Addition
Addition (often signified by the plus symbol "+") is one of the four
basic operations of arithmetic; the others are subtraction,
multiplication and division. The addition of two whole numbers is the
total amount of those quantities combined. For example, in the
adjacent picture, there is a combination of three apples and two
apples together, making a total of five apples. This observation is
equivalent to the mathematical expression "3 + 2 = 5" i.e., "3 add 2
is equal to 5".
Besides counting items, addition can also be defined on other types of
numbers, such as integers, real numbers and complex numbers. This is
part of arithmetic, a branch of mathematics. In algebra, another area
of mathematics, addition can be performed on abstract objects such as
vectors and matrices.
Addition
Addition has several important properties. It is commutative, meaning
that order does not matter, and it is associative, meaning that when
one adds more than two numbers, the order in which addition is
performed does not matter (see Summation). Repeated addition of 1 is
the same as counting; addition of 0 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. 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. In primary education, students
are taught to add numbers in the decimal system, starting 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.
Contents
1 Notation and terminology
2 Interpretations
2.1 Combining sets
2.2 Extending a length
3 Properties
3.1 Commutativity
3.2 Associativity
3.3 Identity element
3.4 Successor
3.5 Units
4 Performing addition
4.1 Innate ability
4.2 Learning addition as children
4.2.1
Addition
Addition table
4.3
Decimal
Decimal system
4.3.1 Carry
4.3.2
Addition
Addition of decimal fractions
4.3.3 Scientific notation
4.4
Addition
Addition in other bases
4.5 Computers
5
Addition
Addition of numbers
5.1 Natural numbers
5.2 Integers
5.3 Rational numbers (fractions)
5.4 Real numbers
5.5 Complex numbers
6 Generalizations
6.1
Addition
Addition in abstract algebra
6.1.1 Vector addition
6.1.2 Matrix addition
6.1.3 Modular arithmetic
6.1.4 General addition
6.2
Addition
Addition in set theory and category theory
7 Related operations
7.1 Arithmetic
7.2 Ordering
7.3 Other ways to add
8 Notes
9 Footnotes
10 References
11 Further reading
Notation and terminology[edit]
The plus sign
Addition
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,
1
+
1
=
2
displaystyle 1+1=2
("one plus one equals two")
2
+
2
=
4
displaystyle 2+2=4
("two plus two equals four")
3
+
3
=
6
displaystyle 3+3=6
("three plus three equals six")
5
+
4
+
2
=
11
displaystyle 5+4+2=11
(see "associativity" below)
3
+
3
+
3
+
3
=
12
displaystyle 3+3+3+3=12
(see "multiplication" below)
Columnar addition – the numbers in the column are to be added, with
the sum written below the underlined number.
There are also 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.[2] For example,
3½ = 3 + ½ = 3.5.
This notation can cause confusion since in most other contexts
juxtaposition denotes multiplication instead.[3]
The sum of a series of related numbers can be expressed through
capital sigma notation, which compactly denotes iteration. For
example,
∑
k
=
1
5
k
2
=
1
2
+
2
2
+
3
2
+
4
2
+
5
2
=
55.
displaystyle sum _ k=1 ^ 5 k^ 2 =1^ 2 +2^ 2 +3^ 2 +4^ 2 +5^ 2
=55.
The numbers or the objects to be added in general addition are
collectively referred to as the terms,[4] the addends[5][6][7] or the
summands;[8] 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.[5][6][7] 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.[9]
All of the above terminology derives from Latin. "Addition" and "add"
are English words derived from the
Latin
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.[9] Using the
gerundive suffix -nd results in "addend", "thing to be added".[a]
Likewise from augere "to increase", one gets "augend", "thing to be
increased".
Redrawn illustration from The Art of Nombryng, one of the first
English arithmetic texts, in the 15th century.[10]
"Sum" and "summand" derive from the
Latin
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.[11] Addere and summare date back at least to
Boethius, if not to earlier Roman writers such as
Vitruvius
Vitruvius and
Frontinus; Boethius also used several other terms for the addition
operation. The later
Middle English
Middle English terms "adden" and "adding" were
popularized by Chaucer.[12]
The plus sign "+" (Unicode:U+002B; ASCII: +) is an
abbreviation of the
Latin
Latin word et, meaning "and".[13] It appears in
mathematical works dating back to at least 1489.[14]
Interpretations[edit]
Addition
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[edit]
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 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.[15]
One possible fix is to consider collections of objects that can be
easily divided, such as pies or, still better, segmented rods.[16]
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[edit]
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.
