Draw Circles Around Sums in the Table

Arithmetic operation

3 + 2 = five with apples, a popular choice in textbooks^[1]

Addition (ordinarily signified by the plus symbol +) is i of the 4 basic operations of arithmetic, the other three existence subtraction, multiplication and division. The addition of two whole numbers results in the full corporeality or sum of those values combined. The instance in the adjacent image shows a combination of three apples and two apples, making a full of five apples. This ascertainment is equivalent to the mathematical expression "3 + ii = 5" (that is, "three plus two is equal to 5").

As well counting items, addition can likewise be defined and executed without referring to concrete objects, using abstractions called numbers instead, such equally integers, real numbers and complex numbers. Addition belongs to arithmetics, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects such as vectors, matrices, subspaces and subgroups.

Addition has several important backdrop. It is commutative, pregnant that order does not matter, and it is associative, meaning that when one adds more than 2 numbers, the order in which addition is performed does not affair (see Summation). Repeated addition of one is the same as counting. Addition of 0 does not alter a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication.

Performing add-on is 1 of the simplest numerical tasks. Improver of very small numbers is attainable to toddlers; the most basic task, 1 + 1, tin can be performed past infants as young as 5 months, and even some members of other animal species. In principal pedagogy, students are taught to add together numbers in the decimal system, starting with single digits and progressively tackling more difficult bug. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of add-on continues to this mean solar day.

Notation and terminology [edit]

Addition is written using the plus sign "+" between the terms;^[ii] that is, in infix notation. The result is expressed with an equals sign. For example,

${\displaystyle 1+i=ii}$ ("ane plus 1 equals two")

${\displaystyle 2+2=4}$ ("ii plus two equals four")

${\displaystyle ane+two=three}$ ("one plus two equals three")

${\displaystyle 5+4+2=11}$ (see "associativity" below)

${\displaystyle 3+iii+3+3=12}$ (encounter "multiplication" below)

Columnar addition – the numbers in the cavalcade are to exist added, with the sum written below the underlined number.

At that place are also situations where add-on is "understood", even though no symbol appears:

• A whole number followed immediately by a fraction indicates the sum of the two, chosen a mixed number.^[three] For case,

  $${\displaystyle 3{\frac {ane}{2}}=3+{\frac {i}{2}}=3.5.}$$

  

  This notation can cause confusion, since in well-nigh other contexts, juxtaposition denotes multiplication instead.^[4]

The sum of a series of related numbers can be expressed through capital sigma note, which compactly denotes iteration. For instance,

${\displaystyle \sum _{k=ane}^{5}k^{2}=one^{2}+2^{2}+3^{ii}+4^{two}+5^{2}=55.}$

The numbers or the objects to be added in general addition are collectively referred to as the terms,^[5] the addends ^{[half-dozen] [7] [8]} or the summands;^[9] this terminology carries over to the summation of multiple terms. This is to exist distinguished from factors, which are multiplied. Some authors call the kickoff addend the augend.^{[6] [seven] [8]} In fact, during the Renaissance, many authors did not consider the first addend an "addend" at all. Today, due to the commutative property of add-on, "augend" is rarely used, and both terms are more often than not called addends.^[ten]

All of the above terminology derives from Latin. "Addition" and "add" are English words derived from the Latin verb addere, which is in turn a compound of ad "to" and dare "to give", from the Proto-Indo-European root *deh₃- "to give"; thus to add is to give to.^[10] Using the gerundive suffix -nd results in "addend", "affair to exist added".^[a] Likewise from augere "to increment", one gets "augend", "thing to be increased".

Redrawn illustration from The Art of Nombryng, i of the beginning English language arithmetic texts, in the 15th century.^[11]

"Sum" and "summand" derive from the Latin noun summa "the highest, the meridian" and associated verb summare. This is advisable not only because the sum of 2 positive numbers is greater than either, but because information technology was common for the ancient Greeks and Romans to add up, reverse to the modernistic practice of adding downwards, so that a sum was literally higher than the addends.^[12] Addere and summare appointment dorsum at least to Boethius, if not to earlier Roman writers such every bit Vitruvius and Frontinus; Boethius besides used several other terms for the addition performance. The later Middle English language terms "adden" and "adding" were popularized by Chaucer.^[13]

The plus sign "+" (Unicode:U+002B; ASCII: +) is an abbreviation of the Latin word et, meaning "and".^[14] It appears in mathematical works dating dorsum to at least 1489.^[15]

Interpretations [edit]

Addition is used to model many physical processes. Even for the elementary case of calculation natural numbers, at that place are many possible interpretations and even more visual representations.

Combining sets [edit]

Maybe the nearly key interpretation of addition lies in combining sets:

• When 2 or more disjoint collections are combined into a single collection, the number of objects in the single collection is the sum of the numbers of objects in the original collections.

This estimation is easy to visualize, with little danger of ambiguity. Information technology is also useful in higher mathematics (for the rigorous definition it inspires, run across § Natural numbers below). However, information technology is non obvious how 1 should extend this version of addition to include fractional numbers or negative numbers.^[16]

I possible fix is to consider collections of objects that can exist hands divided, such every bit pies or, still better, segmented rods.^[17] Rather than solely combining collections of segments, rods can be joined end-to-end, which illustrates another formulation of addition: adding non the rods but the lengths of the rods.

Extending a length [edit]

A number-line visualization of the algebraic addition ii + 4 = half dozen. A translation by 2 followed by a translation past 4 is the same as a translation past 6.

A number-line visualization of the unary addition ii + 4 = vi. A translation by 4 is equivalent to four translations by 1.

A second interpretation of addition comes from extending an initial length by a given length:

• When an original length is extended by a given amount, the terminal length is the sum of the original length and the length of the extension.^[18]

The sum a + b can exist interpreted equally a binary operation that combines a and b, in an algebraic sense, or it can be interpreted as the add-on of b more units to a. Nether the latter interpretation, the parts of a sum a + b play asymmetric roles, and the operation a + b is viewed as applying the unary operation +b to a.^[19] Instead of calling both a and b addends, it is more appropriate to call a the augend in this case, since a plays a passive role. The unary view is also useful when discussing subtraction, because each unary addition operation has an inverse unary subtraction performance, and vice versa.

Properties [edit]

Commutativity [edit]

4 + 2 = 2 + iv with blocks

Addition is commutative, pregnant that one tin change the social club of the terms in a sum, simply still get the same result. Symbolically, if a and b are any ii numbers, and so

a + b = b + a.

The fact that improver is commutative is known as the "commutative law of addition" or "commutative property of addition". Some other binary operations are commutative, such as multiplication, just many others are non, such as subtraction and division.

Associativity [edit]

2 + (1 + 3) = (ii + 1) + 3 with segmented rods

Addition is associative, which means that when three or more numbers are added together, the guild of operations does not change the consequence.

As an example, should the expression a + b + c be defined to mean (a + b) + c or a + (b + c)? Given that addition is associative, the choice of definition is irrelevant. For whatever iii numbers a, b, and c, information technology is truthful that (a + b) + c = a + (b + c). For example, (1 + 2) + 3 = three + iii = six = 1 + 5 = one + (2 + iii).

