MEng1052
Introduction to Computer and programming
Debrebrhan University By:Endrias Haile

Positional number system
Integers
MEng1052
Introduction to Computer and programming
Debrebrhan University By:Endrias Haile

Positional number system
The following shows the place values for the integer +224 in the decimal system.
Note that the digit 2 in position 1 has the value 20, but the same digit in position 2 has the
value 200. Also note that we normally drop the plus sign, but it is implicit.
MEng1052
Introduction to Computer and programming
Debrebrhan University By:Endrias Haile
Examp
le

Reals
Example 2.3
The following shows the place values for the real number +24.13.

Positional number system
The binary system (base 2)
In this system the
base b = 2
and we use only 2 symbols
S = {0, 1}
The symbols in this system are often referred to as
binary digits
or bits(binary digit).
8 bits = 1
byte
4 bytes = 1 word (in many systems)
MEng1052
Introduction to Computer and programming
Debrebrhan University By:Endrias Haile

2.
94
Integers
Example
The following shows that the number (11001)
2
in
binary is the same as 25 in decimal. The subscript 2
shows that the base is 2.

Reals
Example
The following shows that the number (101.11)
2
in binary is equal
to the number 5.75 in decimal.

Positional number system
The hexadecimal system (base 16)
In this system the
base b = 16
and we use 16 symbols to
represent a number.
The set of symbol is
S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}
Note that the symbols A, B, C, D, E, F are equivalent to 10,
11, 12, 13, 14, and 15 respectively.
The symbols in this system are often referred to as
hexadecimal digits
.
MEng1052
Introduction to Computer and programming
Debrebrhan University By:Endrias Haile

2.
97
Integers
The following shows that the number (2AE)16 in
hexadecimal is equivalent to 686 in decimal.
Example

Positional number system
The octal system (base 8)
In this system the
base b = 8
and we use 8 symbols to
represent a number.
The set of symbol is
S = {0, 1, 2, 3, 4, 5, 6, 7}
The symbols in this system are often referred to as
octal
digits
.
MEng1052
Introduction to Computer and programming
Debrebrhan University By:Endrias Haile

2.
99
Integers
Example
The following shows that the number (1256)
8
in octal is the same as
686 in decimal.

Conversion
We need to know how to convert a number in one
system to the equivalent number in another system.
Since the decimal system is more familiar than the
other systems, we first show how to covert from any
base to decimal. Then we show how to convert from
decimal to any base. Finally, we show how we can
easily convert from binary to hexadecimal or octal and
vice versa.
MEng1052
Introduction to Computer and programming
Debrebrhan University By:Endrias Haile

Decimal to any base

Decimal to any base
MEng1052
Introduction to Computer and programming
Debrebrhan University By:Endrias Haile

Decimal to any base
MEng1052
Introduction to Computer and programming
Debrebrhan University By:Endrias Haile
The following shows how to convert 35 in decimal to binary. We
start with the number in decimal, we move to the left while
continuously finding the quotients and the remainder of division
by 2. The result is 35 = (100011)
2
.

Decimal to any base
MEng1052
Introduction to Computer and programming
Debrebrhan University By:Endrias Haile

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