A number-line visualization of the unary addition 2 + 4 = 6. A
translation by 4 is equivalent to four translations by 1.
A second interpretation of addition comes from extending an initial
length by a given length:
When an original length is extended by a given amount, the final
length is the sum of the original length and the length of the
extension.[17]
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.[18] 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[edit]
Commutativity[edit]
4 + 2 = 2 + 4 with blocks
Addition
Addition is commutative: one can change the order of the terms in a
sum, and the result is 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". Some other binary operations are commutative, such as
multiplication, but many others are not, such as subtraction and
division.
Associativity[edit]
2 + (1 + 3) = (2 + 1) + 3 with segmented rods
Addition
Addition is associative: when adding three or more numbers, the order
of operations does not matter.
As an example, 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).
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.[19]
Identity element[edit]
5 + 0 = 5 with bags of dots
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 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
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.[20]
Successor[edit]
Within 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.[21] For instance, 3
is the successor of 2 and 7 is the successor of 6. Because of this
succession, the value of a + b can also be seen as the bth successor
of a, making addition iterated succession. For examples, 6 + 2 is 8,
because 8 is the successor of 7, which is the successor of 6, making 8
the 2nd successor of 6.
Units[edit]
To numerically add physical quantities with units, they must be
expressed with common units.[22] 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[edit]
Innate ability[edit]
Studies on mathematical development starting around the 1980s have
exploited the phenomenon of habituation: infants look longer at
situations that are unexpected.[23] A seminal experiment by Karen Wynn
in 1992 involving
Mickey Mouse
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.[24] 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.[25]
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 macaque and cottontop tamarin
monkeys performed similarly to human infants. More dramatically, after
being taught the meanings of the
Arabic numerals
Arabic numerals 0 through 4, one
chimpanzee was able to compute the sum of two numerals without further
training.[26] More recently, Asian elephants have demonstrated an
ability to perform basic arithmetic.[27]
Learning addition as children[edit]
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.[28]
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 6 + 6 = 12 and then reason
that 6 + 7 is one more, or 13.[29] 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.[30]
Different nations introduce whole numbers and arithmetic at different
ages, with many countries teaching addition in pre-school.[31]
However, throughout the world, addition is taught by the end of the
first year of elementary school.[32]
Addition
Addition table[edit]
Children are often presented with the addition table of pairs of
numbers from 1 to 10 to memorize. Knowing this, one can perform any
addition.
Addition
Addition table
Addition
Addition table of 1
1
+
0
=
1
1
+
1
=
2
1
+
2
=
3
1
+
3
=
4
1
+
4
=
5
1
+
5
=
6
1
+
6
=
7
1
+
7
=
8
1
+
8
=
9
1
+
9
=
10
1
+
10
=
11
Addition
Addition table of 2
2
+
0
=
2
2
+
1
=
3
2
+
2
=
4
2
+
3
=
5
2
+
4
=
6
2
+
5
=
7
2
+
6
=
8
2
+
7
=
9
2
+
8
=
10
2
+
9
=
11
2
+
10
=
12
Addition
Addition table of 3
3
+
0
=
3
3
+
1
=
4
3
+
2
=
5
3
+
3
=
6
3
+
4
=
7
3
+
5
=
8
3
+
6
=
9
3
+
7
=
10
3
+
8
=
11
3
+
9
=
12
3
+
10
=
13
Addition
Addition table of 4
4
+
0
=
4
4
+
1
=
5
4
+
2
=
6
4
+
3
=
7
4
+
4
=
8
4
+
5
=
9
4
+
6
=
10
4
+
7
=
11
4
+
8
=
12
4
+
9
=
13
4
+
10
=
14
Addition
Addition table of 5
5
+
0
=
5
5
+
1
=
6
5
+
2
=
7
5
+
3
=
8
5
+
4
=
9
5
+
5
=
10
5
+
6
=
11
5
+
7
=
12
5
+
8
=
13
5
+
9
=
14
5
+
10
=
15
Addition
Addition table of 6
6
+
0
=
6
6
+
1
=
7
6
+
2
=
8
6
+
3
=
9
6
+
4
=
10
6
+
5
=
11
6
+
6
=
12
6
+
7
=
13
6
+
8
=
14
6
+
9
=
15
6
+
10
=
16
Addition
Addition table of 7
7
+
0
=
7
7
+
1
=
8
7
+
2
=
9
7
+
3
=
10
7
+
4
=
11
7
+
5
=
12
7
+
6
=
13
7
+
7
=
14
7
+
8
=
15
7
+
9
=
16
7
+
10
=
17
Addition
Addition table of 8
8
+
0
=
8
8
+
1
=
9
8
+
2
=
10
8
+
3
=
11
8
+
4
=
12
8
+
5
=
13
8
+
6
=
14
8
+
7
=
15
8
+
8
=
16
8
+
9
=
17
8
+
10
=
18
Addition
Addition table of 9
9
+
0
=
9
9
+
1
=
10
9
+
2
=
11
9
+
3
=
12
9
+
4
=
13
9
+
5
=
14
9
+
6
=
15
9
+
7
=
16
9
+
8
=
17
9
+
9
=
18
9
+
10
=
19
Addition
Addition table of 10
10
+
0
=
10
10
+
1
=
11
10
+
2
=
12
10
+
3
=
13
10
+
4
=
14
10
+
5
=
15
10
+
6
=
16
10
+
7
=
17
10