When improver is used together with other operations, the order of operations becomes of import. In the standard order of operations, addition is a lower priority than exponentiation, nth roots, multiplication and segmentation, but is given equal priority to subtraction.^[20]

Identity chemical element [edit]

5 + 0 = 5 with bags of dots

Adding naught to any number, does not change the number; this means that zero is the identity element for addition, and is also known as the condiment identity. In symbols, for every a , one has

a + 0 = 0 + a = a .

This law was first identified in Brahmagupta's Brahmasphutasiddhanta in 628 Advertisement, although he wrote it as three separate laws, depending on whether a is negative, positive, or zero itself, and he used words rather than algebraic symbols. Later Indian mathematicians refined the concept; around the year 830, Mahavira wrote, "zero becomes the same equally what is added to information technology", corresponding to the unary statement 0 + a = a . In the 12th century, Bhaskara wrote, "In the addition of nil, or subtraction of information technology, the quantity, positive or negative, remains the same", corresponding to the unary statement a + 0 = a .^[21]

Successor [edit]

Within the context of integers, addition of one also plays a special role: for whatsoever integer a, the integer (a + one) is the least integer greater than a, also known as the successor of a.^[22] For case, 3 is the successor of 2 and seven is the successor of half dozen. Because of this succession, the value of a + b can also be seen every bit the bth successor of a, making addition iterated succession. For example, 6 + 2 is eight, because 8 is the successor of seven, which is the successor of half dozen, making 8 the 2nd successor of 6.

Units [edit]

To numerically add together concrete quantities with units, they must be expressed with common units.^[23] For example, calculation 50 milliliters to 150 milliliters gives 200 milliliters. Yet, if a measure of v feet is extended by 2 inches, the sum is 62 inches, since 60 inches is synonymous with 5 anxiety. On the other hand, information technology is normally meaningless to endeavour to add 3 meters and 4 foursquare meters, since those units are incomparable; this sort of consideration is fundamental in dimensional assay.^[24]

Performing addition [edit]

Innate ability [edit]

Studies on mathematical development starting effectually the 1980s have exploited the miracle of habituation: infants await longer at situations that are unexpected.^[25] A seminal experiment by Karen Wynn in 1992 involving Mickey Mouse dolls manipulated behind a screen demonstrated that five-month-one-time infants wait i + one to be 2, and they are comparatively surprised when a concrete situation seems to imply that one + ane is either one or 3. This finding has since been affirmed by a variety of laboratories using different methodologies.^[26] Some other 1992 experiment with older toddlers, between xviii and 35 months, exploited their development of motor control past allowing them to recall ping-pong balls from a box; the youngest responded well for small numbers, while older subjects were able to compute sums upward to 5.^[27]

Even some nonhuman animals prove a limited power to add, specially primates. In a 1995 experiment imitating Wynn's 1992 result (but using eggplants instead of dolls), rhesus macaque and cottontop tamarin monkeys performed similarly to human infants. More than dramatically, after being taught the meanings of the Arabic numerals 0 through 4, one chimpanzee was able to compute the sum of two numerals without further preparation.^[28] More recently, Asian elephants have demonstrated an power to perform basic arithmetic.^[29]

Childhood learning [edit]

Typically, children first principal counting. When given a problem that requires that two items and three items be combined, young children model the state of affairs with physical objects, ofttimes fingers or a drawing, and so count the total. As they gain experience, they larn or discover the strategy of "counting-on": asked to notice two plus three, children count three past two, saying "iii, iv, v" (usually ticking off fingers), and arriving at five. This strategy seems virtually universal; children can easily pick it upward from peers or teachers.^[30] Almost discover information technology independently. With additional experience, children learn to add more quickly by exploiting the commutativity of improver by counting up from the larger number, in this case, starting with 3 and counting "four, v." Eventually children begin to recall certain addition facts ("number bonds"), either through experience or rote memorization. In one case some facts are committed to memory, children brainstorm to derive unknown facts from known ones. For example, a child asked to add half-dozen and seven may know that 6 + half dozen = 12 and then reason that 6 + 7 is one more, or xiii.^[31] Such derived facts can be constitute very quickly and nigh simple school students eventually rely on a mixture of memorized and derived facts to add fluently.^[32]

Different nations introduce whole numbers and arithmetic at different ages, with many countries teaching addition in pre-school.^[33] However, throughout the earth, addition is taught by the end of the first twelvemonth of elementary school.^[34]

Table [edit]

Children are often presented with the addition table of pairs of numbers from 0 to nine to memorize. Knowing this, children tin can perform any improver.

+	0	1	ii	three	4	5	6	7	8	nine
0	0	1	2	3	iv	5	half dozen	7	viii	ix
1	1	two	3	iv	v	half-dozen	vii	viii	ix	10
2	two	3	four	5	6	7	8	nine	10	11
3	3	four	5	six	7	eight	9	10	11	12
4	four	five	6	vii	8	9	ten	11	12	13
5	5	6	7	8	ix	10	eleven	12	xiii	xiv
half dozen	6	7	8	nine	10	eleven	12	13	14	15
seven	7	viii	9	10	11	12	thirteen	xiv	xv	16
8	8	9	ten	eleven	12	13	xiv	15	sixteen	17
nine	ix	x	xi	12	13	14	15	16	17	18

Decimal organization [edit]

The prerequisite to addition in the decimal organisation is the fluent recall or derivation of the 100 single-digit "add-on facts". One could memorize all the facts past rote, but pattern-based strategies are more enlightening and, for almost people, more efficient:^[35]

• Commutative property: Mentioned above, using the pattern a + b = b + a reduces the number of "addition facts" from 100 to 55.
• One or 2 more than: Adding 1 or 2 is a basic task, and information technology tin can be accomplished through counting on or, ultimately, intuition.^[35]
• Zip: Since zero is the additive identity, adding zero is fiddling. Nonetheless, in the teaching of arithmetic, some students are introduced to improver every bit a process that always increases the addends; give-and-take problems may help rationalize the "exception" of zero.^[35]
• Doubles: Adding a number to itself is related to counting by two and to multiplication. Doubles facts course a backbone for many related facts, and students observe them relatively piece of cake to grasp.^[35]
• Virtually-doubles: Sums such as half-dozen + 7 = xiii tin be quickly derived from the doubles fact vi + six = 12 past adding one more than, or from 7 + 7 = 14 but subtracting i.^[35]
• Five and ten: Sums of the form 5 + x and x + ten are ordinarily memorized early on and tin can be used for deriving other facts. For example, 6 + seven = 13 can exist derived from 5 + 7 = 12 by adding one more than.^[35]
• Making ten: An advanced strategy uses 10 as an intermediate for sums involving 8 or nine; for example, 8 + half dozen = viii + two + 4 = ten + 4 = fourteen.^[35]

Equally students grow older, they commit more facts to retentiveness, and acquire to derive other facts rapidly and fluently. Many students never commit all the facts to memory, but can however observe any bones fact chop-chop.^[32]

Carry [edit]

The standard algorithm for adding multidigit numbers is to align the addends vertically and add the columns, starting from the ones cavalcade on the right. If a column exceeds 9, the extra digit is "carried" into the adjacent cavalcade. For example, in the addition 27 + 59

          ¹   27 + 59 ————   86        

7 + ix = 16, and the digit i is the carry.^[b] An alternating strategy starts calculation from the most meaning digit on the left; this route makes carrying a little clumsier, but it is faster at getting a rough judge of the sum. At that place are many alternative methods.

