Mark Twain Media Inc Publishers Chapter One Review of Number Systems Answers

Arithmetics performance

3 + 2 = 5 with apples, a popular choice in textbooks^[i]

Improver (commonly signified past the plus symbol +) is one of the 4 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 those values combined. The instance in the adjacent epitome shows a combination of three apples and 2 apples, making a total of five apples. This observation is equivalent to the mathematical expression "3 + 2 = v" (that is, "3 plus ii is equal to 5").

Too counting items, addition tin also be divers and executed without referring to concrete objects, using abstractions chosen numbers instead, such as integers, real numbers and complex numbers. Add-on belongs to arithmetic, a branch of mathematics. In algebra, another expanse of mathematics, improver tin can likewise be performed on abstract objects such equally vectors, matrices, subspaces and subgroups.

Addition has several important properties. Information technology is commutative, meaning that gild does not matter, and it is associative, pregnant that when ane adds more than 2 numbers, the society in which addition is performed does not matter (see Summation). Repeated add-on of 1 is the same as counting. Addition of 0 does not alter a number. Add-on also obeys predictable rules concerning related operations such equally subtraction and multiplication.

Performing addition is one of the simplest numerical tasks. Addition of very small numbers is attainable to toddlers; the nigh basic chore, 1 + 1, can be performed past infants equally young as five months, and even some members of other animal species. In chief teaching, students are taught to add together numbers in the decimal organization, starting with single digits and progressively tackling more hard bug. Mechanical aids range from the ancient abacus to the modern computer, where research on the about efficient implementations of addition continues to this twenty-four hour period.

Notation and terminology [edit]

Addition is written using the plus sign "+" between the terms;^[2] that is, in infix notation. The result is expressed with an equals sign. For example,

${\displaystyle 1+1=two}$ ("one plus one equals two")

${\displaystyle 2+ii=4}$ ("two plus ii equals four")

${\displaystyle one+two=iii}$ ("i plus two equals 3")

${\displaystyle 5+4+2=11}$ (run into "associativity" beneath)

${\displaystyle 3+3+3+3=12}$ (see "multiplication" beneath)

Columnar improver – the numbers in the column are to be added, with the sum written beneath the underlined number.

In that location 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 ii, called a mixed number.^[3] For example,

  $${\displaystyle 3{\frac {1}{2}}=3+{\frac {1}{ii}}=3.5.}$$

  

  This notation tin crusade confusion, since in most other contexts, juxtaposition denotes multiplication instead.^[4]

The sum of a series of related numbers tin can be expressed through majuscule sigma notation, which compactly denotes iteration. For example,

${\displaystyle \sum _{one thousand=1}^{five}k^{ii}=1^{2}+2^{ii}+iii^{2}+4^{ii}+5^{2}=55.}$

The numbers or the objects to exist added in general addition are collectively referred to every bit the terms,^[5] the addends ^{[half dozen] [7] [8]} or the summands;^[nine] this terminology carries over to the summation of multiple terms. This is to be distinguished from factors, which are multiplied. Some authors call the starting time addend the augend.^{[vi] [7] [eight]} In fact, during the Renaissance, many authors did not consider the offset addend an "addend" at all. Today, due to the commutative property of addition, "augend" is rarely used, and both terms are by and large called addends.^[x]

All of the above terminology derives from Latin. "Addition" and "add" are English words derived from the Latin verb addere, which is in turn a compound of advertizing "to" and cartel "to requite", from the Proto-Indo-European root *deh₃- "to give"; thus to add is to give to.^[10] Using the gerundive suffix -nd results in "addend", "thing to be added".^[a] Likewise from augere "to increase", i gets "augend", "thing to be increased".

Redrawn illustration from The Art of Nombryng, one of the first English arithmetic texts, in the 15th century.^[11]

"Sum" and "summand" derive from the Latin noun summa "the highest, the top" and associated verb summare. This is appropriate not just because the sum of two positive numbers is greater than either, simply because information technology was common for the ancient Greeks and Romans to add upward, opposite to the modernistic do of calculation downward, and then that a sum was literally higher than the addends.^[12] Addere and summare appointment back at least to Boethius, if not to earlier Roman writers such equally Vitruvius and Frontinus; Boethius also used several other terms for the add-on functioning. The later Middle English terms "adden" and "adding" were popularized by Chaucer.^[13]

The plus sign "+" (Unicode:U+002B; ASCII: +) is an abbreviation of the Latin word et, meaning "and".^[14] It appears in mathematical works dating back to at least 1489.^[15]

Interpretations [edit]

Add-on is used to model many physical processes. Fifty-fifty for the simple case of calculation natural numbers, in that location are many possible interpretations and even more visual representations.