+
8
=
18
10
+
9
=
19
10
+
10
=
20
Decimal
Decimal system[edit]
The prerequisite to addition in the decimal system is the fluent
recall or derivation 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:[33]
Commutative
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.[33]
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.[33]
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 students find them relatively easy to grasp.[33]
Near-doubles: Sums such as 6 + 7 = 13 can be quickly derived from the
doubles fact 6 + 6 = 12 by adding one more, or from 7 + 7 = 14 but
subtracting one.[33]
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 from 5 + 7 = 12 by adding one more.[33]
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.[33]
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.[30]
Carry[edit]
Main article: Carry (arithmetic)
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 27 + 59
¹
27
+ 59
————
86
7 + 9 = 16, and the digit 1 is the carry.[b] 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.
Addition
Addition of decimal fractions[edit]
Decimal
Decimal fractions can be added by a simple modification of the above
process.[34] 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[edit]
Main article:
Scientific notation
Scientific notation § Basic operations
In scientific notation, numbers are written in the form
x
=
a
×
10
b
displaystyle x=atimes 10^ b
, where
a
displaystyle a
is the significand and
10
b
displaystyle 10^ b
is the exponential part.
Addition
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:
2.34
×
10
−
5
+
5.67
×
10
−
6
=
2.34
×
10
−
5
+
0.567
×
10
−
5
=
2.907
×
10
−
5
displaystyle 2.34times 10^ -5 +5.67times 10^ -6 =2.34times 10^ -5
+0.567times 10^ -5 =2.907times 10^ -5
Addition
Addition in other bases[edit]
Main article: Binary addition
Addition
Addition in other bases is very similar to decimal addition. As an
example, one can consider addition in binary.[35] 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.[36] 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:
1 1 1 1 1 (carried digits)
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, 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).
Computers[edit]
Addition
Addition with an op-amp. See Summing amplifier for details.
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.[37]
Addition
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
Difference Engine including the addition and
carry mechanisms.
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.[38]
Blaise Pascal
Blaise Pascal invented the mechanical calculator in 1642;[39] 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[40] and the earliest
automatic, digital computer.
Pascal's calculator
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
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.
"Full adder" logic circuit that adds two binary digits, A and B, along
with a carry input Cin, producing the sum bit, S, and a carry output,
Cout.
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 999 + 1,
but one bypasses the group of 9s and skips to the answer.[41]
In practice, computational addition may be achieved via XOR and AND
bitwise logical operations in conjunction with bitshift operations as
shown in the pseudocode below. Both XOR and AND gates are
straightforward to realize in digital logic allowing the realization
of full adder circuits which in turn may be combined into more complex
logical operations. In modern digital computers, integer addition is
typically the fastest 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 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. Many implementations are, in fact, hybrids of these last
three designs.[42][43] 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
Latin texts often claimed that in the process of adding "a number to a
number", both numbers vanish.[44] In modern times, the ADD instruction
of a microprocessor often replaces the augend with the sum but
preserves the addend.[45] In a high-level programming language,
evaluating a + b 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 a = a + b. Some languages such as C or
C++
C++ allow
this to be abbreviated as a += b.
// Iterative Algorithm
int add(int x, int y)
int carry = 0;
while (y != 0)
carry = AND(x, y); // Logical AND
x = XOR(x, y); // Logical XOR
y = carry << 1; // left bitshift carry by one
return x;
// Recursive Algorithm
int add(int x, int y)
return x if (y == 0) else add(XOR(x, y) , AND(x, y) << 1);
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.[46] The
Year 2000 problem
Year 2000 problem was a series of bugs where overflow errors occurred
due to use of a 2-digit format for years.[47]
Addition
Addition of numbers[edit]
To prove the usual properties of addition, one must first define
addition for the context in question.