Decimal fractions [edit]

Decimal fractions can be added by a simple modification of the to a higher place process.^[36] One aligns two decimal fractions higher up each other, with the decimal point in the same location. If necessary, ane can add together abaft zeros to a shorter decimal to brand it the same length as the longer decimal. Finally, one performs the same improver process as above, except the decimal betoken is placed in the respond, exactly where information technology was placed in the summands.

As an example, 45.1 + 4.34 can be solved as follows:

          four v . 1 0 +  0 4 . 3 4 ————————————    4 9 . four 4        

Scientific notation [edit]

In scientific annotation, numbers are written in the grade ${\displaystyle x=a\times 10^{b}}$ , where ${\displaystyle a}$ is the significand and ${\displaystyle x^{b}}$ is the exponential role. Improver requires two numbers in scientific note to exist represented using the same exponential part, then that the two significands can only be added.

For instance:

${\displaystyle 2.34\times 10^{-5}+5.67\times 10^{-6}=ii.34\times 10^{-5}+0.567\times 10^{-5}=ii.907\times 10^{-5}}$

Non-decimal [edit]

Addition in other bases is very similar to decimal improver. Equally an example, one can consider addition in binary.^[37] Adding two single-digit binary numbers is relatively simple, using a form of carrying:

0 + 0 → 0

0 + 1 → 1

1 + 0 → i

1 + 1 → 0, comport 1 (since one + 1 = 2 = 0 + (one × ii¹))

Calculation two "1" digits produces a digit "0", while 1 must be added to the adjacent column. This is similar to what happens in decimal when certain single-digit numbers are added together; if the issue equals or exceeds the value of the radix (ten), the digit to the left is incremented:

5 + five → 0, conduct 1 (since 5 + v = 10 = 0 + (i × 10¹))

vii + 9 → 6, carry ane (since 7 + 9 = 16 = six + (ane × ten^one))

This is known as carrying.^[38] When the outcome of an add-on exceeds the value of a digit, the procedure is to "carry" the excess amount divided by the radix (that is, 10/ten) to the left, adding it to the next positional value. This is correct since the next position has a weight that is higher past a gene equal to the radix. Carrying works the aforementioned mode in binary:

          1 1 one 1 one    (carried digits)          0 one 1 0 i +   1 0 ane ane 1 —————————————   i 0 0 i 0 0 = 36        

In this example, two numerals are being added together: 01101₂ (xiii_x) and 10111₂ (23₁₀). The height row shows the carry bits used. Starting in the rightmost cavalcade, i + 1 = x₂ . The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. The 2d column from the right is added: 1 + 0 + 1 = 10_two over again; the 1 is carried, and 0 is written at the bottom. The tertiary column: i + 1 + i = 11₂ . This fourth dimension, a 1 is carried, and a 1 is written in the lesser row. Proceeding like this gives the final answer 100100_ii (36₁₀).

Computers [edit]

Analog computers work straight with concrete quantities, then their addition mechanisms depend on the course of the addends. A mechanical adder might represent two addends equally the positions of sliding blocks, in which case they tin be added with an averaging lever. If the addends are the rotation speeds of two shafts, they can be added with a differential. A hydraulic adder can add together the pressures in two chambers by exploiting Newton'due south second law to balance forces on an assembly of pistons. The most common state of affairs for a general-purpose analog computer is to add ii voltages (referenced to basis); this tin be accomplished roughly with a resistor network, but a better blueprint exploits an operational amplifier.^[39]

Addition is also primal to the operation of digital computers, where the efficiency of add-on, in detail the deport machinery, is an of import limitation to overall operation.

Role of Charles Babbage's Difference Engine including the improver and carry mechanisms

The abacus, also called a counting frame, is a computing tool that was in use centuries before the adoption of the written mod numeral system and is still widely used past merchants, traders and clerks in Asia, Africa, and elsewhere; it dates back to at least 2700–2300 BC, when it was used in Sumer.^[xl]

Blaise Pascal invented the mechanical figurer in 1642;^[41] it was the starting time operational adding auto. It fabricated utilize of a gravity-assisted comport machinery. Information technology was the only operational mechanical calculator in the 17th century^[42] and the earliest automated, digital estimator. Pascal'southward calculator was limited by its carry mechanism, which forced its wheels to only turn i manner so it could add together. To subtract, the operator had to employ the Pascal's estimator'due south complement, which required as many steps every bit an addition. Giovanni Poleni followed Pascal, building the second functional mechanical calculator in 1709, a computing clock made of woods that, in one case setup, could multiply two numbers automatically.

"Full adder" logic circuit that adds two binary digits, A and B, along with a carry input C_in , producing the sum bit, Due south, and a comport output, C_out .

Adders execute integer addition in electronic digital computers, usually using binary arithmetic. The simplest architecture is the ripple carry adder, which follows the standard multi-digit algorithm. One slight improvement is the comport skip design, again following human intuition; 1 does not perform all the carries in computing 999 + 1, simply one bypasses the group of 9s and skips to the answer.^[43]

In practice, computational addition may exist achieved via XOR and AND bitwise logical operations in conjunction with bitshift operations as shown in the pseudocode below. Both XOR and AND gates are straightforward to realize in digital logic allowing the realization of full adder circuits which in turn may be combined into more complex logical operations. In modern digital computers, integer addition is typically the fastest arithmetics instruction, yet it has the largest affect on operation, since it underlies all floating-indicate operations likewise as such basic tasks equally address generation during memory admission and fetching instructions during branching. To increase speed, modern designs calculate digits in parallel; these schemes become past such names as carry select, carry lookahead, and the Ling pseudocarry. Many implementations are, in fact, hybrids of these concluding three designs.^{[44] [45]} Different addition on paper, addition on a calculator ofttimes changes the addends. On the ancient abacus and adding lath, both addends are destroyed, leaving only the sum. The influence of the abacus on mathematical thinking was stiff plenty that early Latin texts often claimed that in the process of adding "a number to a number", both numbers vanish.^[46] In modern times, the Add together educational activity of a microprocessor ofttimes replaces the augend with the sum but preserves the addend.^[47] In a loftier-level programming language, evaluating a + b does non alter either a or b; if the goal is to supplant a with the sum this must be explicitly requested, typically with the statement a = a + b . Some languages such as C or C++ allow this to be abbreviated as a += b .

                        // Iterative algorithm            int                                    add together            (            int                                    x            ,                                    int                                    y            )                                    {                                                int                                    carry                                    =                                    0            ;                                                while                                    (            y                                    !=                                    0            )                                    {                                                            carry                                    =                                    AND            (            10            ,                                    y            );                                    // Logical AND                                    10                                    =                                    XOR            (            x            ,                                    y            );                                    // Logical XOR                                    y                                    =                                    carry                                    <<                                    one            ;                                    // left bitshift carry by one                                    }                                                render                                    ten            ;                                    }                        // Recursive algorithm            int                                    add            (            int                                    x            ,                                    int                                    y            )                                    {                                                render                                    10                                    if                                    (            y                                    ==                                    0            )                                    else                                    add together            (            XOR            (            x            ,                                    y            ),                                    AND            (            10            ,                                    y            )                                    <<                                    1            );                        }                      

On a computer, if the result of an add-on is as well large to store, an arithmetic overflow occurs, resulting in an incorrect answer. Unanticipated arithmetic overflow is a adequately common cause of program errors. Such overflow bugs may be hard to observe and diagnose considering they may manifest themselves only for very large input information sets, which are less likely to be used in validation tests.^[48] The Year 2000 problem was a serial of bugs where overflow errors occurred due to utilise of a two-digit format for years.^[49]

Addition of numbers [edit]

To prove the usual properties of improver, one must outset define add-on for the context in question. Addition is kickoff divers on the natural numbers. In set theory, addition is then extended to progressively larger sets that include the natural numbers: the integers, the rational numbers, and the real numbers.^[50] (In mathematics education,^[51] positive fractions are added before negative numbers are even considered; this is also the historical route.^[52])

Natural numbers [edit]

There are ii pop means to ascertain the sum of two natural numbers a and b. If ane defines natural numbers to be the cardinalities of finite sets, (the cardinality of a fix is the number of elements in the ready), then information technology is advisable to ascertain their sum every bit follows:

• Permit N(Southward) be the cardinality of a set S. Take two disjoint sets A and B, with North(A) = a and N(B) = b . Then a + b is defined as ${\displaystyle Due north(A\cup B)}$ .^[53]

Here, A ∪ B is the union of A and B. An alternating version of this definition allows A and B to possibly overlap and then takes their disjoint union, a mechanism that allows mutual elements to be separated out and therefore counted twice.

The other pop definition is recursive:

• Let n ⁺ be the successor of n, that is the number following n in the natural numbers, so 0⁺=1, 1⁺=two. Define a + 0 = a . Define the general sum recursively by a + (b ⁺) = (a + b)⁺ . Hence 1 + 1 = i + 0⁺ = (one + 0)⁺ = one⁺ = 2.^[54]

Again, there are minor variations upon this definition in the literature. Taken literally, the above definition is an application of the recursion theorem on the partially ordered ready Due north ².^[55] On the other hand, some sources prefer to use a restricted recursion theorem that applies but to the set of natural numbers. One then considers a to be temporarily "stock-still", applies recursion on b to define a function "a +", and pastes these unary operations for all a together to form the full binary operation.^[56]

This recursive conception of addition was adult by Dedekind as early as 1854, and he would expand upon it in the following decades.^[57] He proved the associative and commutative properties, amidst others, through mathematical induction.

Integers [edit]

The simplest conception of an integer is that information technology consists of an absolute value (which is a natural number) and a sign (generally either positive or negative). The integer null is a special 3rd example, being neither positive nor negative. The corresponding definition of improver must go on by cases:

• For an integer n, permit |northward| be its absolute value. Permit a and b be integers. If either a or b is zero, treat information technology equally an identity. If a and b are both positive, define a + b = |a| + |b|. If a and b are both negative, define a + b = −(|a| + |b|). If a and b have different signs, ascertain a + b to be the divergence betwixt |a| and |b|, with the sign of the term whose accented value is larger.^[58] As an case, −half dozen + 4 = −ii; because −six and four accept dissimilar signs, their absolute values are subtracted, and since the absolute value of the negative term is larger, the answer is negative.

Although this definition can exist useful for concrete problems, the number of cases to consider complicates proofs unnecessarily. And so the following method is commonly used for defining integers. Information technology is based on the remark that every integer is the difference of ii natural integers and that two such differences, a – b and c – d are equal if and only if a + d = b + c . So, 1 can define formally the integers as the equivalence classes of ordered pairs of natural numbers under the equivalence relation

(a, b) ~ (c, d) if and only if a + d = b + c .

The equivalence form of (a, b) contains either (a – b, 0) if a ≥ b , or (0, b – a) otherwise. If due north is a natural number, i can denote +n the equivalence class of (n, 0), and past –northward the equivalence form of (0, n). This allows identifying the natural number north with the equivalence course +n .

Addition of ordered pairs is washed component-wise:

${\displaystyle (a,b)+(c,d)=(a+c,b+d).}$

A straightforward computation shows that the equivalence class of the result depends only on the equivalences classes of the summands, and thus that this defines an addition of equivalence classes, that is integers.^[59] Some other straightforward ciphering shows that this addition is the same as the to a higher place case definition.

This style of defining integers as equivalence classes of pairs of natural numbers, can be used to embed into a group any commutative semigroup with cancellation property. Here, the semigroup is formed by the natural numbers and the grouping is the condiment group of integers. The rational numbers are synthetic similarly, by taking as semigroup the nonzero integers with multiplication.

This construction has been also generalized nether the name of Grothendieck group to the case of any commutative semigroup. Without the cancellation property the semigroup homomorphism from the semigroup into the grouping may be not-injective. Originally, the Grothendieck grouping was, more specifically, the result of this construction practical to the equivalences classes under isomorphisms of the objects of an abelian category, with the direct sum equally semigroup operation.

Rational numbers (fractions) [edit]

Addition of rational numbers can be computed using the least common denominator, just a conceptually simpler definition involves only integer addition and multiplication:

• Define ${\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {advertizing+bc}{bd}}.}$

As an example, the sum ${\displaystyle {\frac {3}{4}}+{\frac {ane}{8}}={\frac {three\times 8+four\times i}{four\times 8}}={\frac {24+4}{32}}={\frac {28}{32}}={\frac {7}{8}}}$ .

Addition of fractions is much simpler when the denominators are the same; in this case, i can simply add together the numerators while leaving the denominator the same: ${\displaystyle {\frac {a}{c}}+{\frac {b}{c}}={\frac {a+b}{c}}}$ , so ${\displaystyle {\frac {ane}{4}}+{\frac {2}{iv}}={\frac {1+2}{4}}={\frac {3}{4}}}$ .^[60]

The commutativity and associativity of rational addition is an easy upshot of the laws of integer arithmetics.^[61] For a more than rigorous and general word, meet field of fractions.

Real numbers [edit]

Adding π²/vi and due east using Dedekind cuts of rationals.

A common structure of the set of real numbers is the Dedekind completion of the set of rational numbers. A existent number is defined to be a Dedekind cut of rationals: a non-empty set of rationals that is closed downward and has no greatest element. The sum of real numbers a and b is defined element by element:

• Define ${\displaystyle a+b=\{q+r\mid q\in a,r\in b\}.}$ ^[62]

This definition was first published, in a slightly modified course, by Richard Dedekind in 1872.^[63] The commutativity and associativity of real improver are immediate; defining the real number 0 to be the set of negative rationals, information technology is easily seen to exist the additive identity. Probably the trickiest part of this construction pertaining to addition is the definition of additive inverses.^[64]

Adding π²/6 and due east using Cauchy sequences of rationals.

Unfortunately, dealing with multiplication of Dedekind cuts is a time-consuming example-past-case process like to the addition of signed integers.^[65] Some other approach is the metric completion of the rational numbers. A real number is essentially defined to exist the limit of a Cauchy sequence of rationals, lima _n . Add-on is defined term by term:

• Ascertain ${\displaystyle \lim _{n}a_{n}+\lim _{n}b_{n}=\lim _{north}(a_{n}+b_{n}).}$ ^[66]

This definition was first published by Georg Cantor, also in 1872, although his formalism was slightly different.^[67] Ane must bear witness that this operation is well-divers, dealing with co-Cauchy sequences. One time that task is done, all the properties of real addition follow immediately from the backdrop of rational numbers. Furthermore, the other arithmetics operations, including multiplication, have straightforward, coordinating definitions.^[68]

Complex numbers [edit]

Addition of two complex numbers tin can be done geometrically by constructing a parallelogram.

