will go from 0…F, a two digit sequence is from 10…FF and a three digit sequence from 100…FFF and so on.

0    1    2    3    4    5    6    7    8    9    A    B     C    D     E     F    10   11   13…..

#### Converting from hexadecimal to decimal:

Example:

3A4B(16)   =   3×163    +    Ax162       +     4×161      +    Bx160

=     3×163    +    10×162     +     4×161     +    11×160

=    3×4096  +     10×256    +     4×16      +    11×1

=     12288   +     2560       +      64         +    11

=     14923(10)

### Binary numbering system:

In binary numbering systems only two digits 0 and 1 are used to represent any number. It will go like 0  1  10  11  100  101  110  111  1000 and so on. It is the numbering system used in computers. In this system the next position to the left from the decimal point represents units, 2’s, 4’s, 8’s etc. and the next position to the right after the decimal point represents (1/2)’s, (1/4)’s, (1/8)’s etc.

0       1       10       11        100      101        110        111      1000…………

#### Converting from binary to decimal:

Example:

1101001(2)  =  1×26 + 1×25 + 0x24 + 1×23 + 0x22 + 0x21 + 1×20

=  1×64  + 1×32 + 0x16 + 1×8 +  0x4 + 0x2 +1×1

=   64    +   32   +   0   +   8   +   0   +   0   +  1

=   105(2)

Example:

1011.101(2)  =  1×23 +  0x22  + 1×21 + 1×20 + 1×1/21 + 0x1/22 + 1×1/23

=  1×8  +  0x4   +  1×2  + 1×1  + 1×0.5  + 0x0.25 + 1×0.125

=   8    +    0     +   2    +   1    +   0.5    +  0.25   +  0.125

=  11.875

### Converting from decimal to other numbering systems:

To convert a decimal number into binary equal then we need to keep on divide the number by 2 and record all the remainders. Writing all the remainders in reverse order results binary equal.

Likewise we divide by 8 if we want to get octal equal and needs to divide by 16 if we want to get hexadecimal equal. Previous post:

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