This article is about addition in mathematics. For addition reaction in chemistry, see addition reaction. Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. Addition also provides a model for related processes such as joining two collections of objects into one collection. Repeated addition of the number one is the most basic form of counting.
Performing addition is one of the simplest numerical tasks, accessible to infants as young as five months and even some animals.
Notation and terminology
Addition is written using the plus sign "+" between the terms; that is, in infix notation. The result is expressed with an equals sign. For example,
- Failed to parse (Missing texvc executable; please see math/README to configure.): 1 + 1 = 2
(verbally, "one plus one equals two")
- Failed to parse (Missing texvc executable; please see math/README to configure.): 2 + 2 = 4
- Failed to parse (Missing texvc executable; please see math/README to configure.): 5 + 4 + 2 = 11
(see "associativity" below)
- Failed to parse (Missing texvc executable; please see math/README to configure.): 3 + 3 + 3 + 3 = 12
(see "multiplication" below)
There are also situations where addition is "understood" even though no symbol appears:
- A column of numbers, with the last number in the column underlined, usually indicates that the numbers in the column are to be added, with the sum written below the underlined number.
- A whole number followed immediately by a fraction indicates the sum of the two, called a mixed number.[2] For example,
3½ = 3 + ½ = 3.5.
This notation can cause confusion since in most other contexts juxtaposition denotes multiplication instead.
The numbers or the objects to be added are generally called the "terms", the "addends", or the "summands"; this terminology carries over to the summation of multiple terms. This is to be distinguished from factors, which are multiplied. Some authors call the first addend the augend. In fact, during the Renaissance, many authors did not consider the first addend an "addend" at all. Today, due to the symmetry of addition, "augend" is rarely used, and both terms are generally called addends.[3]
All of this 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 Indo-European root do- "to give"; thus to add is to give to.[4] Using the gerundive suffix -nd results in "addend", "thing to be added".[5] Likewise from augere "to increase", one gets "augend", "thing to be increased".
"Sum" and "summand" derive from the Latin noun summa "the highest, the top" and associated verb summare. This is appropriate not only because the sum of two positive numbers is greater than either, but because it was once common to add upward, contrary to the modern practice of adding downward, so that a sum was literally higher than the addends.[7]
Addere and summare date back at least to Boethius, if not to earlier Roman writers such as Vitruvius and Frontinus; Boethius also used several other terms for the addition operation. The later Middle English terms "adden" and "adding" were popularized by Chaucer.[8]
Interpretations
Addition is used to model countless physical processes. Even for the simple case of adding natural numbers, there are many possible interpretations and even more visual representations.
Combining sets
Possibly the most fundamental interpretation of addition lies in combining sets:
- When two or more collections are combined into a single collection, the number of objects in the single collection is the sum of the number 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 version of addition to include fractional numbers or negative numbers.[9]
One possible fix is to consider collections of objects that can be easily divided, such as pies or, still better, segmented rods.[10] Rather than just 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.
Extending a length
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 final length is the sum of the original length and the length of the extension.
Image:AdditionLineAlgebraic.svg
A number-line visualization of the algebraic addition 2 + 4 = 6. A translation by 2 followed by a translation by 4 is the same as a translation by 6.
Image:AdditionLineUnary.svg
A number-line visualization of the unary addition 2 + 4 = 6. A translation by 4 is equivalent to four translations by 1.
The sum a + b can be interpreted as a binary operation that combines a and b, in an algebraic sense, or it can be interpreted as the addition of b more units to a. Under 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. 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 operation. and vice versa.
Properties
Commutativity
Addition is commutative, meaning that one can reverse the terms in a sum left-to-right, and the result will be the same. Symbolically, if a and b are any two numbers, then
- a + b = b + a.
The fact that addition is commutative is known as the "commutative law of addition". This phrase suggests that there are other commutative laws: for example, there is a commutative law of multiplication. However, many binary operations are not commutative, such as subtraction and division, so it is misleading to speak of an unqualified "commutative law".
Associativity
A somewhat subtler property of addition is associativity, which comes up when one tries to define repeated addition. Should the expression
- "a + b + c"
be defined to mean (a + b) + c or a + (b + c)? That addition is associative tells us that the choice of definition is irrelevant. For any three numbers a, b, and c, it is true that
- (a + b) + c = a + (b + c).
For example, (1 + 2) + 3 = 3 + 3 = 6 = 1 + 5 = 1 + (2 + 3). Not all operations are associative, so in expressions with other operations like subtraction, it is important to specify the order of operations.
Zero and one
When adding zero to any number, the quantity does not change; zero is the identity element for addition, also known as the additive identity. In symbols, for any a,
- a + 0 = 0 + a = a.
This law was first identified in Brahmagupta's Brahmasphutasiddhanta in 628, 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 as what is added to it", corresponding to the unary statement 0 + a = a. 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 a + 0 = a.[11]
In the context of integers, addition of one also plays a special role: for any integer a, the integer (a + 1) is the least integer greater than a, also known as the successor of a. Because of this succession, the value of some a + b can also be seen as the Failed to parse (Missing texvc executable; please see math/README to configure.): b^{th}
successor of a, making addition iterated succession.
Units
In order to numerically add physical quantities with units, they must first be expressed with common unit. For example, 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 in dimensional analysis.
