3 + 2 = 5 withapples, a popular choice in textbooks[1]
Addition, usually denoted with theplus sign+, is one of the four basicoperations ofarithmetic, the other three beingsubtraction,multiplication, anddivision. The addition of twowhole numbers results in the total orsum of those values combined. For example, the adjacent image shows two columns of apples, one with three apples and the other with two apples, totaling to five apples. This observation is expressed as"3 + 2 = 5", which is read as "three plus twoequals five".
Addition has several important properties. It iscommutative, meaning that the order of thenumbers being added does not matter, so3 + 2 = 2 + 3, and it isassociative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter. Repeated addition of1 is the same as counting (seeSuccessor function). Addition of0 does not change a number. Addition also obeys rules concerning related operations such as subtraction and multiplication.
Performing addition is one of the simplest numerical tasks to perform. Addition of very small numbers is accessible to toddlers; the most basic task,1 + 1, can be performed by infants as young as five months, and even some members of other animal species. Inprimary education, students are taught to add numbers in thedecimal system, beginning with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancientabacus to the moderncomputer, where research on the most efficient implementations of addition continues to this day.
Addition is written using theplus sign "+"between the terms, and the result is expressed with anequals sign. For example, reads "one plus two equals three".[2] Nonetheless, some situations where addition is "understood", even though no symbol appears: a whole number followed immediately by afraction indicates the sum of the two, called amixed number, with an example,[3] This notation can cause confusion, since in most other contexts,juxtaposition denotesmultiplication instead.[4]
The terms of addends in the operation of an addition
The numbers or the objects to be added in general addition are collectively referred to as theterms,[5] theaddends or thesummands.[2] This terminology carries over to the summation of multiple terms.This is to be distinguished fromfactors, which aremultiplied.Some authors call the first addend theaugend.[6] In fact, during theRenaissance, many authors did not consider the first addend an "addend" at all. Today, due to thecommutative property of addition, "augend" is rarely used, and both terms are generally called addends.[7]
All of the above terminology derives fromLatin. "Addition" and "add" areEnglish words derived from the Latinverbaddere, which is in turn acompound ofad "to" anddare "to give", from theProto-Indo-European root*deh₃- "to give"; thus toadd is togive to.[7] Using thegerundivesuffix-nd results in "addend", "thing to be added".[a] Likewise fromaugere "to increase", one gets "augend", "thing to be increased".[8]
Redrawn illustration fromThe Art of Nombryng, one of the first English arithmetic texts, in the 15th century.[9]
"Sum" and "summand" derive from the Latinnounsumma "the highest" or "the top", used in Medieval Latin phrasesumma linea ("top line") meaning the sum of a column of numerical quantities, following theancient Greek andRoman practice of putting the sum at the top of a column.[10]Addere andsummare date back at least toBoethius, if not to earlier Roman writers such asVitruvius andFrontinus; Boethius also used several other terms for the addition operation. The laterMiddle English terms "adden" and "adding" were popularized byChaucer.[11]
Addition is one of the four basicoperations ofarithmetic, with the other three beingsubtraction,multiplication, anddivision. This operation works by adding two or more terms.[12] An arbitrary of many operation of additions is called thesummation.[13] An infinite summation is a delicate procedure known as aseries,[14] and it can be expressed throughcapital sigma notation, which compactly denotesiteration of the operation of addition based on the given indexes.[15] For example,
Addition is used to model many physical processes. Even for the simple case of addingnatural numbers, there are many possible interpretations and even more visual representations.
One set has three shapes while the other set has two. The total of shapes is five, which is a consequence of the addition of the objects from the two sets:.
Possibly the most basic interpretation of addition lies in combiningsets, that is:[2]
When two 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 interpretation is easy to visualize, with little danger of ambiguity. It is also useful in higher mathematics (for the rigorous definition it inspires, see§ Natural numbers below). However, it is not obvious how one should extend this interpretation to include fractional or negative numbers.[16]
One possibility is to consider collections of objects that can be easily divided, such as pies or, still better, segmented rods. Rather than solely combining collections of segments, rods can be joined end-to-end, which illustrates another conception of addition: adding not the rods but the lengths of the rods.[17]
A number-line visualization of the algebraic addition. A "jump" that has a distance of followed by another that is as long as, is the same as a translation by.
