# Binary subtraction truth table borrow

To form the negative of any number, first complement all bits of the number. The result is the one's complement of the original number. Then, add 1 to the result, thus forming the two's complement of the original number. Arithmetic involving such signed numbers is known generally as two's complement arithmetic. To check the validity of this process, let's take the two's complement of 0. We should logically get a result of 0. So, we start with , and form the one's complement Now add 1 to the result But this won't fit in a 4-bit number, so the extra 1-bit is lost, leaving a result of Remember to discard the carry from the highest-order bit.

Two's complement arithmetic always works this way. Therefore it is not possible to correctly take the two's complement of It will come back again as But if we invert B and add 1 with the low-order C in , we get the result of A - B.

We can use Exclusive-OR gates, as shown to the right, to control whether we will add or subtract on any given occasion. With a control input of 0, the XOR gates will leave the B input number unchanged, and will also apply a logic 0 as the initial input carry.

This is exactly what we want in order to add the two numbers. However, if we apply a logic 1 to the control input, the XOR gates will invert the B input number to form its one's complement, and will also add 1 through the initial input carry. This changes B to its two's complement. Thus, the output result will actually be A - B. Note that in two's complement addition, the output carry is ignored. You can also think of it as an inverted "borrow" bit rather than as a carry, so that a carry of 1 corresponds to a borrow of 0.

That logic also holds for the input carry, which also represents an input borrow bit of 0. When we add or subtract signed numbers, we need to introduce a new concept: Overflow occurs when the result has the wrong sign bit for the operation that was performed. For example, if we add two positive numbers 7 and 6 , we should get a positive result However, using 4-bit binary numbers, we would add to and get as the result. Therefore, an overflow has occurred, where the result would have to have more bits than the original two numbers.

This is not as much of a problem as you might think. These ranges are sufficient for most practical applications. Indeed, the binary MSB is commonly known as the sign bit.

The use of this bit to distinguish between positive and negative numbers also allows us to divide the counting sequence evenly between positive and negative numbers. Now we need to look at the relationship between the binary numbers for positive and negative versions of the same magnitude. If we then increment the result, we get -1 , which is what we wanted. Will this relationship hold for all negative numbers? In fact, it does work, as you can determine for yourself.

To form the negative of any number, first complement all bits of the number. The result is the one's complement of the original number. Then, add 1 to the result, thus forming the two's complement of the original number. Arithmetic involving such signed numbers is known generally as two's complement arithmetic.

To check the validity of this process, let's take the two's complement of 0. We should logically get a result of 0. So, we start with , and form the one's complement Now add 1 to the result But this won't fit in a 4-bit number, so the extra 1-bit is lost, leaving a result of Remember to discard the carry from the highest-order bit.

Two's complement arithmetic always works this way. Therefore it is not possible to correctly take the two's complement of It will come back again as But if we invert B and add 1 with the low-order C in , we get the result of A - B. We can use Exclusive-OR gates, as shown to the right, to control whether we will add or subtract on any given occasion.

With a control input of 0, the XOR gates will leave the B input number unchanged, and will also apply a logic 0 as the initial input carry. This is exactly what we want in order to add the two numbers. However, if we apply a logic 1 to the control input, the XOR gates will invert the B input number to form its one's complement, and will also add 1 through the initial input carry. This changes B to its two's complement. Thus, the output result will actually be A - B.

Note that in two's complement addition, the output carry is ignored. You can also think of it as an inverted "borrow" bit rather than as a carry, so that a carry of 1 corresponds to a borrow of 0. That logic also holds for the input carry, which also represents an input borrow bit of 0. When we add or subtract signed numbers, we need to introduce a new concept: