![Addition01](https://upload.wikimedia.org/wikipedia/commons/5/5c/Addition01.svg)
Addition (usually signified by the
plus symbol
The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, resulti ...
) is one of the four basic
operations
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Man ...
of
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
, the other three being
subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
,
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
and
division. The addition of two
whole numbers results in the total amount or ''
sum
Sum most commonly means the total of two or more numbers added together; see addition.
Sum can also refer to:
Mathematics
* Sum (category theory), the generic concept of summation in mathematics
* Sum, the result of summation, the additio ...
'' of those values combined. The example in the adjacent image shows a combination of three apples and two apples, making a total of five apples. This observation is equivalent to the
mathematical expression (that is, "3 ''plus'' 2 is
equal
Equal(s) may refer to:
Mathematics
* Equality (mathematics).
* Equals sign (=), a mathematical symbol used to indicate equality.
Arts and entertainment
* ''Equals'' (film), a 2015 American science fiction film
* ''Equals'' (game), a board game
...
to 5").
Besides
counting items, addition can also be defined and executed without referring to
concrete object
In common usage and classical mechanics, a physical object or physical body (or simply an object or body) is a collection of matter within a defined contiguous boundary in three-dimensional space. The boundary must be defined and identified by t ...
s, using abstractions called
numbers instead, such as
integers,
real numbers and
complex numbers. Addition belongs to arithmetic, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 properties. It is
commutative, meaning that the order of the
operands does not matter, and it is
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
, meaning that when one adds more than two numbers, the order in which addition is performed does not matter (see ''
Summation
In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, mat ...
''). Repeated addition of is the same as counting (see ''
Successor function
In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by ''S'', so ''S''(''n'') = ''n'' +1. For example, ''S''(1) = 2 and ''S''(2) = 3. The successor functio ...
''). Addition of does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication.
Performing addition is one of the simplest numerical tasks to do. 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
Primary education or elementary education is typically the first stage of formal education, coming after preschool/kindergarten and before secondary school. Primary education takes place in ''primary schools'', ''elementary schools'', or first ...
, students are taught to add numbers in the
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient
abacus to the modern
computer
A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as C ...
, where research on the most efficient implementations of addition continues to this day.
Notation and terminology
![PlusCM128](https://upload.wikimedia.org/wikipedia/commons/f/f7/PlusCM128.svg)
Addition is written using the
plus sign
The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, result ...
"+" between the terms; that is, in
infix notation
Infix notation is the notation commonly used in arithmetical and logical formulae and statements. It is characterized by the placement of operators between operands—" infixed operators"—such as the plus sign in .
Usage
Binary relations a ...
. The result is expressed with an
equals sign. For example,
:
("one plus one equals two")
:
("two plus two equals four")
:
("one plus two equals three")
:
(see "associativity"
below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
Bottom may refer to:
Anatomy and sex
* Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
)
:
(see "multiplication"
below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
Bottom may refer to:
Anatomy and sex
* Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
)
![AdditionVertical](https://upload.wikimedia.org/wikipedia/commons/b/bd/AdditionVertical.svg)
There are also situations where addition is "understood", even though no symbol appears:
* A whole number followed immediately by a
fraction
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
indicates the sum of the two, called a ''mixed number''. For example,
This notation can cause confusion, since in most other contexts,
juxtaposition denotes
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
instead.
The sum of a
series of related numbers can be expressed through
capital sigma notation, which compactly denotes
iteration. For example,
:
Terms
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''.
[ and ] 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.
[Schwartzman p. 19]
All of the above terminology derives from
Latin. "
Addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
" and "
add
Addition (usually signified by the plus symbol ) 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 amount or '' sum'' of ...
" 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 "to give"; thus to ''add'' is to ''give to''.
Using the
gerundive suffix
In linguistics, a suffix is an affix which is placed after the stem of a word. Common examples are case endings, which indicate the grammatical case of nouns, adjectives, and verb endings, which form the conjugation of verbs. Suffixes can carry ...
''-nd'' results in "addend", "thing to be added".
["Addend" is not a Latin word; in Latin it must be further conjugated, as in ''numerus addendus'' "the number to be added".] Likewise from ''augere'' "to increase", one gets "augend", "thing to be increased".
![AdditionNombryng](https://upload.wikimedia.org/wikipedia/commons/0/01/AdditionNombryng.svg)
"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 common for the
ancient Greeks and
Romans to add upward, contrary to the modern practice of adding downward, so that a sum was literally higher than the addends.
''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.
The
plus sign
The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, result ...
"+" (
Unicode:U+002B;
ASCII:
+
) is an abbreviation of the Latin word ''et'', meaning "and". It appears in mathematical works dating back to at least 1489.
Interpretations
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
![AdditionShapes](https://upload.wikimedia.org/wikipedia/commons/a/ab/AdditionShapes.svg)
Possibly the most fundamental interpretation of addition lies in combining
sets:
* 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 below). However, it is not obvious how one should extend this version of addition to include fractional numbers or negative numbers.
One possible fix 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
![AdditionLineAlgebraic](https://upload.wikimedia.org/wikipedia/commons/6/6c/AdditionLineAlgebraic.svg)
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.
The sum ''a'' + ''b'' can be interpreted as a
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
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 play asymmetric roles, and the operation 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
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
, because each unary addition operation has an inverse unary subtraction operation, and ''vice versa''.
Properties
Commutativity
![AdditionComm01](https://upload.wikimedia.org/wikipedia/commons/f/ff/AdditionComm01.svg)
Addition is
commutative, meaning that one can change the order of the terms in a sum, but still get the same result. 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" or "commutative property of addition". Some other
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
s are commutative, such as multiplication, but many others are not, such as subtraction and division.
Associativity
![AdditionAsc](https://upload.wikimedia.org/wikipedia/commons/d/d1/AdditionAsc.svg)
Addition is
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
, which means that when three or more numbers are added together, the
order of operations
In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression.
For exampl ...
does not change the result.
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 any three numbers ''a'', ''b'', and ''c'', it is true that . For example, .
When addition is used together with other operations, the
order of operations
In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression.
For exampl ...
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
![AdditionZero](https://upload.wikimedia.org/wikipedia/commons/f/fd/AdditionZero.svg)
Adding
zero to any number, does not change the number; this means that 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
Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical trea ...
's ''
Brahmasphutasiddhanta'' in 628 AD, 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
chronology of Indian mathematicians spans from the Indus Valley civilisation and the Vedas to Modern India.
Indian mathematicians have made a number of contributions to mathematics that have significantly influenced scientists and mathematicians ...
refined the concept; around the year 830,
Mahavira
Mahavira (Sanskrit: महावीर) also known as Vardhaman, was the 24th ''tirthankara'' (supreme preacher) of Jainism. He was the spiritual successor of the 23rd ''tirthankara'' Parshvanatha. Mahavira was born in the early part of the 6t ...
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
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. I ...
also plays a special role: for any integer ''a'', the integer is the least integer greater than ''a'', also known as the
successor of ''a''. 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 ''b''th successor of ''a'', making addition iterated succession. For example, is 8, because 8 is the successor of 7, which is the successor of 6, making 8 the 2nd 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:
infant
An infant or baby is the very young offspring of human beings. ''Infant'' (from the Latin word ''infans'', meaning 'unable to speak' or 'speechless') is a formal or specialised synonym for the common term ''baby''. The terms may also be used to ...
s look longer at situations that are unexpected. A seminal experiment by
Karen Wynn in 1992 involving
Mickey Mouse
Mickey Mouse is an animated cartoon Character (arts), character co-created in 1928 by Walt Disney and Ub Iwerks. The longtime mascot of The Walt Disney Company, Mickey is an Anthropomorphism, anthropomorphic mouse who typically wears red sho ...
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 (but using
eggplant
Eggplant ( US, Canada), aubergine ( UK, Ireland) or brinjal (Indian subcontinent, Singapore, Malaysia, South Africa) is a plant species in the nightshade family Solanaceae. ''Solanum melongena'' is grown worldwide for its edible fruit.