A number-line visualization of the unary addition. A translation by is equivalent to four translations by.
A second interpretation of addition comes from extending an initial length by a given length:[18]
When an original length is extended by a given amount, the final length is the sum of the original length and the length of the extension.
The sum can be interpreted as abinary operation that combines and algebraically, or it can be interpreted as the addition of more units to. Under the latter interpretation, the parts of a sum play asymmetric roles, and the operation is viewed as applying theunary operation to.[19] Instead of calling both and addends, it is more appropriate to call the "augend" in this case, since plays a passive role. The unary view is also useful when discussingsubtraction, because each unary addition operation has an inverse unary subtraction operation, and vice versa.
Addition iscommutative, meaning that one can change the order of the terms in a sum, but still get the same result. Symbolically, if and are any two numbers, then:[20]The fact that addition is commutative is known as the "commutative law of addition"[21] or "commutative property of addition".[22] Some otherbinary operations are commutative too as inmultiplication,[23] but others are not as insubtraction anddivision.[24]
Addition isassociative, which means that when three or more numbers are added together, theorder of operations does not change the result. For any three numbers,, and, it is true that:[25]For example,.
When addition is used together with other operations, theorder of operations becomes important. In the standard order of operations, addition is a lower priority thanexponentiation,nth roots, multiplication and division, but is given equal priority to subtraction.[26]
Addingzero to any number does not change the number. In other words, zero is theidentity element for addition, and is also known as theadditive identity. In symbols, for every, one has:[25]This law was first identified inBrahmagupta'sBrahmasphutasiddhanta in 628 AD, although he wrote it as three separate laws, depending on whether is negative, positive, or zero itself, and he used words rather than algebraic symbols. LaterIndian mathematicians refined the concept; around the year 830,Mahavira wrote, "zero becomes the same as what is added to it", corresponding to the unary statement. In the 12th century,Bhaskara wrote, "In the addition of cipher, or subtraction of it, the quantity, positive or negative, remains the same", corresponding to the unary statement.[27]
Within the context of integers, addition ofone also plays a special role: for any integer, the integer is the least integer greater than, also known as thesuccessor of. For instance, 3 is the successor of 2, and 7 is the successor of 6. Because of this succession, the value of can also be seen as the-th successor of, making addition an iterated succession. For example,6 + 2 is 8, because 8 is the successor of 7, which is the successor of 6, making 8 the second successor of 6.[28]
To numerically add physical quantities withunits, they must be expressed with common units.[29] For example, adding 50 milliliters to 150 milliliters gives 200 milliliters. However, if a measure of 5 feet is extended by 2 inches, the sum is 62 inches, since 60 inches is synonymous with 5 feet. On the other hand, it is usually meaningless to try to add 3 meters and 4 square meters, since those units are incomparable; this sort of consideration is fundamental indimensional analysis.[30]
Studies on mathematical development starting around the 1980s have exploited the phenomenon ofhabituation:infants look longer at situations that are unexpected.[31] A seminal experiment byKaren Wynn in 1992 involvingMickey Mouse dolls manipulated behind a screen demonstrated that five-month-old infantsexpect1 + 1 to be 2, and they are comparatively surprised when a physical situation seems to imply that1 + 1 is either 1 or 3. This finding has since been affirmed by a variety of laboratories using different methodologies.[32] Another 1992 experiment with oldertoddlers, between 18 and 35 months, exploited their development of motor control by allowing them to retrieveping-pong balls from a box; the youngest responded well for small numbers, while older subjects were able to compute sums up to 5.[33]
Even some nonhuman animals show a limited ability to add, particularlyprimates. In a 1995 experiment imitating Wynn's 1992 result (but usingeggplants instead of dolls),rhesus macaque andcottontop tamarin monkeys performed similarly to human infants. More dramatically, after being taught the meanings of theArabic numerals 0 through 4, onechimpanzee was able to compute the sum of two numerals without further training.[34] More recently,Asian elephants have demonstrated an ability to perform basic arithmetic.[35]
Typically, children first mastercounting. When given a problem that requires that two items and three items be combined, young children model the situation with physical objects, often fingers or a drawing, and then count the total. As they gain experience, they learn or discover the strategy of "counting-on": asked to find two plus three, children count three past two, saying "three, four,five" (usually ticking off fingers), and arriving at five. This strategy seems almost universal; children can easily pick it up from peers or teachers.[36] Most discover it independently. With additional experience, children learn to add more quickly by exploiting the commutativity of addition by counting up from the larger number, in this case, starting with three and counting "four,five." Eventually children begin to recall certain addition facts ("number bonds"), either through experience or rote memorization. Once some facts are committed to memory, children begin to derive unknown facts from known ones. For example, a child asked to add six and seven may know that6 + 6 = 12 and then reason that6 + 7 is one more, or 13.[37] Such derived facts can be found very quickly and most elementary school students eventually rely on a mixture of memorized and derived facts to add fluently.[38]
Different nations introduce whole numbers and arithmetic at different ages, with many countries teaching addition in pre-school.[39] However, throughout the world, addition is taught by the end of the first year of elementary school.[40]
An ability to add a pair of single digits (numbers from 0 to 9) is a prerequisite for addition of arbitrary numbers in thedecimal system. With 10 choices for each of the two digits to be added, this makes 100 single-digit "addition facts", which can be presented in anaddition table.
+
0
1
2
3
4
5
6
7
8
9
0
0
1
2
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
10
2
2
3
4
5
6
7
8
9
10
11
3
3
4
5
6
7
8
9
10
11
12
4
4
5
6
7
8
9
10
11
12
13
5
5
6
7
8
9
10
11
12
13
14
6
6
7
8
9
10
11
12
13
14
15
7
7
8
9
10
11
12
13
14
15
16
8
8
9
10
11
12
13
14
15
16
17
9
9
10
11
12
13
14
15
16
17
18
Learning to fluently and accurately compute single-digit additions is a major focus of early schooling in arithmetic. Sometimes students are encouraged to memorize the full addition table byrote, but pattern-based strategies are typically more enlightening and, for most people, more efficient:[41]
Commutative property: Mentioned above, using the pattern reduces the number of "addition facts" from 100 to 55.
One or two more: Adding 1 or 2 is a basic task, and it can be accomplished through counting on or, ultimately,intuition.[41]
Zero: Since zero is the additive identity, adding zero is trivial. Nonetheless, in the teaching of arithmetic, some students are introduced to addition as a process that always increases the addends;word problems may help rationalize the "exception" of zero.[41]
Doubles: Adding a number to itself is related to counting by two and tomultiplication. Doubles facts form a backbone for many related facts, and students find them relatively easy to grasp.[41]
Near-doubles: Sums such as 6 + 7 = 13 can be quickly derived from the doubles fact6 + 6 = 12 by adding one more, or from7 + 7 = 14 but subtracting one.[41]
Five and ten: Sums of the form 5 +x and 10 +x are usually memorized early and can be used for deriving other facts. For example,6 + 7 = 13 can be derived from5 + 7 = 12 by adding one more.[41]
Making ten: An advanced strategy uses 10 as an intermediate for sums involving 8 or 9; for example,8 + 6 = 8 + 2 + 4 =10 + 4 = 14.[41]
As students grow older, they commit more facts to memory and learn to derive other facts rapidly and fluently. Many students never commit all the facts to memory, but can still find any basic fact quickly.[38]
The standard algorithm for adding multidigit numbers is to align the addends vertically and add the columns by using the above addition table, starting from the ones column on the right. If the result of a column exceeds nine, the extra digit is "carried" into the next column. For example, in the following image, the ones in the addition of59 + 27 is 9 + 7 = 16, and the digit 1 is the carry.[42] An alternate strategy starts adding from the most significant digit on the left; this route makes carrying a little clumsier, but it is faster at getting a rough estimate of the sum.[b]
