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- Subtraction
- Difference of two squares
- Division (mathematics)
- Symmetric difference
- Addition
- Ones' complement
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In arithmetic, **subtraction** is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with. Subtraction is denoted by a minus sign in infix notation, in contrast to the use of the plus sign for addition.
Since subtraction is not a commutative operator, the two operands are named. The traditional names for the parts of the formula
are *minuend* (*c*) − *subtrahend* (*b*) = *difference* (*a*).
Subtraction is used to model four related processes:
In mathematics, it is often useful to view or even define subtraction as a kind of addition, the addition of the additive inverse. We can view 7 − 3 = 4 as the sum of two terms: 7 and -3. This perspective allows us to apply to subtraction all of the familiar rules and nomenclature of addition. Subtraction is not associative or commutative—in fact, it is anticommutative and left-associative—but addition of signed numbers is both.
Imagine a line segment of length *b* with the left end labeled *a* and the right end labeled *c*. Starting from *a*, it takes *b* steps to the right to reach *c*. This movement to the right is modeled mathematically by addition:
From *c*, it takes *b* steps to the *left* to get back to *a*. This movement to the left is modeled by subtraction:
Now, a line segment labeled with the numbers 1, 2, and 3. From position 3, it takes no steps to the left to stay at 3, so 3 − 0 = 3. It takes 2 steps to the left to get to position 1, so 3 − 2 = 1. This picture is inadequate to describe what would happen after going 3 steps to the left of position 3. To represent such an operation, the line must be extended.
To subtract arbitrary natural numbers, one begins with a line containing every natural number (0, 1, 2, 3, 4, 5, 6, ...). From 3, it takes 3 steps to the left to get to 0, so 3 − 3 = 0. But 3 − 4 is still invalid since it again leaves the line. The natural numbers are not a useful context for subtraction.
The solution is to consider the integer number line (..., −3, −2, −1, 0, 1, 2, 3, ...). From 3, it takes 4 steps to the left to get to −1:
There are some cases where subtraction as a separate operation becomes problematic. For example, 3 − (−2) (i.e. subtract −2 from 3) is not immediately obvious from either a natural number view or a number line view, because it is not immediately clear what it means to move −2 steps to the left or to take away −2 apples. One solution is to view subtraction as addition of signed numbers. Extra minus signs simply denote additive inversion. Then we have 3 − (−2) = 3 + 2 = 5. This also helps to keep the ring of integers "simple" by avoiding the introduction of "new" operators such as subtraction. Ordinarily a ring only has two operations defined on it; in the case of the integers, these are addition and multiplication. A ring already has the concept of additive inverses, but it does not have any notion of a separate subtraction operation, so the use of signed addition as subtraction allows us to apply the ring axioms to subtraction without needing to prove anything.
There are various algorithms for subtraction, and they differ in their suitability for various applications. A number of methods are adapted to hand calculation; for example, when making change, no actual subtraction is performed, but rather the change-maker counts forward.
For machine calculation, the method of complements is preferred, whereby the subtraction is replaced by an addition in a modular arithmetic.
Methods used to teach subtraction to elementary school varies from country to country, and within a country, different methods are in fashion at different times. In what is, in the U.S., called traditional mathematics, a specific process is taught to students at the end of the 1st year or during the 2nd year for use with multi-digit whole numbers, and is extended in either the fourth or fifth grade to include decimal representations of fractional numbers.
Some American schools currently teach a method of subtraction using borrowing and a system of markings called crutches][. Although a method of borrowing had been known and published in textbooks prior, apparently the crutches are the invention of William A. Brownell who used them in a study in November 1937][. This system caught on rapidly, displacing the other methods of subtraction in use in America at that time.
Some European schools employ a method of subtraction called the Austrian method, also known as the additions method. There is no borrowing in this method. There are also crutches (markings to aid memory), which vary by country][.
Both these methods break up the subtraction as a process of one digit subtractions by place value. Starting with a least significant digit, a subtraction of subtrahend:
from minuend
where each *s*_{i} and *m*_{i} is a digit, proceeds by writing down *m*_{1} − *s*_{1}, *m*_{2} − *s*_{2}, and so forth, as long as *s*_{i} does not exceed *m*_{i}. Otherwise, *m*_{i} is increased by 10 and some other digit is modified to correct for this increase. The American method corrects by attempting to decrease the minuend digit *m*_{i+1} by one (or continuing the borrow leftwards until there is a non-zero digit from which to borrow). The European method corrects by increasing the subtrahend digit *s*_{i+1} by one.
**Example:** 704 − 512. The minuend is 704, the subtrahend is 512. The minuend digits are *m*_{3} = 7, *m*_{2} = 0 and *m*_{1} = 4. The subtrahend digits are *s*_{3} = 5, *s*_{2} = 1 and *s*_{1} = 2. Beginning at the one's place, 4 is not less than 2 so the difference 2 is written down in the result's one place. In the ten's place, 0 is less than 1, so the 0 is increased to 10, and the difference with 1, which is 9, is written down in the ten's place. The American method corrects for the increase of ten by reducing the digit in the minuend's hundreds place by one. That is, the 7 is struck through and replaced by a 6. The subtraction then proceeds in the hundreds place, where 6 is not less than 5, so the difference is written down in the result's hundred's place. We are now done, the result is 192.
The Austrian method does not reduce the 7 to 6. Rather it increases the subtrahend hundred's digit by one. A small mark is made near or below this digit (depending on the school). Then the subtraction proceeds by asking what number when increased by 1, and 5 is added to it, makes 7. The answer is 1, and is written down in the result's hundred's place.
There is an additional subtlety in that the student always employs a mental subtraction table in the American method. The Austrian method often encourages the student to mentally use the addition table in reverse. In the example above, rather than adding 1 to 5, getting 6, and subtracting that from 7, the student is asked to consider what number, when increased by 1, and 5 is added to it, makes 7.
When subtracting two numbers with units, they must have the same unit. In most cases the difference will have the same unit as the original numbers. One exception is when subtracting two numbers with percentage as unit. In this case, the difference will have percentage points as unit.