Combining sets [edit]

Possibly the most central interpretation of add-on lies in combining sets:

• When two or more disjoint collections are combined into a single collection, the number of objects in the unmarried collection is the sum of the numbers of objects in the original collections.

This estimation is easy to visualize, with little danger of ambiguity. It is besides useful in higher mathematics (for the rigorous definition information technology inspires, meet § Natural numbers below). However, it is not obvious how one should extend this version of addition to include fractional numbers or negative numbers.^[16]

1 possible ready is to consider collections of objects that can be easily divided, such every bit pies or, still better, segmented rods.^[17] Rather than solely combining collections of segments, rods tin be joined cease-to-cease, which illustrates another conception of addition: calculation non 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 two + 4 = 6. A translation by 4 is equivalent to four translations by ane.

A second estimation of add-on comes from extending an initial length by a given length:

• When an original length is extended by a given corporeality, the final length is the sum of the original length and the length of the extension.^[18]

The sum a + b tin can exist 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. Nether the latter estimation, the parts of a sum a + b play asymmetric roles, and the operation a + b is viewed as applying the unary performance +b to a.^[nineteen] Instead of calling both a and b addends, it is more appropriate to call a the augend in this example, since a plays a passive role. The unary view is as well useful when discussing subtraction, considering each unary improver operation has an changed unary subtraction operation, and vice versa.

Backdrop [edit]

Commutativity [edit]

4 + 2 = 2 + 4 with blocks

Improver is commutative, pregnant that one tin can change the lodge of the terms in a sum, just even so 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 equally the "commutative police of add-on" or "commutative belongings of improver". Some other binary operations are commutative, such as multiplication, but many others are non, such as subtraction and sectionalization.

Associativity [edit]

ii + (ane + 3) = (two + 1) + 3 with segmented rods

Addition is associative, which means that when three or more than numbers are added together, the social club of operations 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 option of definition is irrelevant. For whatever three numbers a, b, and c, it is true that (a + b) + c = a + (b + c). For case, (1 + ii) + 3 = three + three = 6 = 1 + 5 = 1 + (2 + 3).

When addition is used together with other operations, the social club of operations becomes important. In the standard club of operations, addition is a lower priority than exponentiation, nth roots, multiplication and division, just is given equal priority to subtraction.^[20]

Identity chemical element [edit]

v + 0 = 5 with bags of dots

Adding zero to any number, does non change the number; this means that nix is the identity chemical element for addition, and is as well known as the additive identity. In symbols, for every a , one has

a + 0 = 0 + a = a .

This police was first identified in Brahmagupta's Brahmasphutasiddhanta in 628 Advertising, although he wrote it equally three split up laws, depending on whether a is negative, positive, or zilch itself, and he used words rather than algebraic symbols. Later on Indian mathematicians refined the concept; around the year 830, Mahavira wrote, "zero becomes the same as what is added to information technology", 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 aforementioned", respective to the unary statement a + 0 = a .^[21]

Successor [edit]

Within the context of integers, addition of ane also plays a special role: for any integer a, the integer (a + i) is the to the lowest degree integer greater than a, also known as the successor of a.^[22] For instance, 3 is the successor of two and 7 is the successor of six. Considering of this succession, the value of a + b tin also be seen equally the bth successor of a, making addition iterated succession. For example, 6 + 2 is 8, because 8 is the successor of 7, which is the successor of vi, making viii the second successor of six.