Decimal fractions can be added by a simple modification of the above process. One aligns two decimal fractions above each other, with the decimal point in the same location. If necessary, one can add trailing zeros to a shorter decimal to make it the same length as the longer decimal. Finally, one performs the same addition process as above, except the decimal point is placed in the answer, exactly where it was placed in the summands.[44] As an example, 45.1 + 4.34 can be solved as follows:
Inscientific notation, numbers are written in the form, where is thesignificand and is the exponential part. To add numbers in scientific notation, they should be expressed with the same exponent, so that the two significands can simply be added.[45]
Addition in other bases is very similar to decimal addition. As an example, one can consider addition in binary.[46] Adding two single-digit binary numbers is relatively simple, using a form of carrying:
Adding two "1" digits produces a digit "0", while 1 must be added to the next column. This is similar to what happens in decimal when certain single-digit numbers are added together; if the result equals or exceeds the value of the radix (10), the digit to the left is incremented:
This is known ascarrying.[47] When the result of an addition exceeds the value of a digit, the procedure is to "carry" the excess amount divided by the radix (that is, 10/10) to the left, adding it to the next positional value. This is correct since the next position has a weight that is higher by a factor equal to the radix. Carrying works the same way in binary:
In this example, two numerals are being added together: 011012 (1310) and 101112 (2310). The top row shows the carry bits used. Starting in the rightmost column,1 + 1 = 102. The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. The second column from the right is added:1 + 0 + 1 = 102 again; the 1 is carried, and 0 is written at the bottom. The third column:1 + 1 + 1 = 112. This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 1001002 (3610).
Analog computers work directly with physical quantities, so their addition mechanisms depend on the form of the addends. A mechanical adder might represent two addends as the positions of sliding blocks, in which case they can be added with anaveraginglever. If the addends are the rotation speeds of twoshafts, they can be added with adifferential. A hydraulic adder can add thepressures in two chambers by exploitingNewton's second law to balance forces on an assembly ofpistons. The most common situation for a general-purpose analog computer is to add twovoltages (referenced toground); this can be accomplished roughly with aresistornetwork, but a better design exploits anoperational amplifier.[48]
Addition is also fundamental to the operation ofdigital computers, where the efficiency of addition, in particular thecarry mechanism, is an important limitation to overall performance.[49]
Part of Charles Babbage'sDifference Engine including the addition and carry mechanisms
Theabacus, also called a counting frame, is a calculating tool that was in use centuries before the adoption of the written modern numeral system and is still widely used by merchants, traders and clerks inAsia,Africa, and elsewhere; it dates back to at least 2700–2300 BC, when it was used inSumer.[50]
Blaise Pascal invented the mechanical calculator in 1642;[51] it was the first operationaladding machine.Pascal's calculator was limited by its gravity-assisted carry mechanism, which forced its wheels to only turn one way so it could add. To subtract, the operator had to use thePascal's calculator's complement, which required as many steps as an addition.[52]Gottfried Leibniz built thestepped reckoner, another mechanical calculator, finished in 1694, andGiovanni Poleni improved on the design in 1709 with a calculating clock made of wood that could perform all four arithmetical operations. These early attempts were not commercially successful but inspired later mechanical calculators of the 19th century.[53]
"Full adder" logic circuit that adds two binary digits,A andB, along with a carry inputCin, producing the sum bit,S, and a carry output,Cout.
Adders execute integer addition in electronic digital computers, usually usingbinary arithmetic. The simplest architecture is the ripple carry adder, which follows the standard multi-digit algorithm. One slight improvement is thecarry skip design, again following human intuition; one does not perform all the carries in computing999 + 1, but one bypasses the group of 9s and skips to the answer.[54]
In practice, computational addition may be achieved viaXOR andAND bitwise logical operations in conjunction with bitshift operations. Both XOR and AND gates are straightforward to realize in digital logic, allowing the realization offull adder circuits, which in turn may be combined into more complex logical operations. In modern digital computers, integer addition is typically the fastest arithmetic instruction, yet it has the largest impact on performance since it underlies allfloating-point operations as well as such basic tasks asaddress generation duringmemory access and fetchinginstructions duringbranching. To increase speed, modern designs calculate digits inparallel; these schemes go by such names as carry select,carry lookahead, and theLing pseudocarry. Many implementations are, in fact, hybrids of these last three designs.[55]
Some decimal computers in the late 1950s and early 1960s used add tables instead of adders, e.g., RCA 301,[56]IBM 1620.[57]
Arithmetic implemented on a computer can deviate from the mathematical ideal in various ways. For example, if the result of an addition is too large for a computer to store, anarithmetic overflow occurs, resulting in an error message and/or an incorrect answer. Unanticipated arithmetic overflow is a fairly common cause ofprogram errors. Such overflow bugs may be hard to discover and diagnose because they may manifest themselves only for very large input data sets, which are less likely to be used in validation tests.[58] TheYear 2000 problem was a series of bugs where overflow errors occurred due to the use of a 2-digit format for years.[59]
Computers have another way of representing numbers, calledfloating-point arithmetic, which is similar to the scientific notation described above and which reduces the overflow problem. Each floating point number has two parts, an exponent and a mantissa. To add two floating-point numbers, the exponents must match, which typically means shifting the mantissa of the smaller number. If the disparity between the larger and smaller numbers is too great, a loss of precision may result. If many smaller numbers are to be added to a large number, it is best to add the smaller numbers together first and then add the total to the larger number, rather than adding small numbers to the large number one at a time. This makes floating-point addition non-associative in general.[60]
To prove the usual properties of addition, one must first define addition for the context in question. Addition is first defined on thenatural numbers. Inset theory, addition is then extended to progressively larger sets that include the natural numbers: theintegers, therational numbers, and thereal numbers.[61] Inmathematics education,[c] positive fractions are added before negative numbers are even considered; this is also the historical route.[63]
There are two popular ways to define the sum of two natural numbers and. If one defines natural numbers to be thecardinalities of finite sets (the cardinality of a set is the number of elements in the set), then it is appropriate to define their sum as follows:[64]
Let be the cardinality of a set. Take two disjoint sets and, with and. Then is defined as.
Here means theunion of and. An alternate version of this definition allows and to possibly overlap and then takes theirdisjoint union, a mechanism that allows common elements to be separated out and therefore counted twice.
Let be the successor of, that is the number following in the natural numbers, so,. Define. Define the general sum recursively by. Hence.
Again, there are minor variations upon this definition in the literature. Taken literally, the above definition is an application of therecursion theorem on thepartially ordered set.[66] On the other hand, some sources prefer to use a restricted recursion theorem that applies only to the set of natural numbers. One then considers to be temporarily "fixed", applies recursion on to define a function "", and pastes these unary operations for all together to form the full binary operation.[67]
This recursive formulation of addition was developed by Dedekind as early as 1854, and he would expand upon it in the following decades. He proved the associative and commutative properties, among others, throughmathematical induction.[68]
The simplest conception of an integer is that it consists of anabsolute value (which is a natural number) and asign (generally eitherpositive ornegative). The integer zero is a special third case, being neither positive nor negative. The corresponding definition of addition must proceed by cases:[69]
For an integer, let be its absolute value. Let and be integers. If either or is zero, treat it as an identity. If and are both positive, define. If and are both negative, define. If and have different signs, define to be the difference between and, with the sign of the term whose absolute value is larger.
As an example,−6 + 4 = −2; because −6 and 4 have different 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 be useful for concrete problems, the number of cases to consider complicates proofs unnecessarily. So the following method is commonly used for defining integers. It is based on the remark that every integer is the difference of two natural integers and that two such differences, and are equal if and only if. So, one can define formally the integers as theequivalence classes ofordered pairs of natural numbers under theequivalence relation if and only if.[70] The equivalence class of contains either if, or if otherwise. Given that is a natural number, then one can denote the equivalence class of, and by the equivalence class of. This allows identifying the natural number with the equivalence class.
The addition of ordered pairs is done component-wise:[71]A straightforward computation shows that the equivalence class of the result depends only on the equivalence classes of the summands, and thus that this defines an addition of equivalence classes, that is, integers.[72] Another straightforward computation shows that this addition is the same as the above case definition.
