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1. Department of Computer
Science & Application
COMPUTER NUMBER SYSTEM
ANIL KUMAR PRAJAPATI

2. What is Number System
 A number system is a system representing numbers. It is also called the system
of numeration and it defines a set of values to represent a quantity. These
numbers are used as digits and the most common ones are 0 and 1, that are used
to represent binary numbers. Digits from 0 to 9 are used to represent other types
of number systems.

3. Types of Number Systems
 There are different types of number systems in which the four main types are:
Binary number system
(Base - 2) like that (110101)2
Octal number system
(Base - 8) like that (826)8
Hexadecimal number system
(Base - 16) like that (27FBE)16
Decimal number system
(Base - 10) like that (102345)10

4. Binary Number System
 The binary number system uses only two digits: 0 and 1. The numbers in this system have
a base of 2. The binary system is applied internally by almost all latest computers and computer-
based devices because of its direct implementation in electronic circuits using logic gates. Every
digit is referred to as a bit. binary number, the rightmost digit is called least significant bit
(LSB) and leftmost digit is called most significant bit (MSB). The binary number are represent
like (110101)2
1 1 0 0 1 1
MSB LSB

5. Binary Number System
 Binary number system conversion into decimal And decimal equivalent of this
number is sum of product of each digit with its positional value.
 Binary to Decimal conversation
1100102 = 1×25 + 1×24 + 0×23 + 0×22 + 1×21 + 0×20
= 32+16 + 0 + 0 + 2 + 0
= 5010

6. Decimal to Binary Conversion
 To convert Decimal to binary numbers, the following steps should be followed:-
 1. Take any decimal number and divide it by “2". After dividing, you will get some results along with the remainder.
 2. If the decimal number chosen by you is even, then the result will be in a whole number and it will give the remainder 0.
 3. If the decimal number chosen by you is odd, then the number will not be divided fully and you will get the remainder “1”.
 4. Continue dividing the number till you get the quotient 0
 5. Now place all the remainders in the series of Least Significant Bit (LSB) at the top and the Most Significant bit (MSB) at
the bottom.

7. Decimal to Binary Conversion Example
 Let us Convert the Decimal Number 75 into a Binary Number.
Division of Decimal
Number by 2
Quotient Remainder Binary
75/2 37 1 (LSB) 1
37/2 18 1 1
18/2 9 0 0
9/2 4 1 1
4/2 2 0 0
2/2 1 0 0
1/2 0 1 (LSB) 1

8. Octal Number System
 Octal number system has eight digits – 0, 1, 2, 3, 4, 5, 6 and 7. Octal number
system is also a positional value system with where each digit has its value
expressed in powers of 8, For example: 458, 538, 2418 are some examples of
numbers in the octal number system. as shown here − (Octal to Decimal)
3268 = 3×82 + 2×81 + 6×80
= 192 + 16 + 6
= 21410

9. Decimal to Octal Conversion
Division Remainder (R)
210 / 8 = 26 2
26 / 8 = 3 2
3 / 8 = 0 3
1. Take the decimal number as dividend.
2. Divide the number by 8 (as 8 is the base of octal so divisor here).
3. Preserve the remainder in an array (and it will be: 0, 1, 2, 3, 4, 5, 6 or 7 because of the divisor 8).
4. Repeat the above two steps until the amount is bigger than zero.
5. Print the mentioned array in the reverse order.
Let us Convert the Decimal Number 210 into a Octal Number. (Therefore, 21010 = 32216)

10. Hexadecimal Number System
 The hexadecimal number system uses sixteen digits/alphabets: 0,1,2,3,4,5,6,7,8,9 and
A,B,C,D,E,F with the base number as 16. Here, A-F of the hexadecimal system means
the numbers 10-15 of the decimal number system respectively. This system is used in
computers to reduce the large-sized strings of the binary system. For example: 7A316,
6E16, 4C2A16 are some examples of numbers in the hexadecimal number system.
(Hexadecimal to Decimal )
17FB16 = 1×163 + 7×162 + 15×161 + 10×160
= 4096 + 1792 + 240 +10
= 613810

11. Decimal to Hexadecimal Conversion
Divide by 16 Quotient Remainder Hex Value
600/ 16 37 8 8
37/ 16 2 5 5
2/ 16 0 2 2
1. Take the decimal number as dividend.
2. Divide the number by 16 (as 16 is the base of octal so divisor here).
3. Preserve the remainder in an array
(and it will be: 0, 1, 2, 3, 4, 5, 6 , 7, 8, 9, A, B, C, D, E, F because of the divisor 16).
1. Repeat the above two steps until the amount is bigger than zero.
2. Print the mentioned array in the reverse order.
Let us Convert the Decimal Number 210 into a Octal Number. (Therefore, 60010 = 25816)

12. Decimal Number System
 Decimal number system is a base 10 number system having 10 digits from 0 to 9. This means
that any numerical quantity can be represented using these 10 digits. Decimal number system is
also a positional value system. This means that the value of digits will depend on its position.
Let us take an example to understand this.
123410 = 1×103 + 2×102 + 3×101 + 4×100
= 1000 + 200 + 30 + 4
= 123410

13. Regards:
Anil Kumar Prajapati