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Binary is a base-2 number system that uses two states 0 and 1 to represent a number. We can also call it to be a true state and a false state. A binary number is built the same way as we build a normal decimal number.
Binary is a base-2 number system that uses two states 0 and 1 to represent a number. We can also call it to be a true state and a false state. A binary number is built the same way as we build a normal decimal number.
Binary arithmetic is an essential part of various digital systems. You can add, subtract, multiply, and divide binary numbers using various methods. These operations are much easier than decimal number arithmetic operations because the binary system has only two digits: 0 and 1.
Binary arithmetic is an essential part of various digital systems. You can add, subtract, multiply, and divide binary numbers using various methods. These operations are much easier than decimal number arithmetic operations because the binary system has only two digits: 0 and 1.
Binary additions and subtractions are performed as same in decimal additions and subtractions. When we perform binary additions, there will be two outputs: Sum (S) and Carry (C).
Binary additions and subtractions are performed as same in decimal additions and subtractions. When we perform binary additions, there will be two outputs: Sum (S) and Carry (C).
Conversion Between Decimal and Binary Representations
Conversion from Binary to Decimal
The conversion of a binary number to decimal can be done in two ways:
Method 1: Multiply each binary number by its position-value, and add the results together. The rightmost bit has a position value 1, the next 2, the value at the third position is 4, fourth 8 and so on (See figure below)
Example: 1101 = 8x1 + 4x1 + 1x1 (or simply, 8+4+1) = 13
Method 2: The advantage of this method is that you need not remember or compute position-values. Starting from leftmost bit, multiply the first bit by 2 and add to the next. Multiply the result by 2 and add to the third bit. Multiply the result by 2 and add to the fourth bit. Continue this process till you have reached the rightmost bit.
Conversion from Decimal to Binary
Here also we give two equivalent methods for converting a decimal number to binary.
Method 1: Subract the largest power of two, and change the bit at that position to 1. Keep subtracting the next largest possible power from the remainder, each time assigning a bit value 1 at the corresponding bit position. Repeat the process till you get a 1 or 0 for the last bit.
Method 2: In this method, we repeatedly divide the given decimal number by 2, and the remainders are placed from right to left:
Example: 43
43 ÷ 2: Quotient 21, remainder 1: Result > 1
21 ÷ 2: Quotient 10, remainder 1: Result > 1 1
10 ÷ 2: Quotient 5, remainder 0: Result > 0 1 1
5 ÷ 2: Quotient 2, remainder 1: Result > 1 0 1 1
2 ÷ 2: Quotient 1, remainder 0: Result > 0 1 0 1 1
1 ÷ 2: Quotient 0, remainder 1: Result > 1 0 1 0 1 1
The process stops when the quotient becomes 0
Binary addition
There are four rules for binary addition:
In fouth 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:
Binary subtraction
Subtraction and Borrow, these two words will be used very frequently for binary Subtraction. There are four rules for binary subtraction:
Example - Subtraction:
Binary multiplication
Binary multiplication is similar to decimal multiplication. It is simpler than decimal multiplication because only 0s and 1s are involved. There are four rules for binary multiplication:
Multiplication is always 0, whenever at least one input is 0.
Example - Multiplication:
Binary division
There are four parts in any division: Dividend, Divisor, quotient, and remainder.
The result is always not defined, whenever the divisor is 0. Binary division is similar to decimal division. It is called as the long division procedure.
Example - division:
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