Addition
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.[48] (In
mathematics education,[49] positive fractions are added before
negative numbers are even considered; this is also the historical
route.[50])
Natural numbers[edit]
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:
Let N(S) be the cardinality of a set S. Take two disjoint sets A and
B, with N(A) = a and N(B) = b. Then a + b is defined as
N
(
A
∪
B
)
displaystyle N(Acup B)
.[51]
Here, A ∪ B 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 a + 0 = a. Define the general
sum recursively by a + (b+) = (a + b)+. Hence 1 + 1 = 1 + 0+ = (1 +
0)+ = 1+ = 2.[52]
Again, there are minor variations upon this definition in the
literature. Taken literally, the above definition is an application of
the
Recursion
Recursion Theorem on the partially ordered set N2.[53] On the
other hand, some sources prefer to use a restricted
Recursion
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.[54]
This recursive formulation of addition was developed by Dedekind as
early as 1854, and he would expand upon it in the following
decades.[55] He proved the associative and commutative properties,
among others, through mathematical induction.
Integers[edit]
Defining (−2) + 1 using only addition of positive numbers: (2 − 4)
+ (3 − 2) = 5 − 6.
Further information: Integer
The simplest conception of an integer is that it consists of an
absolute value (which is a natural number) and a sign (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 a + b = a + b. If a and b are both
negative, define a + b = −(a + b). If a and b have different
signs, define a + b to be the difference between a and b, with the
sign of the term whose absolute value is larger.[56] 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, it is
far too complicated to produce elegant general proofs; there are too
many cases to consider.
A much more convenient conception of the integers is the Grothendieck
group construction. The essential observation is that every integer
can be expressed (not uniquely) as the difference of two natural
numbers, so we may as well define an integer as the difference of two
natural numbers.
Addition
Addition is then defined to be compatible with
subtraction:
Given two integers a − b and c − d, where a, b, c, and d are
natural numbers, define (a − b) + (c − d) = (a + c) − (b +
d).[57]
Rational numbers (fractions)[edit]
Addition
Addition of rational numbers can be computed using the least common
denominator, but a conceptually simpler definition involves only
integer addition and multiplication:
Define
a
b
+
c
d
=
a
d
+
b
c
b
d
.
displaystyle frac a b + frac c d = frac ad+bc bd .
As an example, the sum
3
4
+
1
8
=
3
×
8
+
4
×
1
4
×
8
=
24
+
4
32
=
28
32
=
7
8
displaystyle frac 3 4 + frac 1 8 = frac 3times 8+4times 1
4times 8 = frac 24+4 32 = frac 28 32 = frac 7 8
.
Addition
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:
a
c
+
b
c
=
a
+
b
c
displaystyle frac a c + frac b c = frac a+b c
, so
1
4
+
2
4
=
1
+
2
4
=
3
4
displaystyle frac 1 4 + frac 2 4 = frac 1+2 4 = frac 3
4
.[58]
The commutativity and associativity of rational addition is an easy
consequence of the laws of integer arithmetic.[59] For a more rigorous
and general discussion, see field of fractions.
Real numbers[edit]
Adding π2/6 and e using Dedekind cuts of rationals.
Further information: Construction of the 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
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
a
+
b
=
q
+
r
∣
q
∈
a
,
r
∈
b
.
displaystyle a+b= q+rmid qin a,rin b .
[60]
This definition was first published, in a slightly modified form, by
Richard Dedekind
Richard Dedekind in 1872.[61] 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.[62]
Adding π2/6 and e using Cauchy sequences of rationals.
Unfortunately, dealing with multiplication of Dedekind cuts is a
time-consuming case-by-case process similar to the addition of signed
integers.[63] Another approach is the metric completion of the
rational numbers. A real number is essentially defined to be the limit
of a
Cauchy sequence
Cauchy sequence of rationals, lim an.
Addition
Addition is defined
term by term:
Define
lim
n
a
n
+
lim
n
b
n
=
lim
n
(
a
n
+
b
n
)
.
displaystyle lim _ n a_ n +lim _ n b_ n =lim _ n (a_ n +b_ n ).