Mos ...
s instead of dolls),
rhesus macaque and
cottontop tamarin
The cotton-top tamarin (''Saguinus oedipus'') is a small New World monkey weighing less than . This New World monkey can live up to 24 years, but most of them die by 13 years. One of the smallest primates, the cotton-top tamarin is easily recogn ...
monkeys performed similarly to human infants. More dramatically, after being taught the meanings of the
Arabic numerals
Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write Decimal, decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers ...
0 through 4, one
chimpanzee
The chimpanzee (''Pan troglodytes''), also known as simply the chimp, is a species of great ape native to the forest and savannah of tropical Africa. It has four confirmed subspecies and a fifth proposed subspecies. When its close relative th ...
was able to compute the sum of two numerals without further training. More recently,
Asian elephants have demonstrated an ability to perform basic arithmetic.
Childhood learning
Typically, children first master
counting. 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. 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 that and then reason that is one more, or 13. 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.
Different nations introduce whole numbers and arithmetic at different ages, with many countries teaching addition in pre-school. However, throughout the world, addition is taught by the end of the first year of elementary school.
Table
Children are often presented with the addition table of pairs of numbers from 0 to 9 to memorize. Knowing this, children can perform any addition.
Decimal system
The prerequisite to addition in the
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
system is the fluent recall or derivation of the 100 single-digit "addition facts". One could
memorize
Memorization is the process of committing something to memory. It is a mental process undertaken in order to store in memory for later recall visual, auditory, or tactical information.
The scientific study of memory is part of cognitive neurosc ...
all the facts by
rote
Rote can refer to:
People
*Jason Butler Rote, American TV writer
*Kyle Rote (1928–2002), American football player and father of:
*Kyle Rote, Jr. (born 1950), American soccer player
*Ryan Rote (born 1982), baseball pitcher
*Tobin Rote (1928–200 ...
, but pattern-based strategies are more enlightening and, for most people, more efficient:
[Fosnot and Dolk p. 99]
* ''Commutative property'': Mentioned above, using the pattern ''a + b = b + a'' 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.
* ''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.
* ''Doubles'': Adding a number to itself is related to counting by two and to
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
. Doubles facts form a backbone for many related facts, and students find them relatively easy to grasp.
* ''Near-doubles'': Sums such as 6 + 7 = 13 can be quickly derived from the doubles fact by adding one more, or from but subtracting one.
* ''Five and ten'': Sums of the form 5 + and 10 + are usually memorized early and can be used for deriving other facts. For example, can be derived from by adding one more.
* ''Making ten'': An advanced strategy uses 10 as an intermediate for sums involving 8 or 9; for example, .
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.
[
]
Carry
The standard algorithm for adding multidigit numbers is to align the addends vertically and add the columns, starting from the ones column on the right. If a column exceeds nine, the extra digit is " carried" into the next column. For example, in the addition
¹
27
+ 59
————
86
7 + 9 = 16, and the digit 1 is the carry.[Some authors think that "carry" may be inappropriate for education; Van de Walle (p. 211) calls it "obsolete and conceptually misleading", preferring the word "trade". However, "carry" remains the standard term.] 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 alternative methods.
Since the end of the 20th century, some US programs, including TERC, decided to remove the traditional transfer method from their curriculum. This decision was criticized, which is why some states and counties didn't support this experiment.
Decimal fractions
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.
As an example, 45.1 + 4.34 can be solved as follows:
4 5 . 1 0
+ 0 4 . 3 4
————————————
4 9 . 4 4
Scientific notation
In scientific notation, numbers are written in the form , where is the significand and is the exponential part. Addition requires two numbers in scientific notation to be represented using the same exponential part, so that the two significands can simply be added.
For example:
:
Non-decimal
Addition in other bases is very similar to decimal addition. As an example, one can consider addition in binary. Adding two single-digit binary numbers is relatively simple, using a form of carrying:
:0 + 0 → 0
:0 + 1 → 1
:1 + 0 → 1
:1 + 1 → 0, carry 1 (since 1 + 1 = 2 = 0 + (1 × 21))
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:
:5 + 5 → 0, carry 1 (since 5 + 5 = 10 = 0 + (1 × 101))
:7 + 9 → 6, carry 1 (since 7 + 9 = 16 = 6 + (1 × 101))
This is known as ''carrying''. 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:
0 1 1 0 1
+ 1 0 1 1 1
—————————————
1 0 0 1 0 0 = 36
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, . 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: again; the 1 is carried, and 0 is written at the bottom. The third column: . This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 1001002 (3610).