Complex numbers are added by calculation the real and imaginary parts of the summands.^{[69] [lxx]} That is to say:

${\displaystyle (a+bi)+(c+di)=(a+c)+(b+d)i.}$

Using the visualization of circuitous numbers in the circuitous airplane, the add-on has the post-obit geometric interpretation: the sum of two circuitous numbers A and B, interpreted as points of the circuitous aeroplane, is the point 10 obtained by edifice a parallelogram three of whose vertices are O, A and B. Equivalently, X is the bespeak such that the triangles with vertices O, A, B, and 10, B, A, are congruent.

Generalizations [edit]

In that location are many binary operations that tin can be viewed equally generalizations of the add-on functioning on the real numbers. The field of abstruse algebra is centrally concerned with such generalized operations, and they as well announced in set theory and category theory.

Abstruse algebra [edit]

Vectors [edit]

In linear algebra, a vector space is an algebraic structure that allows for calculation whatever 2 vectors and for scaling vectors. A familiar vector space is the ready of all ordered pairs of real numbers; the ordered pair (a,b) is interpreted as a vector from the origin in the Euclidean airplane to the point (a,b) in the aeroplane. The sum of two vectors is obtained past calculation their individual coordinates:

${\displaystyle (a,b)+(c,d)=(a+c,b+d).}$

This addition operation is central to classical mechanics, in which velocities, accelerations and forces are all represented by vectors.^[71]

Matrices [edit]

Matrix addition is defined for 2 matrices of the same dimensions. The sum of two thousand × n (pronounced "grand by n") matrices A and B, denoted past A + B , is once more an chiliad × due north matrix computed by calculation corresponding elements:^{[72] [73]}

${\displaystyle {\begin{aligned}\mathbf {A} +\mathbf {B} &={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\\\end{bmatrix}}+{\brainstorm{bmatrix}b_{11}&b_{12}&\cdots &b_{1n}\\b_{21}&b_{22}&\cdots &b_{2n}\\\vdots &\vdots &\ddots &\vdots \\b_{m1}&b_{m2}&\cdots &b_{mn}\\\end{bmatrix}}\\&={\begin{bmatrix}a_{11}+b_{11}&a_{12}+b_{12}&\cdots &a_{1n}+b_{1n}\\a_{21}+b_{21}&a_{22}+b_{22}&\cdots &a_{2n}+b_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}+b_{m1}&a_{m2}+b_{m2}&\cdots &a_{mn}+b_{mn}\\\cease{bmatrix}}\\\finish{aligned}}}$

For example:

${\displaystyle {\brainstorm{bmatrix}1&three\\1&0\\one&ii\end{bmatrix}}+{\begin{bmatrix}0&0\\vii&5\\2&ane\end{bmatrix}}={\brainstorm{bmatrix}one+0&3+0\\1+7&0+5\\i+2&2+one\finish{bmatrix}}={\begin{bmatrix}1&three\\8&5\\iii&three\cease{bmatrix}}}$

Modular arithmetic [edit]

In modular arithmetic, the set of available numbers is restricted to a finite subset of the integers, and addition "wraps around" when reaching a certain value, called the modulus. For example, the set of integers modulo 12 has twelve elements; it inherits an addition operation from the integers that is central to musical set theory. The set of integers modulo 2 has but two elements; the addition operation it inherits is known in Boolean logic as the "exclusive or" office. A similar "wrap around" operation arises in geometry, where the sum of two bending measures is oftentimes taken to be their sum as real numbers modulo 2π. This amounts to an addition performance on the circle, which in turn generalizes to add-on operations on many-dimensional tori.

General theory [edit]

The general theory of abstract algebra allows an "addition" operation to exist whatsoever associative and commutative performance on a set. Basic algebraic structures with such an addition functioning include commutative monoids and abelian groups.

Set theory and category theory [edit]

A far-reaching generalization of addition of natural numbers is the add-on of ordinal numbers and cardinal numbers in set theory. These give two unlike generalizations of addition of natural numbers to the transfinite. Unlike most addition operations, add-on of ordinal numbers is non commutative.^[74] Addition of cardinal numbers, even so, is a commutative operation closely related to the disjoint union operation.

In category theory, disjoint marriage is seen equally a particular case of the coproduct operation,^[75] and full general coproducts are possibly the well-nigh abstract of all the generalizations of improver. Some coproducts, such as direct sum and wedge sum, are named to evoke their connection with addition.

[edit]

Improver, along with subtraction, multiplication and division, is considered one of the bones operations and is used in elementary arithmetic.

Arithmetic [edit]

Subtraction can exist idea of as a kind of improver—that is, the addition of an condiment inverse. Subtraction is itself a sort of inverse to improver, in that adding x and subtracting x are changed functions.

Given a prepare with an add-on operation, one cannot always define a corresponding subtraction operation on that set; the set of natural numbers is a simple example. On the other paw, a subtraction operation uniquely determines an improver performance, an condiment changed operation, and an condiment identity; for this reason, an condiment group can be described as a set that is closed nether subtraction.^[76]

Multiplication can be idea of as repeated addition. If a single term x appears in a sum due north times, so the sum is the product of n and x. If n is non a natural number, the production may still brand sense; for example, multiplication by −1 yields the additive inverse of a number.

In the real and complex numbers, addition and multiplication can be interchanged by the exponential function:^[77]

${\displaystyle e^{a+b}=e^{a}e^{b}.}$

This identity allows multiplication to be carried out by consulting a table of logarithms and computing add-on by hand; it also enables multiplication on a slide rule. The formula is still a practiced offset-club approximation in the wide context of Lie groups, where it relates multiplication of minute grouping elements with addition of vectors in the associated Prevarication algebra.^[78]

There are fifty-fifty more generalizations of multiplication than addition.^[79] In full general, multiplication operations e'er distribute over addition; this requirement is formalized in the definition of a band. In some contexts, such as the integers, distributivity over addition and the being of a multiplicative identity is enough to uniquely determine the multiplication operation. The distributive property likewise provides information about improver; by expanding the production (one + 1)(a + b) in both ways, one concludes that add-on is forced to be commutative. For this reason, ring addition is commutative in general.^[lxxx]

Sectionalisation is an arithmetic operation remotely related to addition. Since a/b = a(b ⁻¹), division is right distributive over addition: (a + b) / c = a/c + b/c .^[81] Yet, division is not left distributive over improver; 1 / (two + ii) is not the same as i/two + 1/ii.

Ordering [edit]

The maximum operation "max (a, b)" is a binary operation like to improver. In fact, if two nonnegative numbers a and b are of different orders of magnitude, and so their sum is approximately equal to their maximum. This approximation is extremely useful in the applications of mathematics, for instance in truncating Taylor series. Even so, it presents a perpetual difficulty in numerical analysis, essentially since "max" is non invertible. If b is much greater than a, then a straightforward calculation of (a + b) − b can accrue an unacceptable circular-off error, perhaps even returning zero. Come across also Loss of significance.

