Number System

 Important Number System for computer are :-

• Decimal Number System (0-9)
• Binary Number System (0-1)
• Hexadecimal Number System (0-9, A-F)
• Octal Number System (0-7)

1.) Decimal Number System : 

      Base : 10

      Values : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

2.) Binary Number System : 

      Base : 2

      Values : 0, 1

3.) Hexadecimal Number System : 

      Base : 16

      Values : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A(10), B(11), C(12), D(13), E(14), F(15)

     

4.) Octal Number System : 

      Base : 8

      Values : 0, 1, 2, 3, 4, 5, 6, 7

Binary to Decimal Conversion:

Binary (base-2) numbers use only two digits: 0 and 1. Decimal (base-10) numbers use ten digits: 0 through 9. The conversion process relies on the idea of place value, just like in decimal numbers.

In decimal, each digit's position represents a power of 10 (e.g., in 123, 3 is $3\times 10^0$, 2 is $2\times 10^1$, 1 is $1\times 10^2$).

In binary, each digit's position represents a power of 2.

The Step-by-Step Process

1. Write down the Binary Number: Start by writing the binary number you want to convert.

2. Assign Place Values (Powers of 2): Starting from the rightmost digit of the binary number and moving to the left, assign increasing powers of 2 to each position.

   • The rightmost digit gets 2^0 (which is 1).

   • The next digit to the left gets 2^1 (which is 2).

   • The next gets 2^2 (which is 4), and so on.

3. Multiply and Sum:

   • For each digit in the binary number:

     • If the binary digit is '1', multiply it by its corresponding power of 2.

     • If the binary digit is '0', the result for that position is 0 (as $0\times \text{any number}=0$).

   • Add up all the results from the multiplications. This sum is your decimal equivalent.

Example

Let's convert the binary number 11010 to decimal.

1. Binary Number: 11010

2. Assign Place Values (from right to left):

   • Rightmost '0' is at position 2^0

   • Next '1' is at position 2^1

   • Next '0' is at position 2^2

   • Next '1' is at position 2^3

   • Leftmost '1' is at position 2^4

3. Multiply and Sum:

   Binary Digit	Position	Power of 2	Calculation	Result
   1	${2^}^4$	$16$	$1\times 16$	$16$
   1	${2^}^3$	$8$	$1\times 8$	$8$
   0	${2^}^2$	$4$	$0\times 4$	$0$
   1	${2^}^1$	$2$	$1\times 2$	$2$
   0	${2^}^0$	$1$	$0\times 1$	$0$
   			Sum:	$16+8+0+2+0=26$