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 piston
A piston is a component of reciprocating engines, reciprocating pumps, gas compressors, hydraulic cylinders and pneumatic cylinders, among other similar mechanisms. It is the moving component that is contained by a cylinder and is made gas-tig ...
s. The most common situation for a general-purpose analog computer is to add two voltages (referenced to ground
Ground may refer to:
Geology
* Land, the surface of the Earth not covered by water
* Soil, a mixture of clay, sand and organic matter present on the surface of the Earth
Electricity
* Ground (electricity), the reference point in an electrical c ...
); this can be accomplished roughly with a resistor
A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
network, but a better design exploits an operational amplifier.
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.
The abacus, 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 in Asia, Africa, and elsewhere; it dates back to at least 2700–2300 BC, when it was used in Sumer
Sumer () is the earliest known civilization in the historical region of southern Mesopotamia (south-central Iraq), emerging during the Chalcolithic and early Bronze Ages between the sixth and fifth millennium BC. It is one of the cradles of c ...
.
Blaise Pascal
Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic Church, Catholic writer.
He was a child prodigy who was educated by his father, a tax collector in Rouen. Pa ...
invented the mechanical calculator in 1642;[ Jean Marguin, p. 48 (1994) ; Quoting ]René Taton
René Taton (4 April 1915 – 9 August 2004) was a French author, historian of science, and long co-editor (along with Suzanne Delorme) of the ''Revue d'histoire des sciences''.
Life
He was born on 4 April 1915 in L'Échelle.
He died on 9 ...
(1963) it was the first operational adding machine. It made use of a gravity-assisted carry mechanism. It was the only operational mechanical calculator in the 17th century and the earliest automatic, digital computer. Pascal's calculator was limited by its carry mechanism, which forced its wheels to only turn one way so it could add. To subtract, the operator had to use the Pascal's calculator's complement, which required as many steps as an addition. Giovanni Poleni followed Pascal, building the second functional mechanical calculator in 1709, a calculating clock made of wood that, once setup, could multiply two numbers automatically.
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 carry skip design, again following human intuition; one does not perform all the carries in computing , but one bypasses the group of 9s and skips to the answer.
In practice, computational addition may be achieved via XOR
Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false).
It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, , ...
and AND
or AND may refer to:
Logic, grammar, and computing
* Conjunction (grammar), connecting two words, phrases, or clauses
* Logical conjunction in mathematical logic, notated as "∧", "⋅", "&", or simple juxtaposition
* Bitwise AND, a boole ...
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 arithmetic instruction, yet it has the largest impact on performance, since it underlies all floating-point operations as well as such basic tasks as address generation during memory access and fetching instructions
Instruction or instructions may refer to:
Computing
* Instruction, one operation of a processor within a computer architecture instruction set
* Computer program, a collection of instructions
Music
* Instruction (band), a 2002 rock band from Ne ...
during branching. To increase speed, modern designs calculate digits in parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster of IBM ...
; these schemes go by such names as carry select, carry lookahead, and the Ling
Ling may refer to:
Fictional characters
* Ling, an ally of James Bond's from the film ''You Only Live Twice''
* Ling, a character in the ''Mulan'' franchise
* Ling, a playable character from the mobile game '' Mobile Legends: Bang Bang''
* Ling ...
pseudocarry. Many implementations are, in fact, hybrids of these last three designs. 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. In modern times, the ADD instruction of a microprocessor often replaces the augend with the sum but preserves the addend. In a high-level programming language, evaluating does not change either ''a'' or ''b''; if the goal is to replace ''a'' with the sum this must be explicitly requested, typically with the statement . Some languages such as C or C++ allow this to be abbreviated as .