Units [edit]

To numerically add concrete quantities with units, they must exist expressed with mutual units.^[23] For instance, adding 50 milliliters to 150 milliliters gives 200 milliliters. However, if a measure of 5 feet is extended past 2 inches, the sum is 62 inches, since 60 inches is synonymous with five anxiety. On the other hand, it is ordinarily meaningless to try to add 3 meters and four foursquare meters, since those units are incomparable; this sort of consideration is fundamental in dimensional analysis.^[24]

Performing addition [edit]

Innate ability [edit]

Studies on mathematical development starting effectually the 1980s have exploited the miracle of habituation: infants look longer at situations that are unexpected.^[25] A seminal experiment past Karen Wynn in 1992 involving Mickey Mouse dolls manipulated behind a screen demonstrated that five-month-old infants look 1 + 1 to exist ii, and they are insufficiently surprised when a physical situation seems to imply that ane + 1 is either 1 or 3. This finding has since been affirmed by a variety of laboratories using different methodologies.^[26] Another 1992 experiment with older toddlers, between eighteen and 35 months, exploited their development of motor control by assuasive them to retrieve ping-pong balls from a box; the youngest responded well for small-scale numbers, while older subjects were able to compute sums upwards to five.^[27]

Fifty-fifty some nonhuman animals show a express power to add, particularly primates. In a 1995 experiment imitating Wynn's 1992 consequence (but using eggplants instead of dolls), rhesus macaque and cottontop tamarin monkeys performed similarly to homo infants. More dramatically, afterward existence taught the meanings of the Arabic numerals 0 through 4, one chimpanzee was able to compute the sum of two numerals without farther training.^[28] More recently, Asian elephants take demonstrated an ability to perform basic arithmetic.^[29]

Childhood learning [edit]

Typically, children start master counting. When given a problem that requires that two items and 3 items be combined, immature children model the state of affairs with physical objects, often fingers or a drawing, and so count the total. As they proceeds experience, they learn or find 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 v. This strategy seems nearly universal; children tin easily pick it upward from peers or teachers.^[thirty] About discover it independently. With additional experience, children learn to add more than quickly by exploiting the commutativity of add-on by counting up from the larger number, in this case, starting with three and counting "iv, five." Eventually children begin to think sure addition facts ("number bonds"), either through experience or rote memorization. One time some facts are committed to memory, children begin to derive unknown facts from known ones. For instance, a child asked to add together six and seven may know that 6 + six = 12 and so reason that 6 + 7 is one more, or xiii.^[31] Such derived facts tin be found very quickly and almost elementary school students eventually rely on a mixture of memorized and derived facts to add together fluently.^[32]

Different nations introduce whole numbers and arithmetics at different ages, with many countries education addition in pre-school.^[33] Still, throughout the earth, addition is taught by the end of the get-go year of elementary school.^[34]

Table [edit]

Children are oft presented with the addition tabular array of pairs of numbers from 0 to 9 to memorize. Knowing this, children can perform any addition.

+	0	1	2	3	4	5	six	7	8	9
0	0	ane	2	three	4	five	6	seven	8	ix
1	1	2	3	iv	5	6	7	8	9	10
two	2	iii	4	5	6	7	8	9	10	11
three	3	4	5	vi	7	8	9	10	11	12
4	four	v	6	7	8	9	10	11	12	xiii
five	5	half dozen	7	eight	ix	10	eleven	12	13	14
6	6	seven	8	9	10	xi	12	xiii	14	15
7	7	8	9	10	11	12	13	14	15	16
viii	eight	9	10	11	12	thirteen	xiv	15	16	17
9	9	10	xi	12	xiii	14	fifteen	xvi	17	eighteen

Decimal system [edit]

The prerequisite to addition in the decimal system is the fluent remember or derivation of the 100 single-digit "addition facts". One could memorize all the facts by rote, only pattern-based strategies are more enlightening and, for most people, more efficient:^[35]

• Commutative property: Mentioned above, using the pattern a + b = b + a reduces the number of "addition facts" from 100 to 55.
• I or two more: Adding 1 or two is a basic task, and it tin can be accomplished through counting on or, ultimately, intuition.^[35]
• Zero: Since zero is the condiment identity, adding zero is little. Nevertheless, in the education of arithmetic, some students are introduced to addition every bit a process that always increases the addends; word problems may assistance rationalize the "exception" of zero.^[35]
• Doubles: Adding a number to itself is related to counting past two and to multiplication. Doubles facts class a backbone for many related facts, and students observe them relatively easy to grasp.^[35]
• Near-doubles: Sums such as 6 + 7 = 13 can be quickly derived from the doubles fact 6 + 6 = 12 by calculation 1 more, or from 7 + vii = 14 merely subtracting one.^[35]
• Five and ten: Sums of the form five + 10 and ten + x are normally memorized early and tin can exist used for deriving other facts. For instance, 6 + vii = 13 can exist derived from 5 + vii = 12 by adding ane more.^[35]
• Making ten: An advanced strategy uses 10 as an intermediate for sums involving 8 or ix; for example, eight + 6 = 8 + 2 + 4 = 10 + 4 = fourteen.^[35]