[64]
This definition was first published by Georg Cantor, also in 1872,
although his formalism was slightly different.[65] 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.[66]
Complex numbers[edit]
Addition
Addition of two complex numbers can be done geometrically by
constructing a parallelogram.
Complex numbers
Complex numbers are added by adding the real and imaginary parts of
the summands.[67][68] That is to say:
(
a
+
b
i
)
+
(
c
+
d
i
)
=
(
a
+
c
)
+
(
b
+
d
)
i
.
displaystyle (a+bi)+(c+di)=(a+c)+(b+d)i.
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 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[edit]
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.
Addition
Addition in abstract algebra[edit]
Vector addition[edit]
Main article: Vector addition
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:
(a,b) + (c,d) = (a+c,b+d).
This addition operation is central to classical mechanics, in which
vectors are interpreted as forces.
Matrix addition[edit]
Main article: Matrix addition
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 A + B, is again an m × n matrix computed by adding corresponding
elements:[69][70]
A
+
B
=
[
a
11
a
12
⋯
a
1
n
a
21
a
22
⋯
a
2
n
⋮
⋮
⋱
⋮
a
m
1
a
m
2
⋯
a
m
n
]
+
[
b
11
b
12
⋯
b
1
n
b
21
b
22
⋯
b
2
n
⋮
⋮
⋱
⋮
b
m
1
b
m
2
⋯
b
m
n
]
=
[
a
11
+
b
11
a
12
+
b
12
⋯
a
1
n
+
b
1
n
a
21
+
b
21
a
22
+
b
22
⋯
a
2
n
+
b
2
n
⋮
⋮
⋱
⋮
a
m
1
+
b
m
1
a
m
2
+
b
m
2
⋯
a
m
n
+
b
m
n
]
displaystyle begin aligned mathbf A + mathbf B &=
begin bmatrix a_ 11 &a_ 12 &cdots &a_ 1n \a_ 21 &a_ 22
&cdots &a_ 2n \vdots &vdots &ddots &vdots \a_ m1
&a_ m2 &cdots &a_ mn \end bmatrix + begin bmatrix b_ 11
&b_ 12 &cdots &b_ 1n \b_ 21 &b_ 22 &cdots &b_
2n \vdots &vdots &ddots &vdots \b_ m1 &b_ m2
&cdots &b_ mn \end bmatrix \&= begin bmatrix a_ 11 +b_ 11
&a_ 12 +b_ 12 &cdots &a_ 1n +b_ 1n \a_ 21 +b_ 21 &a_
22 +b_ 22 &cdots &a_ 2n +b_ 2n \vdots &vdots &ddots
&vdots \a_ m1 +b_ m1 &a_ m2 +b_ m2 &cdots &a_ mn +b_
mn \end bmatrix \end aligned
For example:
[
1
3
1
0
1
2
]
+
[
0
0
7
5
2
1
]
=
[
1
+
0
3
+
0
1
+
7
0
+
5
1
+
2
2
+
1
]
=
[
1
3
8
5
3
3
]
displaystyle begin bmatrix 1&3\1&0\1&2end bmatrix +
begin bmatrix 0&0\7&5\2&1end bmatrix = begin bmatrix
1+0&3+0\1+7&0+5\1+2&2+1end bmatrix = begin bmatrix
1&3\8&5\3&3end bmatrix
Modular arithmetic[edit]
Main article: Modular arithmetic
In modular arithmetic, 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
Boolean logic as the "exclusive or" function. In geometry, 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 addition[edit]
The general theory of abstract algebra allows an "addition" operation
to be any associative and commutative operation on a set. Basic
algebraic structures with such an addition operation include
commutative monoids and abelian groups.
Addition
Addition in set theory and category theory[edit]
A far-reaching generalization of addition of natural numbers is the
addition of ordinal numbers 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
Addition of cardinal numbers, however, is
a commutative operation closely related to the disjoint union
operation.
In category theory, 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 and Wedge sum, are named to evoke their connection with
addition.
Related operations[edit]
Addition, along with subtraction, multiplication and division, is
considered one of the basic operations and is used in elementary
arithmetic.
Arithmetic[edit]
Subtraction
Subtraction can be thought of as a kind of addition—that is, the
addition of an additive inverse.
Subtraction
Subtraction is itself a sort of
inverse to addition, in that adding x and subtracting x 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.[71]
Multiplication
Multiplication can be thought of as repeated addition. If a single
term x appears in a sum n times, then the sum is the product of n and
x. If n is not a natural number, the product may still make sense; for
example, multiplication by −1 yields the additive inverse of a
number.