The approximation becomes exact in a kind of infinite limit; if either a or b is an infinite cardinal number, their key sum is exactly equal to the greater of the 2.^[83] Accordingly, in that location is no subtraction operation for infinite cardinals.^[84]

Maximization is commutative and associative, like addition. Furthermore, since addition preserves the ordering of existent numbers, addition distributes over "max" in the aforementioned way that multiplication distributes over addition:

${\displaystyle a+\max(b,c)=\max(a+b,a+c).}$

For these reasons, in tropical geometry one replaces multiplication with addition and addition with maximization. In this context, improver is called "tropical multiplication", maximization is called "tropical addition", and the tropical "condiment identity" is negative infinity.^[85] Some authors adopt to replace addition with minimization; then the condiment identity is positive infinity.^[86]

Tying these observations together, tropical add-on is approximately related to regular improver through the logarithm:

${\displaystyle \log(a+b)\approx \max(\log a,\log b),}$

which becomes more authentic as the base of the logarithm increases.^[87] The approximation tin be made exact past extracting a abiding h, named by analogy with Planck's constant from quantum mechanics,^[88] and taking the "classical limit" as h tends to null:

${\displaystyle \max(a,b)=\lim _{h\to 0}h\log(e^{a/h}+eastward^{b/h}).}$

In this sense, the maximum functioning is a dequantized version of addition.^[89]

Other means to add together [edit]

Incrementation, also known as the successor operation, is the add-on of 1 to a number.

Summation describes the addition of arbitrarily many numbers, usually more just two. It includes the idea of the sum of a single number, which is itself, and the empty sum, which is zero.^[90] An space summation is a delicate procedure known as a serial.^[91]

Counting a finite set is equivalent to summing 1 over the set.

Integration is a kind of "summation" over a continuum, or more precisely and more often than not, over a differentiable manifold. Integration over a zero-dimensional manifold reduces to summation.

Linear combinations combine multiplication and summation; they are sums in which each term has a multiplier, commonly a existent or complex number. Linear combinations are especially useful in contexts where straightforward addition would violate some normalization rule, such as mixing of strategies in game theory or superposition of states in quantum mechanics.^[92]

Convolution is used to add together two independent random variables defined past distribution functions. Its usual definition combines integration, subtraction, and multiplication.^[93] In full general, convolution is useful every bit a kind of domain-side addition; by contrast, vector addition is a kind of range-side improver.

Come across likewise [edit]

• Lunar arithmetic
• Mental arithmetics
• Parallel improver (mathematics)
• Verbal arithmetics (besides known equally cryptarithms), puzzles involving addition

Notes [edit]

1. ^ "Addend" is not a Latin word; in Latin it must exist farther conjugated, as in numerus addendus "the number to exist added".
2. ^ Some authors think that "carry" may be inappropriate for instruction; Van de Walle (p. 211) calls it "obsolete and conceptually misleading", preferring the word "trade". However, "carry" remains the standard term.

Footnotes [edit]