Addition

+

**Subtraction**

**−**

Multiplication

×

Division

÷

Addition

+

Multiplication

×

Division

÷

In mathematics, the **difference of two squares**, or the difference of perfect squares, is a squared (multiplied by itself) number subtracted from another squared number. It refers to the identity
in elementary algebra.
The proof is straightforward. Starting from the right-hand side, apply the distributive law to get
and set
as an application of the commutative law. The resulting identity is one of the most commonly used in mathematics. Among many uses, it gives a simple proof of the AM–GM inequality in two variables.
The proof just given indicates the scope of the identity in abstract algebra: it will hold in any commutative ring *R*.
Conversely, if this identity holds in a ring *R* for all pairs of elements *a* and *b* of the ring, then *R* is commutative. To see this, we apply the distributive law to the right-hand side of the original equation and get
and for this to be equal to , we must have
for all pairs *a*, *b* of elements of *R*, so the ring *R* is commutative.
The difference of two squares can also be illustrated geometrically as the difference of two square areas in a plane. In the diagram, the shaded part represents the difference between the areas of the two squares, i.e. . The area of the shaded part can be found by adding the areas of the two rectangles; , which can be factorized to . Therefore
Another geometric proof proceeds as follows: We start with the figure shown in the first diagram below, a large square with a smaller square removed from it. The side of the entire square is a, and the side of the small removed square is b. The area of the shaded region is . A cut is made, splitting the region into two rectangular pieces, as shown in the second diagram. The larger piece, at the top, has width a and height a-b. The smaller piece, at the bottom, has width a-b and height b. Now the smaller piece can be detached, rotated, and placed to the right of the larger piece. In this new arrangement, shown in the last diagram below, the two pieces together form a rectangle, whose width is and whose height is . This rectangle's area is . Since this rectangle came from rearranging the original figure, it must have the same area as the original figure. Therefore, .Any odd number can be expressed as difference of two squares.
The difference of two squares is used to find the linear factors of the *sum* of two squares, using complex number coefficients.
For example, the root of can be found using difference of two squares:
Therefore the linear factors are and .
Since the two factors found by this method are Complex conjugates, we can use this in reverse as a method of multiplying a complex number to get a real number. This is used to get real denominators in complex fractions.
The difference of two squares can also be used in the rationalising of irrational denominators. This is a method for removing surds from expressions (or at least moving them), applying to division by some combinations involving square roots.
For example: The denominator of can be rationalised as follows:
Here, the irrational denominator has been rationalised to . Any odd number can be expressed as difference of two squares.
The difference of two squares can also be used as an arithmetical short cut. If you are multiplying two numbers whose average is a number which is easily squared the difference of two squares can be used to give you the product of the original two numbers.
For example:
Which means using the difference of two squares can be restated as
which is .
The identity also holds in inner product spaces over the field of real numbers, such as for dot product of Euclidean vectors:
The proof is identical. By the way, assuming that and have equal norms (which means that their dot squares are equal), it demonstrates analytically the fact that two diagonals of a rhombus are perpendicular.