A circular slide rule
In the real and complex numbers, addition and multiplication can be
interchanged by the exponential function:
ea + b = ea eb.[72]
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 groups, where it
relates multiplication of infinitesimal group elements with addition
of vectors in the associated Lie algebra.[73]
There are even more generalizations of multiplication than
addition.[74] In general, multiplication operations always distribute
over addition; this requirement is formalized in the definition of a
ring. 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 (1 + 1)(a + b) in both ways, one concludes that addition is
forced to be commutative. For this reason, ring addition is
commutative in general.[75]
Division is an arithmetic operation remotely related to addition.
Since a/b = a(b−1), division is right distributive over addition: (a
+ b) / c = a/c + b/c.[76] However, division is not left distributive
over addition; 1 / (2 + 2) is not the same as 1/2 + 1/2.
Ordering[edit]
Log-log plot
Log-log plot of x + 1 and max (x, 1) from x = 0.001 to 1000[77]
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 (a + b) − b can accumulate an
unacceptable round-off error, 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.[78] Accordingly, there is no
subtraction operation for infinite cardinals.[79]
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:
a + max (b, c) = max (a + b, a + c).
For these reasons, in tropical geometry one replaces multiplication
with addition and addition with maximization. In this context,
addition is called "tropical multiplication", maximization is called
"tropical addition", and the tropical "additive identity" is negative
infinity.[80] Some authors prefer to replace addition with
minimization; then the additive identity is positive infinity.[81]
Tying these observations together, tropical addition is approximately
related to regular addition through the logarithm:
log (a + b) ≈ max (log a, log b),
which becomes more accurate as the base of the logarithm
increases.[82] The approximation can be made exact by extracting a
constant h, named by analogy with
Planck's constant
Planck's constant from quantum
mechanics,[83] and taking the "classical limit" as h tends to zero:
max
(
a
,
b
)
=
lim
h
→
0
h
log
(
e
a
/
h
+
e
b
/
h
)
.
displaystyle max(a,b)=lim _ hto 0 hlog(e^ a/h +e^ b/h ).
In this sense, the maximum operation is a dequantized version of
addition.[84]
Other ways to add[edit]
Incrementation, also known as the successor operation, is the addition
of 1 to a number.
Summation 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.[85] An
infinite summation is a delicate procedure known as a series.[86]
Counting
Counting a finite set is equivalent to summing 1 over the set.
Integration is a kind of "summation" over a continuum, or more
precisely and generally, over a differentiable manifold. Integration
over a zero-dimensional manifold reduces to summation.
Linear combinations combine multiplication and summation; they are
sums in which each term has a multiplier, 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 or superposition of states in
quantum mechanics.
Convolution
Convolution is used to add two independent random variables 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.
Notes[edit]
^ "Addend" is not a
Latin
Latin word; in
Latin
Latin it must be further
conjugated, as in numerus addendus "the number to be added".
^ 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.
Footnotes[edit]
^ From Enderton (p.138): "...select two sets K and L with card K = 2
and card L = 3. Sets of fingers are handy; sets of apples are
preferred by textbooks."
^ Devine et al. p.263
^ Mazur, Joseph. Enlightening Symbols: A Short History of Mathematical
Notation and Its Hidden Powers. Princeton University Press, 2014. p.
161
^ Department of the Army (1961) Army Technical Manual TM 11-684:
Principles and Applications of Mathematics for
Communications-Electronics. Section 5.1
^ a b Shmerko, V. P.; Yanushkevich [Ânuškevič], Svetlana N.
[Svitlana N.]; Lyshevski, S. E. (2009).
Computer
Computer arithmetics for
nanoelectronics. CRC Press. p. 80.
^ a b Schmid, Hermann (1974).
Decimal
Decimal Computation (1 ed.). Binghamton,
New York, USA: John Wiley & Sons. ISBN 0-471-76180-X.
^ a b Schmid, Hermann (1983) [1974].
Decimal
Decimal Computation (1 (reprint)
ed.). Malabar, Florida, USA: Robert E. Krieger Publishing Company.
ISBN 0-89874-318-4.
^ Hosch, W. L. (Ed.). (2010). The Britannica Guide to Numbers and
Measurement. The Rosen Publishing Group. p.38
^ a b Schwartzman p.19
^ Karpinski pp.56–57, reproduced on p.104
^ Schwartzman (p.212) attributes adding upwards to the Greeks and
Romans, saying it was about as common as adding downwards. On the
other hand, Karpinski (p.103) writes that
Leonard of Pisa
Leonard of Pisa "introduces
the novelty of writing the sum above the addends"; it is unclear
whether Karpinski is claiming this as an original invention or simply
the introduction of the practice to Europe.
