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Spring 2017 / 20181
Numbering systems
• The decimal numbering system is widely used, because the people Accustomed(اعتاد) to use the hand fingers in their counting.
• But with the development of the computer science another numbering systems are founded. Such as:– Binary, Octal, Hexadecimal, or any base r > 2.
• Numbering systems classification– Positional numbering systems: that depends on the position of
the digit in the number. Which means that the same digit can take different values according to his position [decimal, Binary, Octal, Hexadecimal]
– None positional numbering systems: that don’t use digit waits in the numbers. [ROMAN numbering system]
Spring 2017 / 20182
Numbering Systems Radix
• The base or the radix of any numbering system is equal to the number of symbols that used to represent its numbers
• The decimal number system is said to be of Base (Radix), 10 because it uses 10 digits
[ 0,1,2,…….,9]
• The Binary number system is said to be of Base (Radix), 2 because it uses 2 digits [ 1,0 ]
• The Octal number system is said to be of Base (Radix), 8 because it uses 8 digits [0,1,2,3….,7]
• The Hexadecimal number system is said to be of Base (Radix), 16 because it uses 16 digits [0,1,2,3…..,A,B,C,D,E,F]
Spring 2017 / 20183
decimal numbers
– The decimal number representation
X= (djdj-1….d3d2d1d0.d-1d-2d-3,…d-(k-1)d-k)10, where
dj (digit): 0,1,2….9 ,
j ( position): …3,2,1,0 ,-1,-2,-3…..
– X as a decimal number with j digits can be represented as
X=10j*dj+10j-1*dj-1….+102*d2+101*d1+100*d0+10-1*d-1+10-2*d-2+….10-(k-1)*d-(k-
1)+10-k*d-k
– the integer part of the number is represented by an ascending positive power
– The fraction part of the number is represented by an ascending negative power
Spring 2017 / 20185
Positional representation rule for numbers
• All positional numbering systems are the same in that they are depends on the wait of the digits in the number, and different only in the base.
• So we can use the same method of representing decimal numbers to be accepted by any other numbering system. As follows:
•
N=
• Where : R- base of the system,
• ai- the digits of the number,
• m- the number of the fraction digits,
• n- number of the integer digits - 1
i=-m
i=n
ai*Ri
Spring 2017 / 20186
Binary system
• Is of base 2 and has two digits (0,1)
• It is the system that used by the computer
• It is represented in the electronic circuits as two states:
on – 1
off – 0
Spring 2017 / 20187
Binary representation
• Using the previous rule we can represent
any binary number as
x=….b2*22+b1*2
1+b0*20+b-1*2
-1+b-2*2-2…..
Where: bi=0,1
Spring 2017 / 20189
Octal system
• The base of this system is 8.
• Its symbols are (0,1,2….7)• Using the previous rule we can represent any octal
number as
x=….o2*82+o1*8
1+o0*80+o-1*8
-1+o-2*8-2…..
Where: oi=0,1,2,3,4,5,6,7
Spring 2017 / 201811
Hexadecimal system
• The base of this system is 16
• it has 16 digits(0,1,2….9,A,B,C,D,E,F)
• A=10,B=11,C=12,D=13,E=14,F=15• Using the previous rule we can represent any
hexadecimal number as
x=….H2*162+H1*161+H0*160+H-1*16-1+H-2*16-2…..
Where: Hi=0,1,2,3,…..9, A,B,C,D,E,F
Spring 2017 / 201813
Number base conversions
• Base R to decimal conversion• A number expressed in base R can be converted to its decimal equivalent
by using the Positional representation rule [ multiplying each coefficient
with the corresponding power of R and adding]
N=i=-m
i=n
ai*Ri
Spring 2017 / 201815
Number base conversions
• Octal To Decimal Conversion
• Examples:
• (312.1)8=3*82+1*81+2*80+1*8-1=(3X64)+(8)+(2)+(1/8)=(202.125)10
• (752)8=7*82+5*81+2*80=(7X64)+(5X8)+(2X1)=(490)10
Spring 2017 / 201817
Number base conversions
• Decimal to base R conversion
• The conversions from decimal to any base R number system is more convenient if the number is separated into an integer part and a fraction part and the conversion of each part is done separately.
• Example
• (d3d2d1d0.d-1d-2d-3(10 ( )r
• (d3d2d1d0(10 ( )r
• (.d-1d-2d-3(10 ( )r
• Integer part divided by base R
• Fraction part multiplied by base R
Spring 2017 / 201818
From decimal to binary
• We use the remainder method as follows:a) Integer part
1. The integer part of the number is divided by 2
2. The reminder calculated
3. The quotient resulted from step 1 divided by 2
4. The reminder calculated
5. The previous steps are repeated until the
quotient becomes equal to 0
6. The required binary number is the collection of
reminders ordered from the last reminder to
the first one
Spring 2017 / 201819
From decimal to binary
b) fraction part
• Converting fraction part is done using the multiplication
instead of division as follows:
1. multiply the number by 2
2. Get the integer part from the result obtained from step 1
3. Multiply the fraction obtained in step1 by 2
4. Get the integer part from the result obtained from step 3
5. Repeat the above steps until we have the fraction part of the
multiplication equals to 0 or we reached the required precision
6. The required number is the collection of integer digits obtained
after each multiplication.
Spring 2017 / 201821
Examples
• Example : Convert the following decimal numbers to Binary numbers
• 1- (29)10 2-(53)10 3-(41)10
1: Division Remainder
29/2 1 LSB
14/2 0
7/2 1
3/2 1
1/2 1 MSB
0
(29)10 =(11101)2
2: (53)10 Division Remainder
53/2 1 LSB
26/2 0
13/2 1
6 /2 0
3 /2 1
1 /2 1 MSB
0
(53)10 =(110101)2
3 : (41)10
41/2 1 LSB
20/2 0
10/2 0
5 /2 1
2 /2 0
1 /2 1 MSB
0
(41)10 =(101001)2
Spring 2017 / 201823
Examples
• Example: Convert the following decimal number into binary Number
• 1: (0.828125 )10
0.828125 x2 = 1+ 0.65625 1 MSB
0.65625 x 2 = 1+ 0.3125 1
0.3125 x 2 = 0+ 0.625 0
0.625 x 2 = 1+ 0.25 1
0.25 x 2 = 0+ 0.5 0
0.5 x 2 = 1+ 0 1 LSB
(0.828125)10 =(0.110101)2
• 2: (41. 625 )10 =(41)10 + (0.625 )10
from the above example (41)10 is represented by (101001)2 while
(0.625 )10 is
0.625 x 2 = 1+ 0.25 1 MSB
0.25 x 2 = 0+ 0.5 0
0.5 x 2 = 1+ 0 1 LSB
(0.625 )10 = (101)2
so (41. 625 )10 = (101001.101)2
Spring 2017 / 201824
Decimal to octal
• Example: Convert the following decimal number into
octal Number (153.6875)10
• (231.54)8
Spring 2017 / 201825
Decimal to hexadecimal
• Example: Convert the following decimal number into
hexadecimal Number (125.34375)10
Spring 2017 / 201826
Binary to octal or hexadecimal and vise
versa
• Each octal digit corresponds to 3 binary digits
• Base 8 = 23
• Each hexadecimal digit corresponds to 4 binary
digits
• Base 16=24
Spring 2017 / 201831
Octal or hexadecimal to binary
• Each octal digit is converted to 3 binary digits according
to the table
Spring 2017 / 201832
Octal or hexadecimal to binary
• Each hexadecimal digit is converted to 4 binary digits
according to the table