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 / 20184

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 / 20188

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 / 201810

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 / 201812

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 / 201814

Number base conversions

• Binary to decimal

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 / 201816

Number base conversions

• Hexadecimal to Decimal Conversion

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 / 201820

Examples

( )2

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 / 201822

Examples

• Convert the following decimal number to a

binary (0.78125)10

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 / 201827

Binary to octal

• Group each 3 binary digits from the radix point

Spring 2017 / 201828

Binary to octal

Spring 2017 / 201829

Binary to hexadecimal

• Group each 4 binary digits from the radix point

Spring 2017 / 201830

Binary to hexadecimal

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

Spring 2017 / 201833

Examples

Spring 2017 / 201834

Examples