Hexadecimal and Binary Arithmetic
Hexadecimal
The number with base sixteen is called hexadecimal number. We can generate these numbers with the combination of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A,B, C, D, E, F. Where A=10, B=11, C=13, D=14, E=15, F=16. We can represent these numbers with suffix sixteen. E.g. (12AB)_{16 }Where A=10, B=11. The 4bit format of binary is used for hexadecimal to binary conversion.
Weighted value
Decimal  Octal  Hexadecimal  Binary 
0  0  0  0000 
1  1  1  0001 
2  2  2  0010 
3  3  3  0011 
4  4  4  0100 
5  5  5  0101 
6  6  6  0110 
7  7  7  0111 
8  8  1000  
9  9  1001  
10  A  1010  
11  B  1011  
12  C  1100  
13  D  1101  
14  E  1110  
15  F  1111 
Decimal to Hexadecimal Conversion
The decimal number is repetitively divided by sixteen and remainders are collected to represent hexadecimal numbers.
Example
 Convert following in hexadecimal number: (1047)_{10}= (417)_{16}
16 1047 7 16 65 1 4 =(417)_{16}
 Convert (333)_{10}into hexadecimal
16 333 13 16 20 4 4 (333)_{10}= (14D)_{16}Where D=13
Hexadecimal to Decimal
Each hexadecimal digit is multiplied by weighted positions, and sum of product is equal to decimal value.
Example
 (A37E)_{16}=(?)_{10}
A= 10
E= 14
=Ax16^{3}+ 3 x 16^{2}+ 7 x 16^{1}+ E x 16^{0}
=10x16^{3}+ 3 x 16^{2}+ 7 x 16^{1}+ 14 x 16^{0}
=40960 + 768 + 112 + 14
(41852)_{10}
Binary to Hexadecimal Conversion
The binary numbers are broken into sections of 4bit digits from last bit and its hexadecimal equivalent is assigned for each section.
Example
 Convert (11 10 11)_{2}into base 16.
(11 10 11)_{2}= 11 1011
0011= 3
1011= 11= B
(3B)_{16}
Note: You have to add 00 before first group to make four bits group. (11 to 0011)
Hexadecimal to Binary Conversion
Binary equivalent of each hexadecimal digit is written in 4bit format or section.
Example
Convert following in Binary numbers:
Algorithm
 Convert each Hexadecimal bit into equivalent binary number by making four bits group.
 Arrange all bits to make hexadecimal number.
(45AF)_{16}
4= 100= 0100 (Make four digit by adding 0 before the bits)
5= 101= 0101
A=10= 1010
F= 15= 1111
=(010001011010111)_{2}
(23AB)_{16}= (0010 0011 1010 1011)_{2}
= 0010, 3= 0011, A= 10, B=11= 1011
=(0010 0011 1010 1011)_{2}
Hexadecimal to Octal
Algorithm
 Convert hexadecimal into binary.
 make group bits from last bit.
 convert each into decimal numbers.
Example
 (ABC)_{16} to (?)_{2}
(ABC)_{16}
A=10= 1010
B=11= 1011
C= 12= 1100
=(101010111100)_{2}
Binary Arithmetic
You have to learn addition, subtraction, multiplication, and division of binary number. In brain, you have to keep that in the arithmetic of binary number, carry is written in binary (2) just like as 10 is used in decimal system for carry.
Addition  Subtraction  Multiplication  Division 
0 + 0 = 0 1 = 0 = 1  0  0 = 0 1  0 = 1  0 * 0 = 0 1 * 0 = 0  0 · 1 = 0 1 · 0 = not defined 
Example Hence, 11+ 11= 110  Example Here, 01 (right most) = 1 because we take carry 2 from left column and left remains 0. Hence, 10 01 = 01  Example  Example 11)1 1 0(10 
1  0  0  1  
+  0  +  1  +  0  +  1  
1  1  0  10 
Addition Example
1  0  1  0  First number  
1  0  0  1  Second number  
1  0  0  1  1 
Add following binary numbers
 1100 + 1111= 11011
1 Carry 1 1 0 0 1 1 1 1 11 0 1 1  110011+ 111100 + 100110= 10010110
1 1 1 1 1 Carry 1 1 0 0 1 1 1 1 1 1 0 0 1 0 0 1 1 1 1 0 0 1 0 1 1 0 Subtraction
The subtraction of binary is more interesting, but less complex for novice students, but not fear, jump to complement methods when confusing takes place in the traditional methods of subtraction.
Example
11011=011
 The first step is to equalize digits placing zero to the left side and make columns. You take right most columns and solve 01.
1 1 0 0 1 1 1  Next step, come to second column from where you have to solve again 01.
1 1 0 0 1 1 0 1 1
Example
100011= 0101
1 0 0 0 0 1 1 1 0 0 0 1 Example
1000 1= 111