// Iterative algorithm
int add(int x, int y)
// Recursive algorithm
int add(int x, int y)
On a computer, if the result of an addition is too large to store, an arithmetic overflow occurs, resulting in an incorrect answer. Unanticipated arithmetic overflow is a fairly common cause of program 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. The Year 2000 problem was a series of bugs where overflow errors occurred due to use of a 2-digit format for years.
Addition of numbers
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. (In mathematics education, positive fractions are added before negative numbers are even considered; this is also the historical route.)
Natural numbers
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:
* Let N(''S'') be the cardinality of a set ''S''. Take two disjoint sets ''A'' and ''B'', with and . Then is defined as .
Here, is the union of ''A'' and ''B''. An alternate version of this definition allows ''A'' and ''B'' to possibly overlap and then takes their disjoint union, a mechanism that allows common elements to be separated out and therefore counted twice.
The other popular definition is recursive:
* Let ''n''+ be the successor of ''n'', that is the number following ''n'' in the natural numbers, so 0+=1, 1+=2. 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 the recursion theorem on the partially ordered set N2. 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 ''a'' to be temporarily "fixed", applies recursion on ''b'' to define a function "''a'' +", and pastes these unary operations for all ''a'' together to form the full binary operation.
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, through mathematical induction.
Integers
The simplest conception of an integer is that it consists of an absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
(which is a natural number) and a sign
A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...
(generally either positive or negative). The integer zero is a special third case, being neither positive nor negative. The corresponding definition of addition must proceed by cases:
* For an integer ''n'', let , ''n'', be its absolute value. Let ''a'' and ''b'' be integers. If either ''a'' or ''b'' is zero, treat it as an identity. If ''a'' and ''b'' are both positive, define . If ''a'' and ''b'' are both negative, define . If ''a'' and ''b'' have different signs, define to be the difference between , ''a'', and , ''b'', , with the sign of the term whose absolute value is larger. As an example, ; 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 the equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of ordered pair
In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
s of natural numbers under the equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
: if and only if .
The equivalence class of contains either if , or otherwise. If is a natural number, one can denote the equivalence class of , and by the equivalence class of . This allows identifying the natural number with the equivalence class .
Addition of ordered pairs is done component-wise:
:
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. Another straightforward computation shows that this addition is the same as the above case definition.
This way 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 group is the additive group of integers. The rational numbers are constructed similarly, by taking as semigroup the nonzero integers with multiplication.
This construction has been also generalized under the name of Grothendieck group
In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic i ...
to the case of any commutative semigroup. Without the cancellation property the semigroup homomorphism from the semigroup into the group may be non-injective. Originally, the ''Grothendieck group'' was, more specifically, the result of this construction applied to the equivalences classes under isomorphisms of the objects of an abelian category, with the direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
as semigroup operation.
Rational numbers (fractions)
Addition of rational numbers can be computed using the least common denominator
In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the lowest common multiple of the denominators of a set of fractions. It simplifies adding, subtracting, and comparing fractions.
Description
The l ...
, but a conceptually simpler definition involves only integer addition and multiplication:
* Define
As an example, the sum .
Addition of fractions is much simpler when the denominators are the same; in this case, one can simply add the numerators while leaving the denominator the same: , so .
The commutativity and associativity of rational addition is an easy consequence of the laws of integer arithmetic. For a more rigorous and general discussion, see '' field of fractions''.
Real numbers
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 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
This definition was first published, in a slightly modified form, by Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...
in 1872.
The commutativity and associativity of real addition are immediate; defining the real number 0 to be the set of negative rationals, it is easily seen to be the additive identity. Probably the trickiest part of this construction pertaining to addition is the definition of additive inverses.
Unfortunately, dealing with multiplication of Dedekind cuts is a time-consuming case-by-case process similar to the addition of signed integers. Another approach is the metric completion of the rational numbers. A real number is essentially defined to be the limit of a Cauchy sequence of rationals, lim ''a''''n''. Addition is defined term by term:
* Define
This definition was first published by Georg Cantor, also in 1872, although his formalism was slightly different.
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.
