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Two's complement is a mathematical operation on binary numbers , best known for its role in computing as a method of signed number representation. For this reason, it is the most important example of a radix complement.
The two's complement of an N -bit number is defined as its complement with respect to 2 N. Two's complement is the most common method of representing signed integers on computers. Compared to other systems for representing signed numbers e. This property makes the system simpler to implement, especially for higher-precision arithmetic. Unlike ones' complement systems, two's complement has no representation for negative zero , and thus does not suffer from its associated difficulties.
Conveniently, another way of finding the two's complement of a number is to take its ones' complement and add one: The method of complements had long been used to perform subtraction in decimal adding machines and mechanical calculators.
John von Neumann suggested use of two's complement binary representation in his First Draft of a Report on the EDVAC proposal for an electronic stored-program digital computer. The first minicomputer, the PDP-8 introduced in , uses two's complement arithmetic as do the Data General Nova , the PDP , and almost all subsequent minicomputers and microcomputers. The term two's complement can mean either a number format or a mathematical operator.
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. The result would be correct if treated as unsigned integer. 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.
This provides another method of detecting overflow—which is equivalent to the method of comparing the carry bits—but which may be easier to implement in some situations, because it does not require access to the internals of the addition. Computers usually use the method of complements to implement subtraction. Using complements for subtraction is closely related to using complements for representing negative numbers, since the combination allows all signs of operands and results; direct subtraction works with two's-complement numbers as well.
Like addition, the advantage of using two's complement is the elimination of examining the signs of the operands to determine whether addition or subtraction is needed. Overflow is detected the same way as for addition, by examining the two leftmost most significant bits of the borrows; overflow has occurred if they are different. Another example is a subtraction operation where the result is negative: As for addition, overflow in subtraction may be avoided or detected after the operation by first sign-extending both inputs by an extra bit.
The product of two N -bit numbers requires 2 N bits to contain all possible values. If the precision of the two operands using two's complement is doubled before the multiplication, direct multiplication discarding any excess bits beyond that precision will provide the correct result. First, the precision is extended from four bits to eight. Then the numbers are multiplied, discarding the bits beyond the eighth bit as shown by " x ":.
This is very inefficient; by doubling the precision ahead of time, all additions must be double-precision and at least twice as many partial products are needed than for the more efficient algorithms actually implemented in computers. Some multiplication algorithms are designed for two's complement, notably Booth's multiplication algorithm. Methods for multiplying sign-magnitude numbers don't work with two's-complement numbers without adaptation.
There isn't usually a problem when the multiplicand the one being repeatedly added to form the product is negative; the issue is setting the initial bits of the product correctly when the multiplier is negative. Two methods for adapting algorithms to handle two's-complement numbers are common:. As an example of the second method, take the common add-and-shift algorithm for multiplication.
Instead of shifting partial products to the left as is done with pencil and paper, the accumulated product is shifted right, into a second register that will eventually hold the least significant half of the product. Since the least significant bits are not changed once they are calculated, the additions can be single precision, accumulating in the register that will eventually hold the most significant half of the product. Comparison is often implemented with a dummy subtraction, where the flags in the computer's status register are checked, but the main result is ignored.
The zero flag indicates if two values compared equal. If the exclusive-or of the sign and overflow flags is 1, the subtraction result was less than zero, otherwise the result was zero or greater. These checks are often implemented in computers in conditional branch instructions.
Unsigned binary numbers can be ordered by a simple lexicographic ordering , where the bit value 0 is defined as less than the bit value 1.