^ Karpinski pp.150–153
^ Cajori, Florian (1928). "Origin and meanings of the signs + and -".
A History of Mathematical Notations, Vol. 1. The Open Court Company,
Publishers.
^ "plus".
Oxford English Dictionary
Oxford English Dictionary (3rd ed.). Oxford University
Press. September 2005. (Subscription or UK public library
membership required.)
^ See Viro 2001 for an example of the sophistication involved in
adding with sets of "fractional cardinality".
^ Adding it up (p.73) compares adding measuring rods to adding sets of
cats: "For example, inches can be subdivided into parts, which are
hard to tell from the wholes, except that they are shorter; whereas it
is painful to cats to divide them into parts, and it seriously changes
their nature."
^ Mosley, F. (2001). Using number lines with 5-8 year olds. Nelson
Thornes. p.8
^ Li, Y., & Lappan, G. (2014). Mathematics curriculum in school
education. Springer. p. 204
^ "Order of Operations Lessons". Algebra.Help. Retrieved 5 March
2012.
^ Kaplan pp.69–71
^ Hempel, C. G. (2001). The philosophy of Carl G. Hempel: studies in
science, explanation, and rationality. p. 7
^ R. Fierro (2012) Mathematics for Elementary School Teachers. Cengage
Learning. Sec 2.3
^ Wynn p.5
^ Wynn p.15
^ Wynn p.17
^ Wynn p.19
^ Randerson, James (21 August 2008). "Elephants have a head for
figures". The Guardian. Retrieved 29 March 2015.
^ F. Smith p.130
^ Carpenter, Thomas; Fennema, Elizabeth; Franke, Megan Loef; Levi,
Linda; Empson, Susan (1999). Children's mathematics: Cognitively
guided instruction. Portsmouth, NH: Heinemann.
ISBN 0-325-00137-5.
^ a b Henry, Valerie J.; Brown, Richard S. (2008). "First-grade basic
facts: An investigation into teaching and learning of an accelerated,
high-demand memorization standard". Journal for Research in
Mathematics Education. 39 (2): 153–183. doi:10.2307/30034895.
^ 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.
^ a b c d e f g Fosnot and Dolk p. 99
^ Rebecca Wingard-Nelson (2014) Decimals and Fractions: It's Easy
Enslow Publishers, Inc.
^ 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
^ Truitt and Rogers pp.1;44–49 and pp.2;77–78
^ Ifrah, Georges (2001). The Universal History of Computing: From the
Abacus
Abacus to the Quantum Computer. New York, NY: John Wiley & Sons,
Inc. ISBN 978-0471396710. p.11
^ Jean Marguin, p. 48 (1994) ; Quoting René Taton (1963)
^ See Competing designs in
Pascal's calculator
Pascal's calculator article
^ Flynn and Overman pp.2, 8
^ Flynn and Overman pp.1–9
^ Yeo, Sang-Soo, et al., eds. Algorithms and Architectures for
Parallel Processing: 10th International Conference, ICA3PP 2010,
Busan, Korea, May 21–23, 2010. Proceedings. Vol. 1. Springer, 2010.
p. 194
^ Karpinski pp.102–103
^ The identity of the augend and addend varies with architecture. For
ADD in x86 see Horowitz and Hill p.679; for ADD in
68k
68k see p.767.
^ Joshua Bloch, "Extra, Extra - Read All About It: Nearly All Binary
Searches and Mergesorts are Broken". Official Google Research Blog,
June 2, 2006.
^ "The Risks Digest Volume 4: Issue 45". The Risks Digest.
^ Enderton chapters 4 and 5, for example, follow this development.
^ According to a survey of the nations with highest TIMSS mathematics
test scores; see Schmidt, W., Houang, R., & Cogan, L. (2002). A
coherent curriculum. American educator, 26(2), p. 4.
^ Baez (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!"
^ Begle p.49, Johnson p.120, Devine et al. p.75
^ Enderton p.79
^ For a version that applies to any poset with the descending chain
condition, see Bergman p.100.
^ Enderton (p.79) observes, "But we want one binary operation +, not
all these little one-place functions."
^ Ferreirós p.223
^ K. Smith p.234, Sparks and Rees p.66
^ Enderton p.92
^ Schyrlet Cameron, and Carolyn Craig (2013)Adding and Subtracting
Fractions, Grades 5 - 8 Mark Twain, Inc.