Addition ofrational numbers involves thefractions. The computation can be done by using theleast common denominator, but a conceptually simpler definition involves only integer addition and multiplication:As an example, the sum.[73]
Addition of fractions is much simpler when thedenominators are the same; in this case, one can simply add the numerators while leaving the denominator the same:so.[73]
The commutativity and associativity of rational addition are easy consequences of the laws of integer arithmetic.[74]
A common construction of the set of real numbers is the Dedekind completion of the set of rational numbers. A real number is defined to be aDedekind cut of rationals: anon-empty set of rationals that is closed downward and has nogreatest element. The sum of real numbersa andb is defined element by element:[75]This definition was first published, in a slightly modified form, byRichard Dedekind in 1872.[76]The commutativity and associativity of real addition are immediate; defining the real number 0 as the set of negative rationals, it is easily seen as the additive identity. Probably the trickiest part of this construction pertaining to addition is the definition of additive inverses.[77]
Adding and using Cauchy sequences of rationals.
Unfortunately, dealing with the multiplication of Dedekind cuts is a time-consuming case-by-case process similar to the addition of signed integers.[78] Another approach is the metric completion of the rational numbers. A real number is essentially defined to be the limit of aCauchy sequence of rationals, lim an. Addition is defined term by term:[79]This definition was first published byGeorg Cantor, also in 1872, although his formalism was slightly different.[80]One must prove that this operation is well-defined, dealing with co-Cauchy sequences. Once that task is done, all the properties of real addition follow immediately from the properties of rational numbers. Furthermore, the other arithmetic operations, including multiplication, have straightforward, analogous definitions.[81]
Addition of two complex numbers can be done geometrically by constructing a parallelogram.
Complex numbers are added by adding the real and imaginary parts of the summands.[82][83] That is to say:
Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbersA andB, interpreted as points of the complex plane, is the pointX obtained by building aparallelogram three of whose vertices areO,A andB.[84]
Many binary operations can be viewed as generalizations of the addition operation on the real numbers. The field of algebra is centrally concerned with such generalized operations, and they also appear inset theory andcategory theory.
Ingroup theory, aGroup is an algebraic structure that allows for composing any two elements.
In the special case where the order does not matter, the composition operator is sometimes called addition. Such groups are referred to as Abelian or commutative; the composition operator is often written as "+".
Inlinear algebra, avector space is an algebraic structure that allows for adding any twovectors and for scaling vectors. A familiar vector space is the set of all ordered pairs of real numbers; the ordered pair is interpreted as a vector from the origin in the Euclidean plane to the point in the plane. The sum of two vectors is obtained by adding their individual coordinates:This addition operation is central toclassical mechanics, in whichvelocities,accelerations andforces are all represented by vectors.[85]
Matrix addition is defined for two matrices of the same dimensions. The sum of twom ×n (pronounced "m by n") matricesA andB, denoted byA +B, is again anm ×n matrix computed by adding corresponding elements:[86][87]
For example:
Inmodular 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.[88] For example, the set of integers modulo 12 has twelve elements; it inherits an addition operation from the integers that is central tomusical set theory.[89] The set of integers modulo 2 has just two elements; the addition operation it inherits is known inBoolean logic as the "exclusive or" function.[90] A similar "wrap around" operation arises ingeometry, where the sum of twoangle measures is often taken to be their sum as real numbers modulo 2π. This amounts to an addition operation on thecircle, which in turn generalizes to the operations of higher-dimensionalLie groups.[91]
A far-reaching generalization of the addition of natural numbers is the addition ofordinal numbers andcardinal numbers in set theory. These give two different generalizations of the addition of natural numbers to thetransfinite. Unlike most addition operations, the addition of ordinal numbers is not commutative.[94] Addition of cardinal numbers, however, is a commutative operation closely related to thedisjoint union operation.[95]
Incategory theory, disjoint union is seen as a particular case of thecoproduct operation,[96] and general coproducts are perhaps the most abstract of all the generalizations of addition. The coproduct such asdirect sum is named to evoke their connection with addition.[97]