1000 10= 110Multiplication
The multiplication of binary number is also like as decimal multiplication.
Example
110 x 11 = 101001 1 0 0 x 1 1 1 1 0 0 1 1 0 0 1 0 0 1 0 0  The first step is to equalize digits placing zero to the left side and make columns. You take right most columns and solve 01.
 The number with base sixteen is called hexadecimal number.
 Each hexadecimal digit is multiplied by weighted positions, and sum of product is equal to decimal value.
 The decimal number is repetitively divided by sixteen and remainders are collected to represent hexadecimal numbers.
 The binary numbers are broken into sections of 4bit digits from last bit and its hexadecimal equivalent is assigned for each section.
 Binary equivalent of each hexadecimal digit is written in 4bit format or section.
A number system that uses sixteen different digits to represent different values is known as hexadecimal number system. The base of hexadecimal number system is 16 because it consist sixteen digits from 0 to 9 and A to F to represent values from ten to fifteen. The digits A, B, C, D, E and F of hexadecimal number represent the decimal numbers 10, 11, 12, 13, 14 and 16 respectively.
The decimal equivalent of a hexadecimal number is the sum of the digits multiplied by 16 with their corresponding weights.
Solution:
2 1 0 (weight)
Hexadecimal Number: A 2 E
Decimal Equivalent = A×16^{2} + 2×16^{1} + E×16^{0}
= 10×256 + 2×16 + 14×1
= 2560 + 32 + 14
= 2606
Hence, (A2E)_{16} = (2606)_{10}
Hexadecimal digit is represented in 4 bits. A hexadecimal number is converted to its binary equivalent by just substituting the respective binary value for each digit of the hexadecimal number.
Solution:
Hexadecimal Number: 4 A 5
Binary Equivalent: 100 1010 0101 (From Binary Table)
Hence, (4A5)_{16} = (10010100101)_{2}There is no any direct method to convert hexadecimal number into octal. So, at first the given hexadecimal number is converted into its binary equivalent then the result will be converted into octal as in previous methods.
Solution:
Hexadecimal Number: 4 2 C
Binary Equivalent: 100 0010 1100 (From Binary Table)
Again, Paired Binary
For octal (3 bits): 10 000 101 100 (pairing from rightmost digit)
Hexadecimal Equivalent: 2 0 5 4
Hence, (42C)_{16} = (2054)_{8}Binary numbers are added to the same manner as decimal numbers. There are only four possible combinations resulting from the addition of two binary digits.
Rules for binary addition:
A  B  A+B 
0  0  0 
0  1  1 
1  0  1 
1  1  10 
The rules for subtraction are the same in the binary system as in the decimal system. The result of the subtraction of binary digit is either 0 or 1.
Rules for binary subtraction:
A  B  AB 
0  0  0 
1  0  1 
1  1  0 
0  1  1 
Binary numbers are multiplied in the same manner as decimal numbers. We need to remember the rules of binary addition.
Rules for binary multiplication:
A  B  A×B 
0  0  0 
1  0  0 
0  1  0 
1  1  1 
Binary numbers are divided in the same manner as decimal numbers. We need to remember the rules of binary subtraction.
Rules for binary division:
A  B  A/B 
0  0  0 
1  0  1 
1  1  Not defined 
0  1  Not defined 

The number with base sixteen is called __________.
decimal number
binary number
hexa decimal number
octal number

What is the decimal value of the hexadecimal number 777?
1191
1199
1911
1919

What is the hexadecimal value of the binary number 11 10 11 ?
5B
3C
8B
3B

What is the binary value of the hexadecimal number 45AF?
101011010111
10001011010111
10001111010111
100010110111

What is the octal value of the hexadecimal number 7?
8
7
9
5

What is the binary value of the hexadecimal number F?
1000
1010
1100
1111

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

What is the binary value of the hexadecimal number 8?
1001
1000
1101
1111

The base of hexadecimal number system is _________.
2
16
8
10

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