^ The verifications are carried out in Enderton p.104 and sketched for
a general field of fractions over a commutative ring in Dummit and
Foote p.263.
^ Enderton p.114
^ Ferreirós p.135; see section 6 of Stetigkeit und irrationale Zahlen
Archived 2005-10-31 at the Wayback Machine..
^ The intuitive approach, inverting every element of a cut and taking
its complement, works only for irrational numbers; see Enderton 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
Computer Science (1995).
^ Textbook constructions are usually not so cavalier with the "lim"
symbol; see Burrill (p. 138) for a more careful, drawn-out development
of addition with Cauchy sequences.
^ Ferreirós p.128
^ Burrill p.140
^ Conway, John B. (1986), Functions of One Complex Variable I,
Springer, ISBN 0-387-90328-3
^ Joshi, Kapil D. (1989), Foundations of Discrete Mathematics, New
York: John Wiley & Sons, ISBN 978-0-470-21152-6
^ Lipschutz, S., & Lipson, M. (2001). Schaum's outline of theory
and problems of linear algebra. Erlangga.
^ Riley, K.F.; Hobson, M.P.; Bence, S.J. (2010). Mathematical methods
for physics and engineering. Cambridge University Press.
ISBN 978-0-521-86153-3.
^ The set still must be nonempty. Dummit and Foote (p.48) discuss this
criterion written multiplicatively.
^ Rudin p.178
^ Lee p.526, Proposition 20.9
^ Linderholm (p.49) observes, "By multiplication, properly speaking, a
mathematician may mean practically anything. By addition he may mean a
great variety of things, but not so great a variety as he will mean by
'multiplication'."
^ Dummit and Foote p.224. For this argument to work, one still must
assume that addition is a group operation and that multiplication has
an identity.
^ For an example of left and right distributivity, see Loday,
especially p.15.
^ Compare Viro Figure 1 (p.2)
^ Enderton calls this statement the "Absorption Law of Cardinal
Arithmetic"; it depends on the comparability of cardinals and
therefore on the Axiom of Choice.
^ Enderton p.164
^ Mikhalkin p.1
^ Akian et al. p.4
^ Mikhalkin p.2
^ Litvinov et al. p.3
^ Viro p.4
^ Martin p.49
^ Stewart p.8
References[edit]
History
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Karpinski, Louis (1925). The History of Arithmetic. Rand McNally.
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Sparks, F.; Rees C. (1979). A Survey of Basic Mathematics.
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Advanced mathematics
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Constructions (2.3 ed.). General Printing.
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Dummit, D.; Foote, R. (1999). Abstract
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Algebra (2 ed.). Wiley.
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Lee, John (2003). Introduction to Smooth Manifolds. Springer.
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Idempotent mathematics and interval analysis. Reliable Computing,
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(in French). Presses universitaires de France. pp. 20–28.
Further reading[edit]
Baroody, Arthur; Tiilikainen, Sirpa (2003). The Development of
Arithmetic
Arithmetic Concepts and Skills. Two perspectives on addition
development. Routledge. p. 75. ISBN 0-8058-3155-X.
Davison, David M.; Landau, Marsha S.; McCracken, Leah; Thompson, Linda
(1999). Mathematics: Explorations & Applications (TE ed.).
Prentice Hall. ISBN 0-13-435817-1.
Bunt, Lucas N. H.; Jones, Phillip S.; Bedient, Jack D. (1976). The
Historical roots of Elementary Mathematics. Prentice-Hall.
ISBN 0-13-389015-5.
Kaplan, Robert (2000). The Nothing That Is: A Natural History of Zero.
Oxford UP. ISBN 0-19-512842-7.
Poonen, Bjorn (2010). "Addition". Girls'
Angle
Angle Bulletin. Girls' Angle.
3 (3-5). ISSN 2151-5743.
Weaver, J. Fred (1982).
Addition
Addition and Subtraction: A Cognitive
Perspective. Interpretations of Number Operations and Symbolic
Representations of
Addition
Addition and Subtraction. Taylor & Francis.
p. 60. ISBN 0-89859-171-6.
Williams, Michael (1985). A History of Computing Technology.
Prentice-Hall. ISBN 0-13-389917-9.
v
t
e
Elementary arithmetic
Addition
Addition (+)
Subtraction
Subtraction (−)
Multiplication
Multiplication (× or ·)