In mathematics, especially in elementary arithmetic, **division** (÷) is an arithmetic operation. Specifically, if *b* times *c* equals *a*, written:
where *b* is not zero, then *a* divided by *b* equals *c*, written:
For instance,
since
In the expression a ÷ b = c, *a* is called the **dividend** or **numerator**, *b* the **divisor** or **denominator** and the result *c* is called the **quotient**.
Conceptually, division describes two distinct but related settings. *Partitioning* involves taking a set of size *a* and forming *b* groups that are equal in size. The size of each group formed, *c*, is the quotient of *a* and *b*. *Quotative* division involves taking a set of size *a* and forming groups of size *c*. The number of groups of this size that can be formed, *b*, is the quotient of *a* and *c*.
Teaching division usually leads to the concept of fractions being introduced to students. Unlike addition, subtraction, and multiplication, the set of all integers is not closed under division. Dividing two integers may result in a remainder. To complete the division of the remainder, the number system is extended to include fractions or rational numbers as they are more generally called.
Division is often shown in algebra and science by placing the *dividend* over the *divisor* with a horizontal line, also called a vinculum or fraction bar, between them. For example, *a* divided by *b* is written
This can be read out loud as "a divided by b", "a by b" or "a over b". A way to express division all on one line is to write the *dividend* (or numerator), then a slash, then the *divisor* (or denominator), like this:
This is the usual way to specify division in most computer programming languages since it can easily be typed as a simple sequence of ASCII characters.
A typographical variation halfway between these two forms uses a solidus (fraction slash) but elevates the dividend, and lowers the divisor:
Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (although typically called the *numerator* and *denominator*), and there is no implication that the division must be evaluated further. A second way to show division is to use the obelus (or division sign), common in arithmetic, in this manner:
This form is infrequent except in elementary arithmetic. ISO 80000-2-9.6 states it should not be used. The obelus is also used alone to represent the division operation itself, as for instance as a label on a key of a calculator.
In some non-English-speaking cultures, "a divided by b" is written *a* : *b*. This notation was introduced in 1631 by William Oughtred in his *Clavis Mathematicae* and later popularized by Gottfried Wilhelm Leibniz. However, in English usage the colon is restricted to expressing the related concept of ratios (then "a is to b").
In elementary mathematics the notation or is used to denote *a* divided by *b*. This notation was first introduced by Michael Stifel in *Arithmetica integra*, published in 1544.
Division is often introduced through the notion of "sharing out" a set of objects, for example a pile of sweets, into a number of equal portions. Distributing the objects several at a time in each round of sharing to each portion leads to the idea of "chunking", i.e., division by repeated subtraction.
More systematic and more efficient (but also more formalised and more rule-based, and more removed from an overall holistic picture of what division is achieving), a person who knows the multiplication tables can divide two integers using pencil and paper using the method of short division, if the divisor is simple. Long division is used for larger integer divisors. If the dividend has a fractional part (expressed as a decimal fraction), one can continue the algorithm past the ones place as far as desired. If the divisor has a fractional part, we can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction.
A person can calculate division with an abacus by repeatedly placing the dividend on the abacus, and then subtracting the divisor the offset of each digit in the result, counting the number of divisions possible at each offset.
A person can use logarithm tables to divide two numbers, by subtracting the two numbers' logarithms, then looking up the antilogarithm of the result.
A person can calculate division with a slide rule by aligning the divisor on the C scale with the dividend on the D scale. The quotient can be found on the D scale where it is aligned with the left index on the C scale. The user is responsible, however, for mentally keeping track of the decimal point.
Modern computers compute division by methods that are faster than long division: see Division algorithm.
In modular arithmetic, some numbers have a multiplicative inverse with respect to the modulus. We can calculate division by multiplication in such a case. This approach is useful in computers that do not have a fast division instruction.
