Addition of binary numbers pdf
The statement "convert x to two's complement" may be ambiguous, since it could describe either the process of representing x in two's-complement notation without changing its value, or the calculation of the two's complement, which is the arithmetic negative of x if two's complement representation is used. A two's-complement number system encodes positive and negative numbers in a binary number representation.
The weight of each bit is a power of two, except for the most significant bit , whose weight is the negative of the corresponding power of two. The most significant bit determines the sign of the number and is sometimes called the sign bit.
The following Python code shows a simple function which will convert an unsigned input integer to a two's complement signed integer using the above logic with bitwise operators:. In two's complement notation, a non-negative number is represented by its ordinary binary representation ; in this case, the most significant bit is 0. Though, the range of numbers represented is not the same as with unsigned binary numbers. For example, an 8-bit unsigned number can represent the values 0 to The two's complement operation is the additive inverse operation, so negative numbers are represented by the two's complement of the absolute value.
To get the two's complement of a binary number, the bits are inverted, or "flipped", by using the bitwise NOT operation; the value of 1 is then added to the resulting value, ignoring the overflow which occurs when taking the two's complement of 0. The most significant bit is 0, so the pattern represents a non-negative value. To obtain the two's complement, 1 is added to the result, giving:.
The most significant bit is 1, so the value represented is negative. The two's complement of a negative number is the corresponding positive value. The two's complement of zero is zero: Furthermore, the two's complement of the most negative number representable e. Hence, there appears to be an 'extra' negative number. Then adding a number to its two's complement results in the N lowest bits set to 0 and the carry bit 1, where the latter has the weight reading it as an unsigned binary number of 2 N.
This shortcut allows a person to convert a number to its two's complement without first forming its ones' complement. In computer circuitry, this method is no faster than the "complement and add one" method; both methods require working sequentially from right to left, propagating logic changes. The method of complementing and adding one can be sped up by a standard carry look-ahead adder circuit; the LSB towards MSB method can be sped up by a similar logic transformation.
When turning a two's-complement number with a certain number of bits into one with more bits e. Some processors do this in a single instruction; on other processors, a conditional must be used followed by code to set the relevant bits or bytes. Similarly, when a two's-complement number is shifted to the right, the most-significant bit, which contains magnitude and the sign information, must be maintained. However, when shifted to the left, a 0 is shifted in.
These rules preserve the common semantics that left shifts multiply the number by two and right shifts divide the number by two. Both shifting and doubling the precision are important for some multiplication algorithms. Note that unlike addition and subtraction, width extension and right shifting are done differently for signed and unsigned numbers. With only one exception, starting with any number in two's-complement representation, if all the bits are flipped and 1 added, the two's-complement representation of the negative of that number is obtained.
The two's complement of the minimum number in the range will not have the desired effect of negating the number. This is because a positive value of cannot be represented with an 8-bit signed binary numeral. This phenomenon is fundamentally about the mathematics of binary numbers, not the details of the representation as two's complement.
Mathematically, this is complementary to the fact that the negative of 0 is again 0. For a given number of bits k there is an even number of binary numbers 2 k , taking negatives is a group action of the group of order 2 on binary numbers, and since the orbit of zero has order 1, at least one other number must have an orbit of order 1 for the orders of the orbits to add up to the order of the set.
Thus some other number must be invariant under taking negatives formally, by the orbit-stabilizer theorem. Note that this negative being the same number is detected as an overflow condition since there was a carry into but not out of the most-significant bit. This can lead to unexpected bugs in that an unchecked implementation of absolute value could return a negative number in the case of the minimum negative.
The abs family of integer functions in C typically has this behaviour. This is also true for Java. The most negative number in two's complement is sometimes called "the weird number," because it is the only exception. Although the number is an exception, it is a valid number in regular two's complement systems. All arithmetic operations work with it both as an operand and unless there was an overflow a result. For example, with eight bits, the unsigned bytes are 0 to Fundamentally, the system represents negative integers by counting backward and wrapping around.
The boundary between positive and negative numbers is arbitrary, but by convention all negative numbers have a left-most bit most significant bit of one. Negating a two's complement number is simple: Invert all the bits and add one to the result.
The system is useful in simplifying the implementation of arithmetic on computer hardware. Overflow checks still must exist to catch operations such as summing and The system therefore allows addition of negative operands without a subtraction circuit and a circuit that detects the sign of a number. Moreover, that addition circuit can also perform subtraction by taking the two's complement of a number see below , which only requires an additional cycle or its own adder circuit.
To perform this, the circuit merely pretends an extra left-most bit of 1 exists. Adding two's-complement numbers requires no special processing even if the operands have opposite signs: This process depends upon restricting to 8 bits of precision; a carry to the nonexistent 9th most significant bit is ignored, resulting in the arithmetically correct result of 10 The last two bits of the carry row reading right-to-left contain vital information: An overflow condition exists when these last two bits are different from one another.
As mentioned above, the sign of the number is encoded in the MSB of the result. In other terms, if the left two carry bits the ones on the far left of the top row in these examples are both 1s or both 0s, the result is valid; if the left two carry bits are "1 0" or "0 1", a sign overflow has occurred. Conveniently, an XOR operation on these two bits can quickly determine if an overflow condition exists. As an example, consider the signed 4-bit addition of 7 and In this case, the far left two MSB carry bits are "01", which means there was a two's-complement addition overflow.
It is then possible, if desired, to 'truncate' the result back to N bits while preserving the value if and only if the discarded bit is a proper sign extension of the retained result bits. If the carry extends past the end of the word it is said to have "wrapped around", a condition called an " end-around carry ". When this occurs, the bit must be added back in at the right-most bit. This phenomenon does not occur in two's complement arithmetic. Subtraction is similar, except that borrows, rather than carries, are propagated to the left.
If the borrow extends past the end of the word it is said to have "wrapped around", a condition called an " end-around borrow ". When this occurs, the bit must be subtracted from the right-most bit. It is easy to demonstrate that the bit complement of a positive value is the negative magnitude of the positive value. Negative zero is the condition where all bits in a signed word are 1. This follows the ones' complement rules that a value is negative when the left-most bit is 1, and that a negative number is the bit complement of the number's magnitude.
The value also behaves as zero when computing. Negative zero is easily produced in a 1's complement adder. Simply add the positive and negative of the same magnitude. Although the math always produces the correct results, a side effect of negative zero is that software must test for negative zero. The generation of negative zero becomes a non-issue if addition is achieved with a complementing subtractor. The first operand is passed to the subtract unmodified, the second operand is complemented, and the subtraction generates the correct result, avoiding negative zero.
If the second operand is negative zero it is inverted and the original value of the first operand is the result. The result can be only 1 of two cases. In case 2, the subtraction will generate a value that is 1 larger than operand 1 and an end-around borrow.
Completing the borrow generates the same value as operand 1. From Wikipedia, the free encyclopedia.