Addition
Addition, usually denoted with the plus sign, is one of the four basic operations of arithmetic, the other three being subtraction, multiplication, and division. The addition of two whole numbers results in the total or sum 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, which is read as "three plus two equals five".
Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers, and complex numbers. Addition belongs to arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects such as vectors, matrices, and elements of additive groups.
Addition has several important properties. It is commutative, meaning that the order of the numbers being added does not matter, so, and it is associative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter. Repeated addition of is the same as counting. Addition of 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,, can be performed by infants as young as five months, and even some members of other animal species. In primary education, students are taught to add numbers in the decimal system, beginning with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of addition continues to this day.
Notation and terminology
Addition is written using the plus sign "+" between the terms, and the result is expressed with an equals sign. For example, reads "one plus two equals three". Nonetheless, some situations where addition is "understood", even though no symbol appears: a whole number followed immediately by a fraction indicates the sum of the two, called a mixed number, with an example, This notation can cause confusion, since in most other contexts, juxtaposition denotes multiplication instead.The numbers or the objects to be added in general addition are collectively referred to as 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 commutative property of addition, "augend" is rarely used, and both terms are generally called addends.
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. Using the gerundive suffix -nd results in "addend", "thing to be added". Likewise from augere "to increase", one gets "augend", "thing to be increased".
"Sum" and "summand" derive from the Latin noun summa "the highest" or "the top", used in Medieval Latin phrase summa linea meaning the sum of a column of numerical quantities, following the ancient Greek and Roman practice of putting the sum at the top of a column.
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.
Definition and interpretations
Addition is one of the four basic operations of arithmetic, with the other three being subtraction, multiplication, and division. This operation works by adding two or more terms. An arbitrary number of addition operations is called a summation. An infinite summation is a delicate procedure known as a series, and it can be expressed through capital sigma notation, which compactly denotes iteration of the addition operation based on the given indexes. For example,Addition is used to model many 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 basic interpretation of addition lies in combining sets, that is:This interpretation is easy to visualize, with little danger of ambiguity. It is also useful in higher mathematics. However, it is not obvious how one should extend this interpretation to include fractional or negative numbers.
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.
Extending a length
A second interpretation of addition comes from extending an initial length by a given length:The sum can be interpreted as a binary 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 the unary operation to. 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 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 change the order of the terms in a sum, but still get the same result. Symbolically, if and are any two numbers, then:The fact that addition is commutative is known as the "commutative law of addition" or "commutative property of addition". Some other binary operations are commutative too as in multiplication, but others are not as in subtraction and division.
Associativity
Addition is associative, which means that when three or more numbers are added together, the order of operations does not change the result. For any three numbers,, and, it is true that:For example,.
When addition is used together with other operations, the order of operations becomes important. In the standard order of operations, addition is a lower priority than exponentiation, nth roots, multiplication and division, but is given equal priority to subtraction.
Identity element
Adding zero to any number does not change the number. In other words, zero is the identity element for addition, and is also known as the additive identity. In symbols, for every, one has:This law was first identified in Brahmagupta's Brahmasphutasiddhanta 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. Later, other 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. 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.
Successor
Within the context of integers, addition of one also plays a special role: for any integer, the integer is the least integer greater than, also known as the successor 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, is 8, because 8 is the successor of 7, which is the successor of 6, making 8 the second successor of 6.Units
To numerically add physical quantities with units, they must be expressed with common units. 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 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. A seminal experiment by Karen Wynn in 1992 involving Mickey Mouse dolls manipulated behind a screen demonstrated that five-month-old infants expect to be 2, and they are comparatively surprised when a physical situation seems to imply that is either 1 or 3. This finding has since been affirmed by a variety of laboratories using different methodologies. Another 1992 experiment with older toddlers, between 18 and 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.Even some nonhuman animals show a limited ability to add, particularly primates. In a 1995 experiment imitating Wynn's 1992 result, rhesus macaque and cottontop tamarin monkeys 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. More recently, Asian elephants have demonstrated an ability to perform basic arithmetic.