The division algorithm is a mathematical theorem that precisely expresses the outcome of the usual process of division of integers. In particular, the theorem asserts that integers called the quotient *q* and remainder *r* always exist and that they are uniquely determined by the dividend *a* and divisor *d*, with *d* ≠ 0. Formally, the theorem is stated as follows: There exist unique integers *q* and *r* such that *a* = *qd* + *r* and 0 ≤ *r* < | *d* |, where | *d* | denotes the absolute value of *d*.
Division of integers is not closed. Apart from division by zero being undefined, the quotient is not an integer unless the dividend is an integer multiple of the divisor. For example 26 cannot be divided by 11 to give an integer. Such a case uses one of five approaches:
Dividing integers in a computer program requires special care. Some programming languages, such as C, treat integer division as in case 5 above, so the answer is an integer. Other languages, such as MATLAB and every computer algebra system return a rational number as the answer, as in case 3 above. These languages provide also functions to get the results of the other cases, either directly of from the result of case 3.
Names and symbols used for integer division include div, /, \, and %. Definitions vary regarding integer division when the dividend or the divisor is negative: rounding may be toward zero (so called T-division) or toward −∞ (F-division); rarer styles can occur – see Modulo operation for the details.
Divisibility rules can sometimes be used to quickly determine whether one integer divides exactly into another.
The result of dividing two rational numbers is another rational number when the divisor is not 0. We may define division of two rational numbers *p*/*q* and *r*/*s* by
All four quantities are integers, and only *p* may be 0. This definition ensures that division is the inverse operation of multiplication.
Division of two real numbers results in another real number when the divisor is not 0. It is defined such *a*/*b* = *c* if and only if *a* = *cb* and *b* ≠ 0.
Division of any number by zero (where the divisor is zero) is undefined. This is because zero multiplied by any finite number always results in a product of zero. Entry of such an expression into most calculators produces an error message.
Dividing two complex numbers results in another complex number when the divisor is not 0, defined thus:
All four quantities are real numbers. *r* and *s* may not both be 0.
Division for complex numbers expressed in polar form is simpler than the definition above:
Again all four quantities are real numbers. *r* may not be 0.
One can define the division operation for polynomials in one variable over a field. Then, as in the case of integers, one has a remainder. See Euclidean division of polynomials, and, for hand-written computation, polynomial long division or synthetic division.
One can define a division operation for matrices. The usual way to do this is to define , where denotes the inverse of *B*, but it is far more common to write out explicitly to avoid confusion.
Because matrix multiplication is not commutative, one can also define a left division or so-called *backslash-division* as . For this to be well defined, need not exist, however does need to exist. To avoid confusion, division as defined by is sometimes called *right division* or *slash-division* in this context.
Note that with left and right division defined this way, is in general not the same as and nor is the same as , but and .
To avoid problems when and/or do not exist, division can also be defined as multiplication with the pseudoinverse, i.e., and , where and denote the pseudoinverse of *A* and *B*.
In abstract algebras such as matrix algebras and quaternion algebras, fractions such as are typically defined as or where is presumed an invertible element (i.e., there exists a multiplicative inverse such that where is the multiplicative identity). In an integral domain where such elements may not exist, *division* can still be performed on equations of the form or by left or right cancellation, respectively. More generally "division" in the sense of "cancellation" can be done in any ring with the aforementioned cancellation properties. If such a ring is finite, then by an application of the pigeonhole principle, every nonzero element of the ring is invertible, so *division* by any nonzero element is possible in such a ring. To learn about when *algebras* (in the technical sense) have a division operation, refer to the page on division algebras. In particular Bott periodicity can be used to show that any real normed division algebra must be isomorphic to either the real numbers **R**, the complex numbers **C**, the quaternions **H**, or the octonions **O**.
The derivative of the quotient of two functions is given by the quotient rule:
There is no general method to integrate the quotient of two functions.

Addition

+

Subtraction

−

Multiplication

×

**Division**

**÷**

Addition

+

Subtraction

−

Multiplication

×

The symmetric difference is

the union without the intersection:

In mathematics, the**symmetric difference** of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets *A* and *B* is commonly denoted by
or
For example, the symmetric difference of the sets and is . The symmetric difference of the set of all students and the set of all females consists of all male students together with all female non-students.
The power set of any set becomes an abelian group under the operation of symmetric difference, with the empty set as the neutral element of the group and every element in this group being its own inverse. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring and intersection as the multiplication of the ring.
The symmetric difference is equivalent to the union of both relative complements, that is:
and it can also be expressed as the union of the two sets, minus their intersection:
or with the XOR operation:
In particular, .
The symmetric difference is commutative and associative:
Thus, the repeated symmetric difference is an operation on a multiset of sets giving the set of elements which are in an odd number of sets.
The symmetric difference of two repeated symmetric differences is the repeated symmetric difference of the join of the two multisets, where for each double set both can be removed. In particular:
This implies a sort of triangle inequality: the symmetric difference of *A* and *C* is contained in the union of the symmetric difference of *A* and *B* and that of *B* and *C*. (But note that for the diameter of the symmetric difference the triangle inequality does not hold.)
The empty set is neutral, and every set is its own inverse:
Taken together, we see that the power set of any set *X* becomes an abelian group if we use the symmetric difference as operation. Because every element in this group is its own inverse, this is in fact a vector space over the field with 2 elements **Z**_{2}. If *X* is finite, then the singletons form a basis of this vector space, and its dimension is therefore equal to the number of elements of *X*. This construction is used in graph theory, to define the cycle space of a graph.
Intersection distributes over symmetric difference:
and this shows that the power set of *X* becomes a ring with symmetric difference as addition and intersection as multiplication. This is the prototypical example of a Boolean ring.
Further properties of the symmetric difference:
The symmetric difference can be defined in any Boolean algebra, by writing
This operation has the same properties as the symmetric difference of sets.
As above, the symmetric difference of a collection of sets contains just elements which are in an odd number of the sets in the collection:
Evidently, this is well-defined only when each element of the union is contributed by a finite number of elements of .
Suppose is a multiset and . Then there is a formula for , the number of elements in , given solely in terms of intersections of elements of :
where is meant to indicate that is a subset of distinct elements of , of which there are .
As long as there is a notion of "how big" a set is, the symmetric difference between two sets can be considered a measure of how "far apart" they are. Formally, if μ is a σ-finite measure defined on a σ-algebra Σ, the function,
is a pseudometric on Σ. *d* becomes a metric if Σ is considered modulo the equivalence relation *X* ~ *Y* if and only if . The resulting metric space is separable if and only if (μ)2L is separable.
Let be some measure space and let and .
Symmetric difference is measurable: .
We write iff . The relation "" is an equivalence relation on the -measurable sets.
We write iff to each there's some such that . The relation "" is a partial order on the family of subsets of .
We write iff and . The relation "" is an equivalence relationship between the subsets of .
The "symmetric closure" of is the collection of all -measurable sets that are to some . The symmetric closure of contains . If is a sub--algebra of , so is the symmetric closure of .
iff -a.e.

the union without the intersection:

In mathematics, the

Addition is used to model countless physical processes. Even for the simple case of adding natural numbers, there are many possible interpretations and even more visual representations. Possibly the most fundamental interpretation of addition lies in combining sets: This interpretation is easy to visualize, with little danger of ambiguity. It is also useful in higher mathematics; for the rigorous definition it inspires, see

