The document discusses decimal and hexadecimal number systems. The decimal system uses 10 symbols (0-9) in a positional notation where the value of each digit depends on its position. Hexadecimal uses 16 symbols (0-9 plus A-F) with a base of 16. To convert between number systems, the integer and fractional parts are converted separately by repeated division or multiplication by the new base. For example, to convert decimal 765.245 to hexadecimal, 765 divides into 16 with remainder 13 and fractional part 0.245 is multiplied by 16 repeatedly.

1. Prepared By:- Nisarg Amin
Topic:- Decimal & Hexadecimal
Number System

2. Decimal Number System
•The decimal number system is what we most commonly use.
It is composed of 10 symbols-0,1,2,3,4 ,5,6, 7,8 and 9. Using
these digits you can express any quantity it is also called the
base 10 system because it makes use of 10 digits. The number
base is also called the radix.
•The decimal system is a positional value system (also called
the positional value system or the place value notation)in
which the value of a digit depends on its position.

3. • For example, consider the whole number 256.
• The rightmost digit position is the ones position (101).
• The numeral in that position indicates how many ones are
present in the number.
• Then next position to the left is the tens, then the hundreds,
ten thousands, and so on.
• Each digit position has a weight that is ten times the weight
of the position to its right.

4. • The right most digit has the least positional value
(weight), therefore, it is called the Least
Significant Digit (LSD).
• The leftmost digit has the maximum positional value
(weight), therefore, it is called the Most
Significant Digit (MSD).
• In the above example, 6 is the LSD and 2 is the MSD.

5. Hexadecimal Number System
• The hexadecimal number system has 16 as the base
number. It has ten numeric digits 0, 1, 2, 3,4,5,6,7,8,9 and
six letters A=10, B=11, C=12, D=13, E=14 & F=15.
• The hexadecimal system is also a positional numbering
system. The value of a hexadecimal digit is expressed as
the power of 16.
• As an example, consider a hexadecimal number with a
fraction-- A65.C2, and the place values or positional values
of each digit as a power of 16.

6. Conversion from Decimal to Binary, Octal
and Hexadecimal
• A decimal number can be an integer, or a mixed number
with an integer part and a fractional part.
• Two processes are required for converting a decimal
number into any other number system, one for the
integer part and the other for the fractional part.
Integer Part:-
• This method involves repeatedly dividing the integer by
the new base until the quotient is zero and recording the
remainder after each step of division.
• Finally, when no more division can occur, write down the
remainders from bottom to top.

7. Fractional Part:-
• Multiply the fractional part by the new base.
• Record the integer part if there is one, else record 0.
• Repeat step 1 with the fractional part of the
previous multiplication and then repeat step until
the fractional part becomes 0. In case of infinite
calculations, generally 3 digits are taken.

8. Decimal to hexadecimal
•The conversion method of decimal to hexadecimal is the
same as that of decimal to binary except that the base
taken is 16 instead of 2.
•For example, to convert 765.24510 to the hexadecimal
equivalent, do the following:
Integer Part
16 765
16 47 - 13
16 2 - 15
0 - 2
0.245
x 16
3.920
x 16
14.720
x 16
11.520
765.24510 = 2FD.3EB16
Fractional Part

9. Conversion of Hexadecimal to
Decimal
• Similarly, to convert a hexadecimal number to its
equivalent decimal number by add up the product of each
digit value (0 to 9, A to F) with its positional value, as
shown below: