Binary System Complement And Arithmatic

Binary system complements

Complements are used in the digital computers in order to simplify the subtraction operation and for the logical manipulations.

As the binary system has base r = 2. So the two types of complements for the binary system are 2's complement and 1's complement.

1's complement

The 1's complement of a number is found by changing all 1's to 0's and all 0's to 1's. This is called as taking complement or 1's complement. Example of 1's Complement is as follows.

2's complement

The 2's complement of binary number is obtained by adding 1 to the Least Significant Bit (LSB) of 1's complement of the number.

2's complement = 1's complement + 1

Example of 2's Complement is as follows.

Binary arithmetic

Binary arithmetic is essential part of all the digital computers and many other digital system.

1. Binary Addition

It is a key for binary subtraction, multiplication, division. There are four rules of binary addition.

In fourth case, a binary addition is creating a sum of (1 + 1 = 10) i.e. 0 is written in the given column and a carry of 1 over to the next column.

Example − Addition

2. Binary Subtraction

Subtraction and Borrow, these two words will be used very frequently for the binary subtraction. There are four rules of binary subtraction.

Example − Subtraction

Shortcut methods For Number System Conversion

There Are many methods to convert number system to other number System here there are few shortcuts methods to covert one number system to another number system   

1. Binary to Octal
2. Octal to Binary
3. Binary to Hexadecimal
4. Hexadecimal to Binary

1. Binary to Octal

Steps

1.  Divide the binary digits into groups of three (starting from the right).
2. Convert each group of three binary digits to one octal digit.

Example

Binary Number  10101₂

Calculating Octal Equivalent 

Step	Binary Number	Octal Number
Step 1	101012	010 101
Step 2	101012	28 58
Step 3	101012	258

 Binary Number  10101₂ = Octal Number 25₈

2. Octal to Binary

Steps

1.  Convert each octal digit to a 3 digit binary number (the octal digits may be treated as decimal for this conversion).
2. Combine all the resulting binary groups (of 3 digits each) into a single binary number.

Example

Octal Number  25₈

Calculating Binary Equivalent 

Step	Octal Number	Binary Number
Step 1	258	210 510
Step 2	258	0102 1012
Step 3	258	0101012

Octal Number 25₈ = Binary Number 10101₂

3. Binary to Hexadecimal

Steps

1. Divide the binary digits into groups of four (starting from the right).
2. Convert each group of four binary digits to one hexadecimal symbol.

Example

Binary Number  10101₂

Calculating hexadecimal Equivalent 

Step	Binary Number	Hexadecimal Number
Step 1	101012	0001 0101
Step 2	101012	110 510
Step 3	101012	1516

Binary Number  10101₂ = Hexadecimal Number  15₁₆

4. Hexadecimal to Binary

Steps

1. Convert each hexadecimal digit to a 4 digit binary number (the hexadecimal digits may be treated as decimal for this conversion).
2. Combine all the resulting binary groups (of 4 digits each) into a single binary number.

Example

Hexadecimal Number − 15₁₆

Calculating Binary Equivalent −

Step	Hexadecimal Number	Binary Number
Step 1	1516	110 510
Step 2	1516	00012 01012
Step 3	1516	000101012

Hexadecimal Number  15₁₆ = Binary Number 10101₂

Number System Conversion

There are many methods or techniques which can be used to convert numbers from one base to another. 

We'll demonstrate here the following

1.           Decimal System to Binary System
2.           Binary System to Decimal System
3.           Octal System to Binary System

1. Decimal to Binary System

Steps

1. Divide the decimal number to be converted by the value of the new base.
2. Get the remainder from Step 1 as the rightmost digit (least significant digit) of new base number.
3. Divide the quotient of the previous divide by the new base.
4. Record the remainder from Step 3 as the next digit (to the left) of the new base number.

Repeat Steps 3 and 4, getting remainders from right to left, until the quotient becomes zero in Step 3.

The last remainder thus obtained will be the Most Significant Digit (MSD) of the new base number.

Example :-

Decimal Number: 29₁₀

Calculating Binary Equivalent 

Steps	Operation	Result	Remainder
Step 1	29 / 2	14	1
Step 2	14 / 2	7	0
Step 3	7 / 2	3	1
Step 4	3 / 2	1	1
Step 5	1 / 2	0	1

As mentioned in Steps 2 and 4, the remainders have to be arranged in the reverse order so that the first remainder becomes the Least Significant Digit (LSD) and the last remainder becomes the Most Significant Digit (MSD).

Decimal Number 29₁₀ = Binary Number 11101₂.

2. Binary System to Decimal System

Steps

1. Determine the column (positional) value of each digit (this depends on the position of the digit and the base of the number system).
2. Multiply the obtained column values (in Step 1) by the digits in the corresponding columns.
3. Sum the products calculated in Step 2. The total is the equivalent value in decimal.

Example :-

Binary Number : 11101₂

Calculating Decimal Equivalent 

Step	Binary Number	Decimal Number
Step 1	111012	((1 × 24) + (1 × 23) + (1 × 22) + (0 × 21) + (1 × 20))10
Step 2	111012	(16 + 8 + 4 + 0 + 1)10
Step 3	111012	2910

Binary Number 11101₂ = Decimal Number  29₁₀

3. Octal System to Binary System

Steps

1. Convert the original number to a decimal number (base 10).
2. Convert the decimal number so obtained to the new base number.

Example :

Octal Number − 25₈

Calculating Binary Equivalent −

Step 1 − Convert to Decimal

Step	Octal Number	Decimal Number
Step 1	258	((2 × 81) + (5 × 80))10
Step 2	258	(16 + 5 )10
Step 3	258	2110

Octal Number  25₈ = Decimal Number  21₁₀

Step 2 – Convert  Decimal to Binary

Step	Operation	Result	Remainder
Step 1	21 / 2	10	1
Step 2	10 / 2	5	0
Step 3	5 / 2	2	1
Step 4	2 / 2	1	0
Step 5	1 / 2	0	1

Decimal Number 21₁₀ = Binary Number  10101₂

Octal Number 25₈ = Binary Number  10101₂