Subtraction

−

Multiplication

×

Division

÷

The **ones' complement** of a binary number is defined as the value obtained by inverting all the bits in the binary representation of the number (swapping 0's for 1's and vice-versa). The ones' complement of the number then behaves like the negative of the original number in some arithmetic operations. To within a constant (of −1), the ones' complement behaves like the negative of the original number with binary addition. However, unlike two's complement, these numbers have not seen widespread use because of issues such as the offset of −1, that negating zero results in a distinct **negative zero** bit pattern, less simplicity with arithmetic borrowing, etc.
A **ones' complement system** or **ones' complement arithmetic** is a system in which negative numbers are represented by the arithmetic negative of the value. In such a system, a number is negated (converted from positive to negative or vice versa) by computing its ones' complement. An N-bit ones' complement numeral system can only represent integers in the range −(2N−1−1) to 2N−1−1 while two's complement can express −2N−1 to 2N−1−1.
The **ones' complement binary** numeral system is characterized by the bit complement of any integer value being the arithmetic negative of the value. That is, inverting all of the bits of a number (the logical complement) produces the same result as subtracting the value from 0.
The early days of digital computing were marked by a lot of competing ideas about both hardware technology and mathematics technology (numbering systems). One of the great debates was the format of negative numbers, with some of the era's most expert people having very strong and different opinions][. One camp supported two's complement, the system that is dominant today. Another camp supported ones' complement, where any positive value is made into its negative equivalent by inverting all of the bits in a word. A third group supported "sign & magnitude" (sign-magnitude), where a value is changed from positive to negative simply by toggling the word's sign (high-order) bit.
There were arguments for and against each of the systems.][ Sign & magnitude allowed for easier tracing of memory dumps (a common process 40 years ago) as numeric values tended to use fewer 1 bits][. Internally, these systems did ones' complement math so numbers would have to be converted to ones' complement values when they were transmitted from a register to the math unit and then converted back to sign-magnitude when the result was transmitted back to the register][. The electronics required more gates than the other systems – a key concern when the cost and packaging of discrete transistors was critical. IBM was one of the early supporters of sign-magnitude, with their 7090 (709x series) computers perhaps the best known architecture to use it.
Ones' complement allowed for somewhat simpler hardware designs as there was no need to convert values when passed to and from the math unit. But it also shared an undesirable characteristic with sign-magnitude – the ability to represent negative zero (−0). Negative zero behaves exactly like positive zero; when used as an operand in any calculation, the result will be the same whether an operand is positive or negative zero. The disadvantage, however, is that the existence of two forms of the same value necessitates two rather than a single comparison when checking for equality with zero. Ones' complement subtraction can also result in an end-around borrow (described below). It can be argued that this makes the addition/subtraction logic more complicated or that it makes it simpler as a subtraction requires simply inverting the bits of the second operand as it is passed to the adder. The CDC 6000 series, UNIVAC 1100 series, and the LINC computer used ones' complement representation.
Two's complement is the easiest to implement in hardware, which may be the ultimate reason for its widespread popularity][. Processors on the early mainframes often consisted of thousands of transistors – eliminating a significant number of transistors was a significant cost savings. The architects of the early integrated circuit-based CPUs (Intel 8080, etc.) chose to use two's complement math. As IC technology advanced, virtually all adopted two's complement technology. Intel, AMD, and Power Architecture chips are all two's complement.][
Positive numbers are the same simple, binary system used by two's complement and sign-magnitude. Negative values are the bit complement of the corresponding positive value. The largest positive value is characterized by the sign (high-order) bit being off (0) and all other bits being on (1). The smallest negative value is characterized by the sign bit being 1, and all other bits being 0. The table below shows all possible values in a 4-bit system, from −7 to +7.
Adding two values is straight forward. Simply align the values on the least significant bit and add, propagating any carry to the bit one position left. 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. This phenomenon does not occur in two's complement arithmetic.
It is easy to demonstrate that the bit complement of a positive value is the negative magnitude of the positive value. The computation of 19 + 3 produces the same result as 19 − (−3).
Add 3 to 19.
Subtract −3 from 19.
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. Adding or subtracting negative zero to/from another value produces the original value.
Adding negative zero:
Subtracting negative zero:
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. The previous example added 22 and −22 and produced −0.
The interesting "corner cases" are when one or both operands are zero and/or negative zero.
Subtracting +0 is trivial (as shown above). If the second operand is negative zero it is inverted and the original value of the first operand is the result. Subtracting −0 is also trivial. The result can be only 1 of two cases. In case 1, operand 1 is −0 so the result is produced simply by subtracting 1 from 1 at every bit position. 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.
The only really interesting case is when both operands are plus or minus zero. Look at this example:
This example shows that of the 4 possible conditions when adding only ±0, an adder will produce −0 in three of them. A complementing subtractor will produce −0 only when both operands are −0.
Donald Knuth: *The Art of Computer Programming*, Volume 2: Seminumerical Algorithms, chapter 4.1

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Dog food 101: What is the difference between natural, organic, and holistic?

So, what do these terms actually mean, and is there a difference? • Natural: AAFCO (American Association of Feed Control Officials) defines the word "natural", when used to describe a pet food as: "A feed or ingredient derived solely from plant ..."Natural" on food labels: What does it really mean?

What does natural really mean? In a fascinating Washington Post article, chemistry professor Robert Wolke describes how baffling the word can be when in reference ... Dietician Erin Coates clears up the difference. She states that “ultimately, the ...- Midpoint, market ratio, mean, median, merit bonus - I don’t know what it is with “m” words in compensation, but there are a lot of them and it can get confusing. Let’s try to simplify things by breaking down the difference between two commonly ...
- but the difference in connotation is, you’ll appreciate from experience, pretty major. So what does the word itself mean? I can tell you what a mother is, and I can tell you what a horse is. But what’s a dude? The first two definitions are historically ...
Rigor: It’s all the rage, but what does it mean?

Our record on high school math and science education is particularly troubling.” But saying a program or curriculum is rigorous does not make it so. The quality of these efforts varies widely. The “R” word has ... over the exact meaning of “rigor ...