Performing addition
Innate ability
Studies on mathematical development starting around the 1980s have exploited the phenomenon of habituation: infants look longer at situations that are unexpected.[12] A seminal experiment by Karen Wynn in 1992 involving Mickey Mouse dolls manipulated behind a screen demonstrated that five-month-old infants expect 1 + 1 to be 2, and they are comparatively surprised when a physical situation seems to imply that 1 + 1 is either 1 or 3. This finding has since been affirmed by a variety of laboratories using different methodologies.[13] Another 1992 experiment with older toddlers, between 18 to 35 months, exploited their development of motor control by allowing them to retrieve ping-pong balls from a box; the youngest responded well for small numbers, while older subjects were able to compute sums up to 5.[14]
Even some nonhuman animals show a limited ability to add, particularly primates. In a 1995 experiment imitating Wynn's 1992 result (but using eggplants instead of dolls), rhesus macaques and cottontop tamarins performed similarly to human infants. More 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 training.[15]
Elementary methods
Typically children master the art of counting first, and this skill extends into a form of addition called "counting-on"; asked to find three plus two, children count two past three, saying "four, five", and arriving at five. This strategy seems almost universal; children can easily pick it up from peers or teachers, and some even invent it independently.[16] Those who count to add also quickly learn to exploit the commutativity of addition by counting up from the larger number.
Decimal system
Image:AdditionTable.svg
Single-digit addition table with various strategies colored: 0 in blue; 1,2 in light blue; (near) doubles in (light) green; making ten in red; 5,10 in gray. [17]
The prerequisitive to addition in the decimal system is the internalization of the 100 single-digit "addition facts". One could memorize all the facts by rote, but pattern-based strategies are more enlightening and, for most people, more efficient:[18]
- One or two more: Adding 1 or 2 is a basic task, and it can be accomplished through counting on or, ultimately, intuition.
- Zero: Since zero is the additive identity, adding zero is trivial. Nonetheless, some children are introduced to addition as a process that always increases the addends; word problems may help rationalize the "exception" of zero.
- Doubles: Adding a number to itself is related to counting by two and to multiplication. Doubles facts form a backbone for many related facts, and fortunately, children find them relatively easy to grasp. near-doubles...
- Five and ten...
- 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.
In traditional mathematics, to add multidigit numbers, one typically aligns the addends vertically and adds the columns, starting from the ones column on the right. If a column exceeds ten, the extra digit is "carried" into the next column.[19] For a more detailed description of this algorithm, see Elementary arithmetic: Addition. 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. There are many different standards-based mathematics methods, but many mathematics curricula such as TERC omit any instruction in traditional methods familiar to parents or mathematics professionals in favor of exploration of new methods.
Computers
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 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 the pressures in two chambers by exploiting Newton's second law to balance forces on an assembly of pistons. The most common situation for a general-purpose analog computer is to add two voltages (referenced to ground); this can be accomplished roughly with a resistor network, but a better design exploits an operational amplifier.[20]
Addition is also fundamental to the operation of digital computers, where the efficiency of addition, in particular the carry mechanism, is an important limitation to overall performance.
Part of Charles Babbage's Difference Engine including the addition and carry mechanisms
Adding machines, mechanical calculators whose primary function was addition, were the earliest automatic, digital computers. Wilhelm Schickard's 1623 Calculating Clock could add and subtract, but it was severely limited by an awkward carry mechanism. As he wrote to Johannes Kepler describing the novel device, "You would burst out laughing if you were present to see how it carries by itself from one column of tens to the next..." Adding 999,999 and 1 on Schickard's machine would require enough force to propagate the carries that the gears might be damaged, so he limited his machines to six digits, even though Kepler's work required more. By 1642 Blaise Pascal independently developed an adding machine with an ingenious gravity-assisted carry mechanism. Pascal's calculator was limited by its carry mechanism in a different sense: its wheels turned only one way, so it could add but not subtract, except by the method of complements. By 1674 Gottfried Leibniz made the first mechanical multiplier; it was still powered, if not motivated, by addition.[21]
" Full adder" logic circuit that adds two binary digits, A and B, along with a carry input Ci, producing the sum bit, S, and a carry output, Co.
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 taught to children. One slight improvement is the carry skip design, again following human intuition; one does not perform all the carries in computing 999 + 1, but one bypasses the group of 9s and skips to the answer.[22]
Since they compute digits one at a time, the above methods are too slow for most modern purposes. In modern digital computers, integer addition is typically the fastest arithmetic instruction, yet it has the largest impact on performance, since it underlies all the floating-point operations as well as such basic tasks as address generation during memory access and fetching instructions during branching. To increase speed, modern designs calculate digits in parallel; these schemes go by such names as carry select, carry lookahead, and the Ling pseudocarry. Almost all modern implementations are, in fact, hybrids of these last three designs.[23]
Unlike addition on paper, addition on a computer often changes the addends. On the ancient abacus and adding board, both addends are destroyed, leaving only the sum. The influence of the abacus on mathematical thinking was strong enough that early Latin texts often claimed that in the process of adding "a number to a number", both numbers vanish.[24] In modern times, the ADD instruction of a microprocessor replaces the augend with the sum but preserves the addend.[25] In a high-level programming language, evaluating a + b does not change either a or b; to change the value of a one uses the addition assignment operator a += b.
Addition of natural and real numbers
In order to prove the usual properties of addition, one must first define addition for the context in question. Addition is first defined 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.[26] (In mathematics education,[27] positive fractions are added before negative numbers are even considered; this is also the historical route.[28])
Natural numbers
- Further information: Natural number
There are two popular ways to define the sum of two natural numbers a and b. If one defines natural numbers to be the cardinalities 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:
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