As students grow older, they commit more facts to retentivity, and learn to derive other facts rapidly and fluently. Many students never commit all the facts to memory, but can still observe any basic fact quickly.^[32]

Acquit [edit]

The standard algorithm for adding multidigit numbers is to marshal the addends vertically and add the columns, starting from the ones cavalcade on the right. If a column exceeds 9, the actress digit is "carried" into the next column. For example, in the addition 27 + 59

          ¹   27 + 59 ————   86        

7 + ix = 16, and the digit 1 is the behave.^[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 gauge of the sum. At that place are many alternative methods.

Decimal fractions [edit]

Decimal fractions can be added by a elementary modification of the above process.^[36] One aligns two decimal fractions above each other, with the decimal indicate 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.

Equally an example, 45.1 + 4.34 can be solved as follows:

          4 5 . 1 0 +  0 4 . 3 4 ————————————    four 9 . iv four        

Scientific notation [edit]

In scientific notation, numbers are written in the form ${\displaystyle ten=a\times 10^{b}}$ , where ${\displaystyle a}$ is the significand and ${\displaystyle ten^{b}}$ is the exponential part. Addition requires two numbers in scientific annotation to exist represented using the same exponential part, so that the two significands can simply exist added.

For case:

${\displaystyle 2.34\times ten^{-5}+5.67\times ten^{-6}=2.34\times x^{-v}+0.567\times 10^{-5}=ii.907\times 10^{-v}}$

Not-decimal [edit]

Addition in other bases is very like to decimal addition. Every bit an example, one can consider improver in binary.^[37] Calculation ii single-digit binary numbers is relatively simple, using a form of conveying:

0 + 0 → 0

0 + 1 → 1

1 + 0 → 1

1 + 1 → 0, bear 1 (since one + ane = 2 = 0 + (ane × twoⁱ))

Adding ii "ane" digits produces a digit "0", while 1 must exist added to the adjacent column. This is like to what happens in decimal when certain unmarried-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 + five → 0, conduct i (since 5 + 5 = x = 0 + (one × 10^one))

vii + nine → 6, carry one (since 7 + 9 = sixteen = 6 + (1 × 10ⁱ))

This is known as carrying.^[38] When the effect 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 information technology to the next positional value. This is right since the next position has a weight that is higher past a factor equal to the radix. Carrying works the same mode in binary:

          ane i 1 1 i    (carried digits)          0 1 1 0 1 +   ane 0 i 1 1 —————————————   ane 0 0 1 0 0 = 36        

In this example, ii numerals are being added together: 01101₂ (13₁₀) and 10111₂ (23₁₀). The top row shows the carry bits used. Starting in the rightmost column, one + 1 = x₂ . The 1 is carried to the left, and the 0 is written at the bottom of the rightmost cavalcade. The second column from the correct is added: 1 + 0 + 1 = ten₂ again; the 1 is carried, and 0 is written at the lesser. The third column: i + ane + i = 11₂ . This time, a ane is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 100100₂ (36₁₀).

Computers [edit]

Analog computers work direct with concrete quantities, so their improver mechanisms depend on the class of the addends. A mechanical adder might represent ii addends as the positions of sliding blocks, in which example they can exist 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 constabulary to remainder forces on an assembly of pistons. The most mutual state of affairs for a general-purpose analog reckoner is to add together two voltages (referenced to ground); this tin can be achieved roughly with a resistor network, but a better blueprint exploits an operational amplifier.^[39]

Addition is also primal to the functioning of digital computers, where the efficiency of addition, in particular the carry mechanism, is an important limitation to overall operation.