1. ^ From Enderton (p. 138): "...select two sets K and L with carte du jour K = 2 and carte du jour Fifty = iii. Sets of fingers are handy; sets of apples are preferred past textbooks."
2. ^ "Addition". world wide web.mathsisfun.com . Retrieved 2020-08-25 .
3. ^ Devine et al. p. 263
4. ^ Mazur, Joseph. Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers. Princeton Academy Press, 2014. p. 161
5. ^ Department of the Army (1961) Army Technical Manual TM eleven-684: Principles and Applications of Mathematics for Communications-Electronics. Section v.1
6. ^ ^{a b} Shmerko, 5.P.; Yanushkevich [Ânuškevič], Svetlana North. [Svitlana Due north.]; Lyshevski, S.E. (2009). Calculator arithmetics for nanoelectronics. CRC Press. p. eighty.
7. ^ ^{a b} Schmid, Hermann (1974). Decimal Computation (1st ed.). Binghamton, NY: John Wiley & Sons. ISBN0-471-76180-X. and Schmid, Hermann (1983) [1974]. Decimal Computation (reprint of 1st ed.). Malabar, FL: Robert E. Krieger Publishing Company. ISBN978-0-89874-318-0.
8. ^ ^{a b} Weisstein, Eric W. "Addition". mathworld.wolfram.com . Retrieved 2020-08-25 .
9. ^ Hosch, W.L. (Ed.). (2010). The Britannica Guide to Numbers and Measurement. The Rosen Publishing Group. p. 38
10. ^ ^{a b} Schwartzman p. 19
11. ^ Karpinski pp. 56–57, reproduced on p. 104
12. ^ Schwartzman (p. 212) attributes adding upwards to the Greeks and Romans, maxim it was nearly as mutual as adding downwards. On the other paw, Karpinski (p. 103) writes that Leonard of Pisa "introduces the novelty of writing the sum in a higher place the addends"; information technology is unclear whether Karpinski is claiming this equally an original invention or merely the introduction of the practice to Europe.
13. ^ Karpinski pp. 150–153
14. ^ Cajori, Florian (1928). "Origin and meanings of the signs + and -". A History of Mathematical Notations, Vol. ane. The Open Court Company, Publishers.
15. ^ "plus". Oxford English language Dictionary (Online ed.). Oxford University Printing. (Subscription or participating institution membership required.)
16. ^ See Viro 2001 for an example of the sophistication involved in calculation with sets of "partial cardinality".
17. ^ Adding information technology upward (p. 73) compares adding measuring rods to adding sets of cats: "For example, inches can be subdivided into parts, which are hard to tell from the wholes, except that they are shorter; whereas it is painful to cats to split up them into parts, and it seriously changes their nature."
18. ^ Mosley, F. (2001). Using number lines with 5–8 year olds. Nelson Thornes. p. 8
19. ^ Li, Y., & Lappan, G. (2014). Mathematics curriculum in schoolhouse education. Springer. p. 204
20. ^ Bronstein, Ilja Nikolaevič; Semendjajew, Konstantin Adolfovič (1987) [1945]. "ii.4.1.1.". In Grosche, Günter; Ziegler, Viktor; Ziegler, Dorothea (eds.). Taschenbuch der Mathematik (in German). Vol. 1. Translated by Ziegler, Viktor. Weiß, Jürgen (23 ed.). Thun and Frankfurt am Primary: Verlag Harri Deutsch (and B.Yard. Teubner Verlagsgesellschaft, Leipzig). pp. 115–120. ISBN978-3-87144-492-0.
21. ^ Kaplan pp. 69–71
22. ^ Hempel, C.1000. (2001). The philosophy of Carl G. Hempel: studies in science, explanation, and rationality. p. seven
23. ^ R. Fierro (2012) Mathematics for Uncomplicated School Teachers. Cengage Learning. Sec 2.three
24. ^ Moebs, William; et al. (2022). "1.4 Dimensional Analysis". Academy Physics Book 1. OpenStax. ISBN978-1-947172-20-three.
25. ^ Wynn p. 5
26. ^ Wynn p. fifteen
27. ^ Wynn p. 17
28. ^ Wynn p. 19
29. ^ Randerson, James (21 August 2008). "Elephants have a caput for figures". The Guardian. Archived from the original on 2 April 2015. Retrieved 29 March 2015.
30. ^ F. Smith p. 130
31. ^ Carpenter, Thomas; Fennema, Elizabeth; Franke, Megan Loef; Levi, Linda; Empson, Susan (1999). Children's mathematics: Cognitively guided instruction . Portsmouth, NH: Heinemann. ISBN978-0-325-00137-1.
32. ^ ^{a b} Henry, Valerie J.; Brown, Richard S. (2008). "Kickoff-grade bones facts: An investigation into educational activity and learning of an accelerated, loftier-demand memorization standard". Periodical for Research in Mathematics Education. 39 (two): 153–183. doi:10.2307/30034895. JSTOR 30034895.
33. ^ Beckmann, S. (2014). The twenty-3rd ICMI study: primary mathematics written report on whole numbers. International Journal of STEM Education, i(ane), one-viii. Chicago
34. ^ Schmidt, West., Houang, R., & Cogan, L. (2002). "A coherent curriculum". American Educator, 26(ii), 1–xviii.
35. ^ ^{a b c d e f thou} Fosnot and Dolk p. 99
36. ^ Rebecca Wingard-Nelson (2014) Decimals and Fractions: Information technology's Piece of cake Enslow Publishers, Inc.
37. ^ Dale R. Patrick, Stephen Due west. Fardo, Vigyan Chandra (2008) Electronic Digital System Fundamentals The Fairmont Press, Inc. p. 155
38. ^ P.E. Bates Bothman (1837) The common school arithmetic. Henry Benton. p. 31
39. ^ Truitt and Rogers pp. 1;44–49 and pp. 2;77–78
40. ^ Ifrah, Georges (2001). The Universal History of Calculating: From the Abacus to the Quantum Figurer. New York: John Wiley & Sons, Inc. ISBN978-0-471-39671-0. p. 11
41. ^ Jean Marguin, p. 48 (1994) ; Quoting René Taton (1963)
42. ^ Come across Competing designs in Pascal'southward computer article
43. ^ Flynn and Overman pp. two, 8
44. ^ Flynn and Overman pp. 1–nine
45. ^ Yeo, Sang-Soo, et al., eds. Algorithms and Architectures for Parallel Processing: 10th International Conference, ICA3PP 2010, Busan, Korea, May 21–23, 2010. Proceedings. Vol. ane. Springer, 2010. p. 194
46. ^ Karpinski pp. 102–103
47. ^ The identity of the augend and addend varies with architecture. For ADD in x86 see Horowitz and Hill p. 679; for Add in 68k see p. 767.
48. ^ Joshua Bloch, "Extra, Extra – Read All About It: Nearly All Binary Searches and Mergesorts are Broken" Archived 2016-04-01 at the Wayback Machine. Official Google Research Blog, June 2, 2006.
49. ^ Neumann, Peter Chiliad. "The Risks Digest Volume four: Issue 45". The Risks Digest. Archived from the original on 2014-12-28. Retrieved 2015-03-xxx .
50. ^ Enderton chapters 4 and 5, for example, follow this development.
51. ^ Co-ordinate to a survey of the nations with highest TIMSS mathematics exam scores; come across Schmidt, Westward., Houang, R., & Cogan, L. (2002). A coherent curriculum. American educator, 26(ii), p. 4.
52. ^ Baez (p. 37) explains the historical development, in "stark contrast" with the fix theory presentation: "Apparently, half an apple is easier to empathize than a negative apple!"
53. ^ Begle p. 49, Johnson p. 120, Devine et al. p. 75
54. ^ Enderton p. 79
55. ^ For a version that applies to whatever poset with the descending chain condition, see Bergman p. 100.
56. ^ Enderton (p. 79) observes, "Merely we want i binary performance +, not all these petty ane-place functions."
57. ^ Ferreirós p. 223
58. ^ K. Smith p. 234, Sparks and Rees p. 66
59. ^ Enderton p. 92
60. ^ Schyrlet Cameron, and Carolyn Craig (2013)Adding and Subtracting Fractions, Grades 5–viii Mark Twain, Inc.
61. ^ The verifications are carried out in Enderton p. 104 and sketched for a general field of fractions over a commutative ring in Dummit and Foote p. 263.
62. ^ Enderton p. 114
63. ^ Ferreirós p. 135; see department six of Stetigkeit und irrationale Zahlen Archived 2005-x-31 at the Wayback Auto.
64. ^ The intuitive arroyo, inverting every element of a cut and taking its complement, works only for irrational numbers; come across Enderton p. 117 for details.
65. ^ Schubert, E. Thomas, Phillip J. Windley, and James Alves-Foss. "Higher Social club Logic Theorem Proving and Its Applications: Proceedings of the 8th International Workshop, volume 971 of." Lecture Notes in Computer science (1995).
66. ^ Textbook constructions are normally non so cavalier with the "lim" symbol; see Burrill (p. 138) for a more careful, drawn-out development of addition with Cauchy sequences.
67. ^ Ferreirós p. 128
68. ^ Burrill p. 140
69. ^ Conway, John B. (1986), Functions of One Complex Variable I, Springer, ISBN978-0-387-90328-half dozen
70. ^ Joshi, Kapil D (1989), Foundations of Discrete Mathematics, New York: John Wiley & Sons, ISBN978-0-470-21152-vi
71. ^ Gbur, p. 1
72. ^ Lipschutz, S., & Lipson, One thousand. (2001). Schaum's outline of theory and problems of linear algebra. Erlangga.
73. ^ Riley, K.F.; Hobson, M.P.; Bence, South.J. (2010). Mathematical methods for physics and technology . Cambridge Academy Printing. ISBN978-0-521-86153-3.
74. ^ Cheng, pp. 124–132
75. ^ Riehl, p. 100
76. ^ The ready still must be nonempty. Dummit and Foote (p. 48) discuss this criterion written multiplicatively.
77. ^ Rudin p. 178
78. ^ Lee p. 526, Suggestion 20.9
79. ^ Linderholm (p. 49) observes, "Past multiplication, properly speaking, a mathematician may mean practically annihilation. Past addition he may mean a great variety of things, but non then great a variety equally he will mean by 'multiplication'."
80. ^ Dummit and Foote p. 224. For this argument to work, ane nonetheless must assume that improver is a grouping performance and that multiplication has an identity.
81. ^ For an case of left and right distributivity, see Loday, especially p. 15.
82. ^ Compare Viro Effigy 1 (p. ii)
83. ^ Enderton calls this argument the "Absorption Police of Key Arithmetic"; information technology depends on the comparability of cardinals and therefore on the Axiom of Choice.
84. ^ Enderton p. 164
85. ^ Mikhalkin p. ane
86. ^ Akian et al. p. 4
87. ^ Mikhalkin p. two
88. ^ Litvinov et al. p. 3
89. ^ Viro p. 4
90. ^ Martin p. 49
91. ^ Stewart p. 8
92. ^ Rieffel and Polak, p. 16
93. ^ Gbur, p. 300

References [edit]

History

• Ferreirós, José (1999). Labyrinth of Idea: A History of Set Theory and Its Office in Modern Mathematics . Birkhäuser. ISBN978-0-8176-5749-9.
• Karpinski, Louis (1925). The History of Arithmetic. Rand McNally. LCC QA21.K3.
• Schwartzman, Steven (1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English language . MAA. ISBN978-0-88385-511-9.
• Williams, Michael (1985). A History of Computing Technology . Prentice-Hall. ISBN978-0-thirteen-389917-seven.

Uncomplicated mathematics

• Sparks, F.; Rees C. (1979). A Survey of Basic Mathematics. McGraw-Colina. ISBN978-0-07-059902-4.

