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 4-bit format of binary is used for hexadecimal to binary conversion.

Weighted value

Decimal to Hexadecimal Conversion

The decimal number is repetitively divided by sixteen and remainders are collected to represent hexadecimal numbers.
Example
1. Convert following in hexadecimal number: (1047)₁₀= (417)₁₆

=(417)₁₆

2. Convert (333)₁₀ into hexadecimal

(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

1. (A37E)₁₆=(?)₁₀
A= 10
E= 14
=Ax16³+ 3 x 16²+ 7 x 16¹+ E x 16⁰
=10x16³+ 3 x 16²+ 7 x 16¹+ 14 x 16⁰
=40960 + 768 + 112 + 14
(41852)₁₀

Binary to Hexadecimal Conversion

The binary numbers are broken into sections of 4-bit digits from last bit and its hexadecimal equivalent is assigned for each section.

Example

1. 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 4-bit 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.

1. (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

2. (23AB)₁₆= (0010 0011 1010 1011)₂
= 0010, 3= 0011, A= 10, B=11= 1011
=(0010 0011 1010 1011)₂

Hexadecimal to Octal

• Convert hexadecimal into binary.
• make group bits from last bit.
• convert each into decimal numbers.

Example

1. (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 + 1 = 1 1 + 1 = 0	0 - 0 = 0 1 - 0 = 1 0 - 1 = 1 and carry 1 1 - 1 = 0	0 * 0 = 0 1 * 0 = 0 0 * 1 = 0 1 * 1 = 1	0 · 1 = 0 1 · 0 = not defined 0 · 0 = 0 1 · 1 = 1
Example Here, 1+1(right most)= 0 and its carry 1 is added to left columns as 1+1 = 11 Hence, 11+ 11= 110	Example Here, 0-1 (right most) = 1 because we take carry 2 from left column and left remains 0. Hence, 10- 01 = 01	Example	Example

Addition Example

Add following binary numbers

1. 1100 + 1111= 11011

2. 110011+ 111100 + 100110= 10010110

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

110-11=011

The first step is to equalize digits placing zero to the left side and make columns. You take right most columns and solve 0-1.

Next step, come to second column from where you have to solve again 0-1.

Example

1000-11= 0101

Example

1000 -1= 111

1000 -10= 110

Multiplication

The multiplication of binary number is also like as decimal multiplication.

Example

110 x 11 = 10100

Division

The division process is like as division of decimal numbers.

Quotient 1 and remainder 100.