Subtraction can be thought of as a kind of addition—that is, the addition of anadditive inverse. Subtraction is itself a sort of inverse to addition, in that adding and subtracting areinverse functions.[98] Given a set with an addition operation, one cannot always define a corresponding subtraction operation on that set; the set of natural numbers is a simple example. On the other hand, a subtraction operation uniquely determines an addition operation, an additive inverse operation, and an additive identity; for this reason, an additive group can be described as a set that is closed under subtraction.[99]
Multiplication can be thought of asrepeated addition. If a single termx appears in a sum times, then the sum is theproduct of andx. Nonetheless, this works only fornatural numbers.[100] By the definition in general, multiplication is the operation between two numbers, called the multiplier and the multiplicand, that are combined into a single number called the product.[101]
A circular slide rule
In the real and complex numbers, addition and multiplication can be interchanged by theexponential function:[102]This identity allows multiplication to be carried out by consulting atable oflogarithms and computing addition by hand; it also enables multiplication on aslide rule. The formula is still a good first-order approximation in the broad context ofLie groups, where it relates multiplication of infinitesimal group elements with addition of vectors in the associatedLie algebra.[103]
There are even more generalizations of multiplication than addition.[104] In general, multiplication operations alwaysdistribute over addition; this requirement is formalized in the definition of aring. In some contexts, integers, distributivity over addition, and the existence of a multiplicative identity are enough to determine the multiplication operation uniquely. The distributive property also provides information about the addition operation; by expanding the product in both ways, one concludes that addition is forced to be commutative. For this reason, ring addition is commutative in general.[105]
Division is an arithmetic operation remotely related to addition. Since, division is right distributive over addition:.[106] However, division is not left distributive over addition, such as is not the same as.[107]
The maximum operation is a binary operation similar to addition. In fact, if two nonnegative numbers and are of differentorders of magnitude, their sum is approximately equal to their maximum. This approximation is extremely useful in the applications of mathematics, for example, in truncatingTaylor series. However, it presents a perpetual difficulty innumerical analysis, essentially since "max" is not invertible. If is much greater than, then a straightforward calculation of can accumulate an unacceptableround-off error, perhaps even returning zero. See alsoLoss of significance.[60]
The approximation becomes exact in a kind of infinite limit; if either or is an infinitecardinal number, their cardinal sum is exactly equal to the greater of the two.[d] Accordingly, there is no subtraction operation for infinite cardinals.[109]
Maximization is commutative and associative, like addition. Furthermore, since addition preserves the ordering of real numbers, addition distributes over "max" in the same way that multiplication distributes over addition:For these reasons, intropical geometry one replaces multiplication with addition and addition with maximization. In this context, addition is called "tropical multiplication", maximization is called "tropical addition", and the tropical "additive identity" isnegative infinity.[110] Some authors prefer to replace addition with minimization; then the additive identity is positive infinity.[111]
Tying these observations together, tropical addition is approximately related to regular addition through thelogarithm:which becomes more accurate as the base of the logarithm increases.[112] The approximation can be made exact by extracting a constant, named by analogy with thePlanck constant fromquantum mechanics,[113] and taking the "classical limit" as tends to zero:In this sense, the maximum operation is adequantized version of addition.[114]
^"Addend" is not a Latin word; in Latin it must be further conjugated, as innumerus addendus "the number to be added".
^For example,al-Khwarizmi performed multi-digit addition in this way from left to right.[43]
^This is according to a survey of the nations with highest TIMSS mathematics test scores.[62]
^Enderton calls this statement the "Absorption Law of Cardinal Arithmetic"; it depends on the comparability of cardinals and therefore on theAxiom of Choice.
^Enderton (1977), p.138: "...select two setsK andL with cardK = 2 and cardL = 3. Sets of fingers are handy; sets of apples are preferred by textbooks."
^Beckmann, S. (2014). The twenty-third ICMI study: primary mathematics study on whole numbers. International Journal of STEM Education, 1(1), 1–8.Chicago
^Schmidt, W., Houang, R., & Cogan, L. (2002). "A coherent curriculum".American Educator, 26(2), 1–18.