Education

• Begle, Edward (1975). The Mathematics of the Elementary Schoolhouse. McGraw-Colina. ISBN978-0-07-004325-1.
• California Country Board of Didactics mathematics content standards Adopted December 1997, accessed December 2005.
• Devine, D.; Olson, J.; Olson, Thousand. (1991). Unproblematic Mathematics for Teachers (2e ed.). Wiley. ISBN978-0-471-85947-5.
• National Research Council (2001). Adding It Upwards: Helping Children Acquire Mathematics. National University Press. doi:10.17226/9822. ISBN978-0-309-06995-3.
• Van de Walle, John (2004). Unproblematic and Middle School Mathematics: Teaching developmentally (5e ed.). Pearson. ISBN978-0-205-38689-5.

Cognitive science

• Fosnot, Catherine T.; Dolk, Maarten (2001). Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction. Heinemann. ISBN978-0-325-00353-5.
• Wynn, Karen (1998). "Numerical competence in infants". The Evolution of Mathematical Skills. Taylor & Francis. ISBN0-86377-816-X.

Mathematical exposition

• Bogomolny, Alexander (1996). "Improver". Interactive Mathematics Miscellany and Puzzles (cut-the-knot.org). Archived from the original on April 26, 2006. Retrieved 3 February 2006.
• Cheng, Eugenia (2017). Beyond Infinity: An Expedition to the Outer Limits of Mathematics. Basic Books. ISBN978-1-541-64413-7.
• Dunham, William (1994). The Mathematical Universe . Wiley. ISBN978-0-471-53656-seven.
• Johnson, Paul (1975). From Sticks and Stones: Personal Adventures in Mathematics. Scientific discipline Inquiry Associates. ISBN978-0-574-19115-i.
• Linderholm, Carl (1971). Mathematics Made Difficult. Wolfe. ISBN978-0-7234-0415-6.
• Smith, Frank (2002). The Glass Wall: Why Mathematics Can Seem Difficult . Teachers College Press. ISBN978-0-8077-4242-half dozen.
• Smith, Karl (1980). The Nature of Modernistic Mathematics (third ed.). Wadsworth. ISBN978-0-8185-0352-viii.

Advanced mathematics

• Bergman, George (2005). An Invitation to General Algebra and Universal Constructions (2.3 ed.). General Printing. ISBN978-0-9655211-4-7.
• Burrill, Claude (1967). Foundations of Real Numbers. McGraw-Hill. LCC QA248.B95.
• Dummit, D.; Foote, R. (1999). Abstract Algebra (two ed.). Wiley. ISBN978-0-471-36857-1.
• Gbur, Greg (2011). Mathematical Methods for Optical Physics and Applied science. Cambridge University Press. ISBN978-0-511-91510-9. OCLC 704518582.
• Enderton, Herbert (1977). Elements of Set Theory. Academic Printing. ISBN978-0-12-238440-0.
• Lee, John (2003). Introduction to Smooth Manifolds. Springer. ISBN978-0-387-95448-6.
• Martin, John (2003). Introduction to Languages and the Theory of Computation (3 ed.). McGraw-Hill. ISBN978-0-07-232200-2.
• Riehl, Emily (2016). Category Theory in Context. Dover. ISBN978-0-486-80903-eight.
• Rudin, Walter (1976). Principles of Mathematical Assay (three ed.). McGraw-Hill. ISBN978-0-07-054235-8.
• Stewart, James (1999). Calculus: Early Transcendentals (4 ed.). Brooks/Cole. ISBN978-0-534-36298-0.

Mathematical inquiry

• Akian, Marianne; Bapat, Ravindra; Gaubert, Stephane (2005). "Min-plus methods in eigenvalue perturbation theory and generalised Lidskii-Vishik-Ljusternik theorem". INRIA Reports. arXiv:math.SP/0402090. Bibcode:2004math......2090A.
• Baez, J.; Dolan, J. (2001). Mathematics Unlimited – 2001 and Beyond. From Finite Sets to Feynman Diagrams. p. 29. arXiv:math.QA/0004133. ISBN3-540-66913-two.
• Litvinov, Grigory; Maslov, Victor; Sobolevskii, Andreii (1999). Idempotent mathematics and interval assay. Reliable Computing, Kluwer.
• Loday, Jean-Louis (2002). "Arithmetree". Journal of Algebra. 258: 275. arXiv:math/0112034. doi:ten.1016/S0021-8693(02)00510-0.
• Mikhalkin, Grigory (2006). Sanz-Solé, Marta (ed.). Proceedings of the International Congress of Mathematicians (ICM), Madrid, Spain, Baronial 22–thirty, 2006. Volume Ii: Invited lectures. Tropical Geometry and its Applications. Zürich: European Mathematical Society. pp. 827–852. arXiv:math.AG/0601041. ISBN978-3-03719-022-7. Zbl 1103.14034.
• Viro, Oleg (2001). Cascuberta, Carles; Miró-Roig, Rosa Maria; Verdera, Joan; Xambó-Descamps, Sebastià (eds.). European Congress of Mathematics: Barcelona, July 10–14, 2000, Volume I. Dequantization of Real Algebraic Geometry on Logarithmic Newspaper. Progress in Mathematics. Vol. 201. Basel: Birkhäuser. pp. 135–146. arXiv:math/0005163. Bibcode:2000math......5163V. ISBN978-3-7643-6417-five. Zbl 1024.14026.

Computing

• Flynn, M.; Oberman, Southward. (2001). Advanced Computer Arithmetic Pattern. Wiley. ISBN978-0-471-41209-0.
• Horowitz, P.; Loma, Westward. (2001). The Art of Electronics (2 ed.). Cambridge Upwards. ISBN978-0-521-37095-0.
• Jackson, Albert (1960). Analog Computation. McGraw-Loma. LCC QA76.four J3.
• Rieffel, Eleanor G.; Polak, Wolfgang H. (iv March 2011). Quantum Calculating: A Gentle Introduction. MIT Press. ISBN978-0-262-01506-6.
• Truitt, T.; Rogers, A. (1960). Nuts of Analog Computers. John F. Rider. LCC QA76.4 T7.
• Marguin, Jean (1994). Histoire des Instruments et Machines à Calculer, Trois Siècles de Mécanique Pensante 1642–1942 (in French). Hermann. ISBN978-2-7056-6166-3.
• Taton, René (1963). Le Calcul Mécanique. Que Sais-Je ? n° 367 (in French). Presses universitaires de French republic. pp. 20–28.

Further reading [edit]

• Baroody, Arthur; Tiilikainen, Sirpa (2003). The Evolution of Arithmetic Concepts and Skills. Two perspectives on improver development. Routledge. p. 75. ISBN0-8058-3155-X.
• Davison, David One thousand.; Landau, Marsha Southward.; McCracken, Leah; Thompson, Linda (1999). Mathematics: Explorations & Applications (TE ed.). Prentice Hall. ISBN978-0-13-435817-eight.
• Bunt, Lucas Due north.H.; Jones, Phillip S.; Bedient, Jack D. (1976). The Historical roots of Elementary Mathematics . Prentice-Hall. ISBN978-0-xiii-389015-0.
• Poonen, Bjorn (2010). "Addition". Girls' Angle Bulletin. three (3–five). ISSN 2151-5743.
• Weaver, J. Fred (1982). "Addition and Subtraction: A Cognitive Perspective". Addition and Subtraction: A Cerebral Perspective. Interpretations of Number Operations and Symbolic Representations of Addition and Subtraction. Taylor & Francis. p. threescore. ISBN0-89859-171-6.

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Source: https://en.wikipedia.org/wiki/Addition

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