Part of Charles Babbage's Difference Engine including the addition and acquit mechanisms

The abacus, besides chosen a counting frame, is a calculating tool that was in use centuries before the adoption of the written modern numeral organization and is still widely used past merchants, traders and clerks in Asia, Africa, and elsewhere; information technology dates back to at least 2700–2300 BC, when it was used in Sumer.^[xl]

Blaise Pascal invented the mechanical calculator in 1642;^[41] it was the first operational calculation auto. It fabricated use of a gravity-assisted acquit mechanism. Information technology was the only operational mechanical computer in the 17th century^[42] and the earliest automated, digital computer. Pascal's estimator was limited by its acquit mechanism, which forced its wheels to only plough one way so it could add. To subtract, the operator had to use the Pascal's calculator'due south complement, which required every bit many steps equally an add-on. Giovanni Poleni followed Pascal, building the second functional mechanical computer in 1709, a calculating clock made of wood that, once setup, could multiply two numbers automatically.

"Total adder" logic circuit that adds 2 binary digits, A and B, along with a behave input C_in , producing the sum bit, S, and a behave output, C_out .

Adders execute integer addition in electronic digital computers, commonly using binary arithmetic. The simplest architecture is the ripple carry adder, which follows the standard multi-digit algorithm. One slight improvement is the bear skip design, over again post-obit human being intuition; i does not perform all the carries in computing 999 + one, but one bypasses the group of 9s and skips to the answer.^[43]

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 teaching, yet it has the largest bear upon on operation, since it underlies all floating-point operations likewise 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 get past such names as bear select, acquit lookahead, and the Ling pseudocarry. Many implementations are, in fact, hybrids of these last three designs.^{[44] [45]} Unlike improver on paper, addition on a computer often changes the addends. On the ancient abacus and adding board, both addends are destroyed, leaving but the sum. The influence of the abacus on mathematical thinking was stiff plenty that early Latin texts oft claimed that in the process of adding "a number to a number", both numbers vanish.^[46] In modern times, the Add together instruction of a microprocessor often replaces the augend with the sum but preserves the addend.^[47] In a high-level programming language, evaluating a + b does non change either a or b; if the goal is to supplant a with the sum this must be explicitly requested, typically with the statement a = a + b . Some languages such as C or C++ allow this to exist abbreviated as a += b .

                        // Iterative algorithm            int                                    add            (            int                                    x            ,                                    int                                    y            )                                    {                                                int                                    behave                                    =                                    0            ;                                                while                                    (            y                                    !=                                    0            )                                    {                                                            carry                                    =                                    AND            (            10            ,                                    y            );                                    // Logical AND                                    x                                    =                                    XOR            (            10            ,                                    y            );                                    // Logical XOR                                    y                                    =                                    acquit                                    <<                                    1            ;                                    // left bitshift conduct past one                                    }                                                return                                    x            ;                                    }                        // Recursive algorithm            int                                    add together            (            int                                    x            ,                                    int                                    y            )                                    {                                                render                                    x                                    if                                    (            y                                    ==                                    0            )                                    else                                    add together            (            XOR            (            ten            ,                                    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 reply. Unanticipated arithmetics overflow is a fairly common cause of program errors. Such overflow bugs may exist difficult to discover and diagnose because they may manifest themselves simply for very large input information sets, which are less likely to exist used in validation tests.^[48] The Year 2000 problem was a serial of bugs where overflow errors occurred due to employ of a 2-digit format for years.^[49]

Addition of numbers [edit]

To prove the usual properties of addition, one must first define improver for the context in question. Improver is first defined on the natural numbers. In ready theory, addition is and then extended to progressively larger sets that include the natural numbers: the integers, the rational numbers, and the real numbers.^[fifty] (In mathematics education,^[51] positive fractions are added before negative numbers are even considered; this is also the historical route.^[52])

Natural numbers [edit]

In that location are two popular ways to ascertain 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 fix), then it is advisable to define their sum as follows:

• Let Due north(S) exist 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 divers as ${\displaystyle N(A\cup B)}$ .^[53]

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 mutual elements to be separated out and therefore counted twice.