Complex numbers
Complex numbers are added by adding the real and imaginary parts of the summands. 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 numbers ''A'' and ''B'', interpreted as points of the complex plane, is the point ''X'' obtained by building a parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...
three of whose vertices are ''O'', ''A'' and ''B''. Equivalently, ''X'' is the point such that the triangles with vertices ''O'', ''A'', ''B'', and ''X'', ''B'', ''A'', are congruent.
Generalizations
There are many binary operations that can be viewed as generalizations of the addition operation on the real numbers. The field of abstract algebra is centrally concerned with such generalized operations, and they also appear in set theory and category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
.
Abstract algebra
Vectors
In linear algebra, a vector space is an algebraic structure that allows for adding any two vectors and for scaling vectors. A familiar vector space is the set of all ordered pairs of real numbers; the ordered pair (''a'',''b'') is interpreted as a vector from the origin in the Euclidean plane to the point (''a'',''b'') in the plane. The sum of two vectors is obtained by adding their individual coordinates:
:
This addition operation is central to classical mechanics, in which velocities, accelerations and force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
s are all represented by vectors.
Matrices
Matrix addition is defined for two matrices of the same dimensions. The sum of two ''m'' × ''n'' (pronounced "m by n") matrices A and B, denoted by , is again an matrix computed by adding corresponding elements:
:
For example:
:
Modular arithmetic
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 just two elements; the addition operation it inherits is known in Boolean logic
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denote ...
as the " exclusive or" function. A similar "wrap around" operation arises in geometry, where the sum of two angle measures is often taken to be their sum as real numbers modulo 2π. This amounts to an addition operation on the circle, which in turn generalizes to addition operations on many-dimensional tori.
General theory
The general theory of abstract algebra allows an "addition" operation to be any associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
and commutative operation on a set. Basic algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
s with such an addition operation include commutative monoids and abelian groups.
Set theory and category theory
A far-reaching generalization of addition of natural numbers is the addition of ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...
s and cardinal numbers in set theory. These give two different generalizations of addition of natural numbers to the transfinite. Unlike most addition operations, addition of ordinal numbers is not commutative. Addition of cardinal numbers, however, is a commutative operation closely related to the disjoint union operation.
In category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, disjoint union is seen as a particular case of the coproduct operation, and general coproducts are perhaps the most abstract of all the generalizations of addition. Some coproducts, such as direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
and wedge sum, are named to evoke their connection with addition.
Related operations
Addition, along with subtraction, multiplication and division, is considered one of the basic operations and is used in elementary arithmetic.
Arithmetic
Subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
can be thought of as a kind of addition—that is, the addition of an additive inverse
In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
. Subtraction is itself a sort of inverse to addition, in that adding and subtracting are inverse functions.
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.
Multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
can be thought of as repeated addition. If a single term appears in a sum ''n'' times, then the sum is the product of ''n'' and . If ''n'' is not a natural number, the product may still make sense; for example, multiplication by yields the additive inverse
In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
of a number.
In the real and complex numbers, addition and multiplication can be interchanged by the exponential function:
:
This identity allows multiplication to be carried out by consulting a table of logarithms and computing addition by hand; it also enables multiplication on a slide rule. The formula is still a good first-order approximation in the broad context of Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s, where it relates multiplication of infinitesimal group elements with addition of vectors in the associated Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
.
There are even more generalizations of multiplication than addition. In general, multiplication operations always distribute over addition; this requirement is formalized in the definition of a ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
. In some contexts, such as the integers, distributivity over addition and the existence of a multiplicative identity is enough to uniquely determine the multiplication operation. The distributive property also provides information about addition; 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.
Division is an arithmetic operation remotely related to addition. Since , division is right distributive over addition: . However, division is not left distributive over addition; is not the same as .
Ordering
The maximum operation "max (''a'', ''b'')" is a binary operation similar to addition. In fact, if two nonnegative numbers ''a'' and ''b'' are of different orders of magnitude, then their sum is approximately equal to their maximum. This approximation is extremely useful in the applications of mathematics, for example in truncating Taylor series. However, it presents a perpetual difficulty in numerical analysis, essentially since "max" is not invertible. If ''b'' is much greater than ''a'', then a straightforward calculation of can accumulate an unacceptable round-off error
A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are d ...
