Coding of numbers and numeral systems

The computer is a calculator: it processes numbers.

Digital information is encoded as numbers.

The computer is made up of electronic circuits.

An electronic circuit can distinguish two electrical states

• Low state: absence of electric current.
• High state: presence of electric current.

A number is coded in a form adapted to be processed by an electronic circuit: we use a coding with two symbols (digits): 0 and 1

• The digit 0 represents the low state (absence of electric current.)
• The digit 1 represents the high state (presence of electric current.)

To encode numbers, we use a numeral system.

Decimal numeral system (base 10)

405
405 =

4
4 x 102

0
+ 0 x 101

5
+ 5 x 100

Let b = 10 (base 10 numeral system)

405 = 4 x b2 + 0 x b1  + 5 x b0

10 symbols (digits) are needed to write numbers in base 10.
digits: { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }

Base b numeral system

To represent a natural integer N in the base b numeral system, we decompose it in the following form:

N = cn.bn + ... + c2.b2 + c1.b1 + c0.b0

The ci are the digits of the base b and fulfill the following condition

0 ≤ ci < b

b digits are needed to write numbers in the base b numeral system.

Binary numeral system (base 2)

Base: b = 2

Digts used: {0, 1}

A digit in the binary numeral system is called a “bit” (binary unit)

Example

23

22

21

20

1001

1

0

0

1

1001 = 

1 x 23 +

0 x 22 +

0 x 21 +

1 x 20

N = (1001)₂ = (9)₁₀

Example

23

22

21

20

1101

1

1

0

1

1001 = 

1 x 23 +

1 x 22 +

0 x 21 +

1 x 20

N = (1101)2 = (13)10

LSB: Least Significant Bit.
MSB: Most Significant Bit.

Conversion from decimal to binary

Example: convert the decimal number 19 to binary.

(19)10 = (10011)2

Octal numeral system (base 8)

Base: b = 8

Digts used: {0, 1, 2, 3, 4, 5, 6, 7}

Example

82

81

80

123

1

2

3

(123)8 = 

1 x 82 +

2 x 81 +

3 x 80

N = (123)8 = (83)10

Hexadecimal numeral system (base 16)

Base: b = 16

Digts used: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}

The digit A represents 10 units, the digit B represents 11 units, etc.

Example

162

161

160

12A

1

2

A

(12A)16 = 

1 x 162 +

2 x 161 +

10 x 160

N = (12A)16 = (298)10

ASCII code

ASCII stands for American Standard Code for Information Interchange

ASCII is a character encoding standard.

It defines 128 characters numbered from 0 to 127 and coded in binary on 7 bits from 0000000 to 1111111.

In a computer, the code of each character of an ASCII text is stored in a byte (8 bits) the 8th bit is set to 0.

The 95 printable characters of the ASCII code:

!"#$%&'()*+,-./0123456789:;<=>?@
ABCDEFGHIJKLMNOPQRSTUVWXYZ[]^_`
abcdefghijklmnopqrstuvwxyz{|}~