Decimal and Binary Number
Number System
History of number starts from primitive age of human being. The development of number system has integrated with the development of human beings. In primitive age, people used to count stones and pebbles. Hindu philosophers have developed the number system called decimal numbers. From the number various other numbers are derived and developed. These numbers are binary numbers, octal numbers, hexadecimal numbers etc.
Decimal (Denary) Numbers
The number of base or radix ten is called decimal numbers. It is first number system in which all the ancient and modern mathematical calculation is done. Other number system is derived from this number. It is generated with the combination of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. We can represent these numbers with suffix two. Eg (9810)_{10}
The following chart displays relations among several number systems derived from the decimal number.
System  Base  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15 
Hexadecimal  16  0  1  2  3  4  5  6  7  8  9  A  B  C  D  E  F 
Decimal  10  0  1  2  3  4  5  6  7  8  9  
Octal  8  0  1  2  3  4  5  6  7  
Binary  2  0  1 
Binary Number
Binary Number is a number of two base numbers. It is represented by 1 and 0. 1 or 0 is called Binary digits.
We can generate this number with the combination of 0 and 1. It is represented with suffix two. Eg (10101)_{2}.
Following table shows some decimal number and their equivalent binary numbers.
Decimal  Binary 
0  0 
1  1 
2  10 
3  11 
4  100 
5  101 
6  110 
7  111 
8  1000 
9  1001 
Conversion of Binary to Decimal and vice versa
We multiply each binary digit by its weighted position, and add each of the weighted value together.
Weighted value:
25  24  23  22  21  20 
32  16  8  4  2  1 
To convert decimal number to binary number,repeated division by two is needed.
Convert following into binary numbers:
(45)_{10}= (101101)_{2}
2  45  1 
2  22  0 
2  11  1 
2  5  1 
2  2  0 
1 
=(101101)_{2}
(225)_{10}= (10111001)_{2}
2  225  1 
2  112  0 
2  56  0 
2  23  1 
2  11  1 
2  5  1 
2  2  0 
1 
= (10111001)_{2}
 The arrangement of number in a computer system is known as computer number system.
 There are different types of computer number system.They are binary number system( have base 2), octal number system(have base 8), decimal number system(have base 10 ) and hexadecimal number system(have base 16).
 Binary Number is a number of two base numbers. It is represented by 1 and 0.
 The number of base or radix ten is called decimal numbers.
A number system act as the symbols used to express quantities on the basis for counting, comparing amounts, performing calculations, and representing value. The different number system use different digits or symbols to represent numbers. The number of digits used in a number system is known as base of the number system. The different number systems have different base number. The different types of number system are: 
 Decimal Number System
 Binary Number System
 Octal Number System
 Hexadecimal Number System
A number system that uses ten different digits to represent different values is known as decimal number system. The base of decimal number system is 10 because it consist ten digits from 0 to 9. Decimal number can be expressed by using powers of 10.
A positive decimal integer can be converted to binary through successive division by 2 till the quotient becomes zero and sequential collection of remainder on last come first basis (i.e. bottom to top).
Solution:
Remainder  
2  349  1 
2  174  0 
2  87  1 
2  43  1 
2  21  1 
2  10  0 
2  5  1 
2  2  0 
2  1  1 
0 
Hence, (349)_{10} = (101011101)_{2}
A positive decimal integer can be converted to octal through successive division by 8 till the quotient becomes zero and sequential collection of remainder on last come first basis (i.e. bottom to top).
Solution:
Remainder  
8  427  3 
8  53  5 
8  6  6 
0 
Hence, (427)_{10} = (653)_{2}
A positive decimal integer can be converted to hexadecimal through successive division by 16 till the quotient becomes zero and sequential collection of remainder on last come first basis (i.e. bottom to top). But the remainder 10 or above is represented by the capital letters from A to F respectively.
Solution:
Remainder  
16  1495  7 
16  93  13 (D) 
16  5  5 
0 
Hence, (1495)_{10} = (5D7)_{16}
A number system that uses two different digits to represent different values is known as binary number system. The base of binary number system is 2 because it consist two digits 0 and 1. Each digit of the binary number system is called Binary Digit (BIT). The binary number system is used in the computer.
The decimal equivalent of a binary number is the sum of the digits multiplied by 2 with their corresponding weights.
Solution:
8 7 6 5 4 3 2 1 0 (weight)
Binary number: 1 0 1 0 1 1 1 0 1
Decimal equivalent: 1×2^{8 }+ 0×2^{7} + 1×2^{6} + 0×2^{5} + 1×2^{4} + 1×2^{3} + 1×2^{2} + 0×2^{1} + 1×2^{0}
=1×256 + 0×128 + 1×64 + 0×32 + 1×16 + 1×8 + 1×4 + 0×2 + 1×1
=256 + 0 + 64 + 0 + 16 + 8 + 4 + 0 + 1
= (349)_{10}
Hence, (1010111101)_{2} = (349)_{10}Octal digit is represented in 3 bits. So, a binary number is converted to its octal equivalent by grouping their successive 3 bits of the binary number starting from the least significant bit (rightmost digit) and then replacing each 3 bit group by its octal equivalent.
Binary Table
Octal  Binary 
0  000 
1  001 
2  010 
3  011 
4  100 
5  101 
6  110 
7  111 
Solution:
Binary Number: 1 0 1 1 1 0 1 1 1 1 0
Paired Binary Digits: 10 111 011 110
Octal Equivalent : 2 7 3 6 (From Binary Table)
Hence, (10111011110)_{2} = (2736)_{8}Hexadecimal digit is represented in 4 bits. So, a binary number is converted to its hexadecimal equivalent by grouping together successively 4 bits of the binary number starting from the least significant bit (rightmost digit) and then replacing each bitgroup by its hexadecimal equivalent.
Binary Table
Hexadecimal  Binary 
0  0000 
1  0001 
2  0010 
3  0011 
4  0100 
5  0101 
6  0110 
7  0111 
8  1000 
9  1001 
A  1010 
B  1011 
C  1101 
D  1110 
E  1111 
Solution:
Binary Number: 1 0 1 1 0 1 1 0 1
Paired Binary Number: 1 0110 1101
Hexadecimal Equivalent: 1 6 D
Hence, (101101101)_{2 }= (16D)_{16}

According to addition rule of binary number system, 1+1 equals to:
2
11
4
10

According to subtraction rule of number system, 01 equals to:
11
1
01
10

According to multiplication rule of number system, 0*1 equals to:
10
11
1
100

Octal number system consists of _________.
10 digits
8 digits
2 digits
16 digits

A number system having two base numbers is:
binary number system
hexadecimal number system
decimal number system
octal number system

What is the decimal value of the binary number 111?
3
8
9
7

What is the decimal value of the binary number 1000?
7
9
6
8

What is the binary value of the decimal number 45?
101111
111101
101101
11101

What is the binary value of the decimal number 226?
11101011
10111011
101111
11101111

The number of base or radix ten is called _________.
decimal number
octal number
hexa decimal number
binary number

The base of decimal number system is _________.
8
2
5
10

The base of binary number system is _________.
10
20
16
2

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