Rules For Binary number Systems
Binary Addition
Binary addition is performed in the same manner as decimal addition. However, since binary system has only two digits, the addition table for binary arithmetic is very simple consisting of only four entries. The complete table for binary addition is as follows:0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0 plus a carry of 1 to the next higher column
Carry-overs are performed in the same manner as in decimal addition. Since 1 is the largest digit in the binary system, any sum greater than 1 requires that a digit be carried over. For instance, 10 plus 10 binary requires the addition of two 1's in the second position. Since 1 + 1 = 0 plus a carry- over of 1, the sum of 10 + 10 is 100 in binary.
Examples :
Add the binary numbers 1011 and 1001.
1011
+ 1001
10100
Add the binary numbers 100111 and 11011.
100111
+ 11011
1000010
Binary Subtraction
The principles of decimal subtraction can as well be applied to subtraction of numbers in other cases. It consists of two steps, which are repeated for each column of the numbers. The first step is to determine if it is necessary to borrow. If the subtrahend (the lower digit) is larger than the minuend (the upper digit), it is necessary to borrow from the column to the left. It is important to note here that the valued borrowed depends upon the base of the number and is always the decimal equivalent of the base. Thus, in decimal, 10 is borrowed; in binary, 2 is borrowed; in octal, 8 is borrowed; in hexadecimal, 16 is borrowed. The second step is to simply to subtract the lower value from the upper value. The complete table for binary subtraction is as follows:
0 - 0 = 0
1 - 0 = 1
1 - 1 = 0
0 - 1 = 1 with a borrow from the next column
Example :
Subtract 01110 from 10101
10101
- 01110
00111
Subtract 1011100 from 0111000
1011100
- 0111000
0100100
We can subtract two binary number by using 1's and 2's complement method.
2's complement method
0 x 0 = 0
0 x 1 = 0
1 x 0 = 0
1 x 1 = 1
Multiply the binary numbers 1010 and 1001
1010
x 1001
1010
0000
0000
1010
1011010
1. Start from the left of the dividend.
2. Perform a series of subtractions in which the divisor is subtracted from the dividend.
3. If subtraction is possible, put a 1 in the quotient and subtract the divisor from the corresponding digits of dividend.
4. If subtraction is not possible(divisor is greater than remainder), record a 0 in the quotient.
5. Bring down the next digit to add to the remainder digits. Proceed as before in a manner similar to long division.
The division process is performed in a manner similar to decimal division. The rules for binary division are ;
Binary Multiplication
Multiplication in the binary system also follows the same general rules as decimal multiplication. However, learning the binary multiplication is a trivial task become the table for binary multiplication is very short, with only four entries instead of the 100 necessary for decimal multiplication. The complete table for binary multiplication is as follows :0 x 0 = 0
0 x 1 = 0
1 x 0 = 0
1 x 1 = 1
Multiply the binary numbers 1010 and 1001
1010
x 1001
1010
0000
0000
1010
1011010
Binary Division
Binary division is again very simple. As in the decimal system (or in any other number system), division by 0 is meaningless. Hence, the complete table for binary division is as follows:1. Start from the left of the dividend.
2. Perform a series of subtractions in which the divisor is subtracted from the dividend.
3. If subtraction is possible, put a 1 in the quotient and subtract the divisor from the corresponding digits of dividend.
4. If subtraction is not possible(divisor is greater than remainder), record a 0 in the quotient.
5. Bring down the next digit to add to the remainder digits. Proceed as before in a manner similar to long division.
The division process is performed in a manner similar to decimal division. The rules for binary division are ;
Example:
1. Divisor greater than 100, so put 0 in quotient.
2. Add digit from dividend to group used above.
3. Subtraction possible so put 1 in the quotient.
4. Remainder from subtraction plus digit from dividend
5. Divisor greater, so put 0 in quotient.
6. Add digit from dividend to group.
7. Subtraction possible, so put 1 in quotient.
2. Add digit from dividend to group used above.
3. Subtraction possible so put 1 in the quotient.
4. Remainder from subtraction plus digit from dividend
5. Divisor greater, so put 0 in quotient.
6. Add digit from dividend to group.
7. Subtraction possible, so put 1 in quotient.
