Binary Addition : Radix or Base for Binary numbers are 2 and having two digits 0 and 1. so while adding we follow following rules:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0 and carry 1 (Decimal 2 and Binary 10)
1 + 1 = 0 and carry 1 (Decimal 2 and Binary 10)
1 + 1 + 1 = 1 and carry 1 (Decimal 3 and Binary 11)
1 + 1 + 1 + 1 = 0 carry 10 (Decimal 4 and Binary 100)
| Example : 1 | Example : 2 | |||
| Carry >> | 1 1 1 1 | Carry >> | 1 1 1 1 1 | |
| 1 1 1 0 1 | 1 0 1 0 . 1 0 1 | |||
0 1 1 1 1 | 1 1 0 1 . 1 0 1 | |||
| Answer : | 1 0 1 1 0 0 | Answer : | 1 1 1 0 0 . 0 1 0 | |
Binary Subtraction
While subtracting we follow the following rules:
0 - 0 = 0
0 + 1 = 1 borrow 1
1 + 0 = 1
1 + 1 = 0
| Example : 1 | Example : 2 | |||
Borrow >> | 1 1 1 | Borrow >> | 1 | |
| 1 1 1 0 1 | 1 0 1 0 . 1 0 1 | |||
0 1 1 1 1 | 1 0 0 1 . 1 0 0 | |||
| Answer : | 0 1 1 1 0 | Answer : | 0 0 0 1 . 0 0 1 | |
Subtraction often is very tricky, you can solve it by using 1's Compliment or 2's Compliment
Like you have 50 - 12 = 38 same 50 + (-12) = 38
Compliment means Invert or all 0's change to 1 and 1's become 0. So Rules for 1's Compliment is :
Make the both side equal by placing leading 0's. Replace all the 1's with 0's and the 0's with 1's, then add both the numbers.
If there was 4 bits and after adding you getting five, the leading bits known as carry bits. If there is a carry bits then remove that from leading position and add to the result of the addition and write the answer down.
Example :(10010)2 - (1001)2 Minuend : 10010 and Subtrahend : 1001. Minuend contain 5 bits where Subtrahend having 4, so it will be 01001 (both side with same numbers of digits and done by adding leading zeros) 01001 (decimal 9) 0110 (one's complement) Add
| Example : 1 | Addition | ||
Carry >> | 1 1 1 | ||
| 18 | 1 0 0 1 0 | ||
+ -9 | 1 0 1 1 0 | ||
| 1 0 1 0 0 0 | |||
| 0 1 0 0 0 | |||
| + 1 | |||
| Answer : | +9 | 0 1 0 0 1 | |
1 is a carry bit because after adding two four bits number we are getting 5 bits, remove the carry from there and add to the remaining part of the result.
Result of : (10010)2 - (1001) = (1001)2 or 18 + (-9) = 9
Result of : (10010)2 - (1001) = (1001)2 or 18 + (-9) = 9
The two's complement of a binary number is obtained by:
1. Replacing all the 1's with 0's and the 0's with 1's (1's Compliment).
2. Adding 1 to the result of 1's compliment using the rules of binary addition.
Now you have the two's complement. The decimal subtraction 18 - 9 = 9 is the same as adding (18) + (-9) = 9
1. Convert the number to be subtracted to its two's complement:(10010)2 - (1001)2 = (10010)2 - (01001)2, 01001 (decimal 9) 10110 (one's complement) + 00001 (add 1 (two's complement)) 10111 (We get)
10111 now represents -9.
10111 now represents -9.
| Example : 1 | Addition | ||
Carry >> | 1 1 1 | ||
| 18 | 1 0 0 1 0 | ||
+ -9 | 1 0 1 1 1 | ||
| Answer : | +9 | 1 0 1 0 0 1 | |
Note that the final carry 1 is ignored and dropped permanently. So you get :Result of : (10010)2 - (1001) = (1001)2 or 18 + (-9) = 9
If no carry that indicate a negative result. When a subtraction produce negative result then you have to re-compliment the result of addition to get the correct answer.
(1000)2 - (1001)2 = (1000)2 - (0110)2, (one's complement).
| Example : 1 | Addition | ||
Carry >> | |||
| 8 | 1 0 0 0 | ||
+ -9 | 0 1 1 0 | ||
| 1 1 1 0 | |||
If no carry bit then re-compliment the result 1110 = 0001 (after 1's compliment) Result of : (1000)2 - (1001) = (-1)2 or 8 + (-9) = -1
Answer : (1000)2 - (1001)2 = (-0001)2
With above example with 2's compliment :(1000)2 - (1001)2 1001 (decimal 9) after 1'compliment we get 0110 + 1 (2's Compliment ) ------- 0111
| Example : 1 | Addition | ||
Carry >> | |||
| 8 | 1 0 0 0 | ||
+ -9 | 0 1 1 1 | ||
| 0 1 1 1 | |||
If no carry bit then re-compliment the result
0111 = 1111 (after 1's compliment) 1111+1 = 0001(2's compliment)
The result will be -0001 Result of : (1000)2 - (1001) = (-1)2 or 8 + (-9) = -1
Binary Multiplication
Binary multiplication is same as normal multiplication only we have to know the rules of addition properly.
(1010.11)2 x (101.10)2
| Example : 1 | |
| 1 0 1 0 . 1 1 | |
x 1 0 1 . 1 0 | |
| 0 0 0 0 0 0 | |
| 1 0 1 0 1 1 | |
| 1 0 1 0 1 1 | |
| 0 0 0 0 0 0 | |
1 0 1 0 1 1 | |
| 1 1 1 0 1 1 . 0 0 1 0 |
Answer : (1010.11)2 x (101.11)2= (1javascript:void(0)11011.0010)2
Binary Division
Binary division too is same as decimal division but binary subtraction rules going used here.
(110011)2 / (111)2
| Example : 1 | |||
| 1 1 1 | 1 1 0 0 1 1 | 1 | |
1 1 1 | |||
| 1 0 0 0 | 1 | ||
1 1 1 | |||
| 1 1 1 | 1 | ||
1 1 1 | |||
| 0 0 0 | |||
| 1 1 1 |
Answer : (110011)2 / (111)2= (111)2
!!!Hope it will be helpful!!!
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