, perhaps even returning zero. See 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 cardinal sum is exactly equal to the greater of the two. Accordingly, there is no subtraction operation for infinite cardinals.
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, in tropical 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" is negative infinity
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...
. Some authors prefer to replace addition with minimization; then the additive identity is positive infinity.
Tying these observations together, tropical addition is approximately related to regular addition through the logarithm:
:
which becomes more accurate as the base of the logarithm increases. The approximation can be made exact by extracting a constant ''h'', named by analogy with Planck's constant from quantum mechanics, and taking the " classical limit" as ''h'' tends to zero:
:
In this sense, the maximum operation is a ''dequantized'' version of addition.
Other ways to add
Incrementation, also known as the successor operation, is the addition of to a number.
Summation
In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, mat ...
describes the addition of arbitrarily many numbers, usually more than just two. It includes the idea of the sum of a single number, which is itself, and the empty sum, which is zero. An infinite summation is a delicate procedure known as a series.
Counting a finite set is equivalent to summing 1 over the set.
Integration
Integration may refer to:
Biology
*Multisensory integration
*Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome into a host genome
*DNA integration, by means of site-specific recombinase technology, ...
is a kind of "summation" over a continuum
Continuum may refer to:
* Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes
Mathematics
* Continuum (set theory), the real line or the corresponding cardinal number ...
, or more precisely and generally, over a differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
. 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, usually a real 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
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
or superposition of states in quantum mechanics.
Convolution is used to add two independent random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s defined by distribution functions. Its usual definition combines integration, subtraction, and multiplication. In general, convolution is useful as a kind of domain-side addition; by contrast, vector addition is a kind of range-side addition.
In music
Addition is also used in the musical set theory. George Perle gives the following example: “do-mi, re-fa♯, mi♭-sol are different requirements of one interval... or other type of equality... and are connected with the axis of symmetry. Do-mi belongs to the family of symmetrically connected dyads as it is shown further:”
''re'' re♯ mi fa fa♯ sol ''sol♯''
''re'' do♯ do si la♯ la ''sol♯''
Axis of the pitches are italicized, the axis is defined with the pitch category.
Thus, do-mi is a part of an interval family-4 and a part of sum family -2 (at G♯ = 0).
A tonal range of Alban Berg’s Lyric Suite is a series of six dyads with their total number being 11. If the line is turned and inverted, then it is with all dyads in total being 6.
The total number of successive dyads of a tonal range in Lyric Suite is 11
''do'' sol re re♯ la♯ ''mi♯''
''si'' mi la sol♯ do♯ ''fa♯''
Axis of the pitches are italicized, the axis is defined by the dyads (interval 1).
See also
* Lunar arithmetic
Lunar arithmetic, formerly called dismal arithmetic, is a version of arithmetic in which the addition and multiplication operations on digits are defined as the max and min operations. Thus, in lunar arithmetic,
:2+7=\max\=7 and 2\times 7 = \min ...
* Mental arithmetic
* Parallel addition (mathematics)
The parallel operator (also known as reduced sum, parallel sum or parallel addition) \, (pronounced "parallel", following the parallel lines notation from geometry) is a mathematical function which is used as a shorthand in electrical e ...
* Verbal arithmetic (also known as cryptarithms), puzzles involving addition
Notes
Footnotes
References
History
*
*
*
*
Elementary mathematics
*
Education
*
California State Board of Education mathematics content standards
Adopted December 1997, accessed December 2005.
*
*
*
Cognitive science
*
*
Mathematical exposition
*
*
*
*
*
*
*
Advanced mathematics
*
*
*
*
*
*
*
*
*
*
Mathematical research
*
*
* Litvinov, Grigory; Maslov, Victor; Sobolevskii, Andreii (1999)
Idempotent mathematics and interval analysis
Reliable Computing
', Kluwer.
*
*
*
Computing
*
*
*
*
*
*
*
Further reading
*
*
*
*
*
{{Authority control
Elementary arithmetic
Mathematical notation
Articles with example C code