^Some authors think that "carry" may be inappropriate for education;van de Walle (2004), p. 211 calls it "obsolete and conceptually misleading", preferring the word "trade". However, "carry" remains the standard term.
^Cassidy, David; Holton, Gerald; Rutherford, James (2002). "Reviewing Units, Mathematics, and Scientific Notation".Understanding Physics. New York: Springer. p. 11.doi:10.1007/0-387-21660-X_3.ISBN978-0-387-98755-2.
^Dale R. Patrick, Stephen W. Fardo, Vigyan Chandra (2008)Electronic Digital System Fundamentals The Fairmont Press, Inc. p. 155
^P.E. Bates Bothman (1837)The common school arithmetic. Henry Benton. p. 31
^Kistermann, F. W. (1998). "Blaise Pascal's adding machine: new findings and conclusions".IEEE Annals of the History of Computing.20 (1):69–76.Bibcode:1998IAHC...20a..69K.doi:10.1109/85.646211.
^Campanile, Benedetta (2024). "La girandola di Poleni: un progetto destinato a scomparire". In Di Mauro, Marco; Romano, Luigi; Zanini, Valeria (eds.).Atti del XLIII Convegno annuale SISFA (in Italian). pp. 151–158.doi:10.6093/978-88-6887-278-6.
^Baez & Dolan (2001), p. 37 explains the historical development, in "stark contrast" with the set theory presentation: "Apparently, half an apple is easier to understand than a negative apple!"
^The verifications are carried out inEnderton (1977), p.104 and sketched for a general field of fractions over a commutative ring inDummit & Foote (1999), p. 263.
^The intuitive approach, inverting every element of a cut and taking its complement, works only for irrational numbers; seeEnderton (1977), p.117 for details.
^Schubert, E. Thomas, Phillip J. Windley, and James Alves-Foss. "Higher Order Logic Theorem Proving and Its Applications: Proceedings of the 8th International Workshop, volume 971 of."Lecture Notes in Computer Science (1995).
^Textbook constructions are usually not so cavalier with the "lim" symbol; seeBurrill (1967), p. 138 for a more careful, drawn-out development of addition with Cauchy sequences.
^Linderholm (1971), p. 49 observes, "Bymultiplication, properly speaking, a mathematician may mean practically anything. Byaddition he may mean a great variety of things, but not so great a variety as he will mean by 'multiplication'."
^Dummit & Foote (1999), p. 224. For this argument to work, one must assume that addition is a group operation and that multiplication has an identity.
^For an example of left and right distributivity, seeLoday (2002), p. 15.
Crossley, J. N.; Henry, A. S. (1990). "Thus spake al-Khwārizmī: A translation of the text of Cambridge University Library Ms. Ii.vi.5".Historia Mathematica.17 (2):103–131.doi:10.1016/0315-0860(90)90048-i.
Department of the Army (1961).Army Technical Manual TM 11-684: Principles and Applications of Mathematics for Communications-Electronics.
Isoda, Masami; Olfos, Raimundo; Noine, Takeshi (2021). "The Teaching of Multidigit Multiplication in the Japanese Approach".Teaching Multiplication with Lesson Study: Japanese and Ibero-American Theories for International Mathematics Education.doi:10.1007/978-3-030-28561-6.ISBN978-3-030-28561-6.
Johnson, Paul (1975).From Sticks and Stones: Personal Adventures in Mathematics. Science Research Associates.ISBN978-0-574-19115-1.
Kaplan, Robert (2000).The Nothing That Is: A Natural History of Zero. Oxford University Press.ISBN0-19-512842-7.
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.
Martin, John (2003).Introduction to Languages and the Theory of Computation (3rd ed.). McGraw-Hill.ISBN978-0-07-232200-2.
Weaver, J. Fred (1982). "Addition and Subtraction: A Cognitive Perspective".Addition and Subtraction: A Cognitive Perspective. Interpretations of Number Operations and Symbolic Representations of Addition and Subtraction. Taylor & Francis. p. 60.ISBN0-89859-171-6.