The other popular definition is recursive:

• Let northward ⁺ be the successor of n, that is the number following northward in the natural numbers, so 0⁺=1, ane⁺=2. Define a + 0 = a . Ascertain the general sum recursively by a + (b ⁺) = (a + b)⁺ . Hence 1 + ane = 1 + 0⁺ = (1 + 0)⁺ = one⁺ = 2.^[54]

Again, at that place 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 N ².^[55] On the other hand, some sources adopt to use a restricted recursion theorem that applies but to the set of natural numbers. One then considers a to exist temporarily "fixed", applies recursion on b to define a role "a +", and pastes these unary operations for all a together to form the full binary operation.^[56]

This recursive conception of add-on was developed past Dedekind every bit early on as 1854, and he would expand upon it in the following decades.^[57] He proved the associative and commutative properties, among others, through mathematical induction.

Integers [edit]

The simplest conception of an integer is that information technology consists of an absolute value (which is a natural number) and a sign (generally either positive or negative). The integer zero is a special tertiary example, being neither positive nor negative. The corresponding definition of addition must proceed by cases:

• For an integer north, let |n| be its accented value. Permit a and b be integers. If either a or b is zero, treat it as an identity. If a and b are both positive, ascertain 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 divergence betwixt |a| and |b|, with the sign of the term whose absolute value is larger.^[58] As an case, −6 + 4 = −two; considering −6 and iv accept dissimilar signs, their absolute values are subtracted, and since the absolute value of the negative term is larger, the answer is negative.

Although this definition tin be useful for physical bug, the number of cases to consider complicates proofs unnecessarily. Then 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, a – b and c – d are equal if and only if a + d = b + c . So, one tin define formally the integers every bit the equivalence classes of ordered pairs of natural numbers under the equivalence relation

(a, b) ~ (c, d) if and only if a + d = b + c .

The equivalence class of (a, b) contains either (a – b, 0) if a ≥ b , or (0, b – a) otherwise. If n is a natural number, one can denote +n the equivalence class of (due north, 0), and by –n the equivalence class of (0, northward). This allows identifying the natural number due north with the equivalence class +n .

Add-on of ordered pairs is done component-wise:

${\displaystyle (a,b)+(c,d)=(a+c,b+d).}$

A straightforward computation shows that the equivalence course of the upshot depends only on the equivalences classes of the summands, and thus that this defines an addition of equivalence classes, that is integers.^[59] Another straightforward ciphering shows that this improver 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 grouping is the additive group of integers. The rational numbers are constructed similarly, by taking equally semigroup the nonzero integers with multiplication.

This construction has been besides generalized under the name of Grothendieck group to the case of whatsoever 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 than specifically, the result of this construction practical to the equivalences classes nether isomorphisms of the objects of an abelian category, with the direct sum as semigroup operation.

Rational numbers (fractions) [edit]

Addition of rational numbers tin exist computed using the to the lowest degree common denominator, but a conceptually simpler definition involves but integer addition and multiplication:

• Ascertain ${\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {advert+bc}{bd}}.}$

As an instance, the sum ${\displaystyle {\frac {three}{four}}+{\frac {1}{8}}={\frac {3\times 8+4\times i}{4\times 8}}={\frac {24+4}{32}}={\frac {28}{32}}={\frac {7}{8}}}$ .

Addition of fractions is much simpler when the denominators are the same; in this case, one tin can just add together the numerators while leaving the denominator the same: ${\displaystyle {\frac {a}{c}}+{\frac {b}{c}}={\frac {a+b}{c}}}$ , then ${\displaystyle {\frac {i}{iv}}+{\frac {ii}{4}}={\frac {1+2}{four}}={\frac {iii}{4}}}$ .^[lx]

The commutativity and associativity of rational addition is an piece of cake consequence of the laws of integer arithmetic.^[61] For a more rigorous and general discussion, meet field of fractions.

Existent numbers [edit]

Adding πⁱⁱ/6 and e using Dedekind cuts of rationals.

A common structure of the set of real numbers is the Dedekind completion of the set of rational numbers. A real number is divers 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:

• Ascertain ${\displaystyle a+b=\{q+r\mid q\in a,r\in b\}.}$ ^[62]

This definition was get-go published, in a slightly modified form, by Richard Dedekind in 1872.^[63] The commutativity and associativity of real addition are immediate; defining the existent number 0 to be the set of negative rationals, it is easily seen to exist the additive identity. Probably the trickiest function of this structure pertaining to improver is the definition of additive inverses.^[64]

Adding π^two/6 and e using Cauchy sequences of rationals.

Unfortunately, dealing with multiplication of Dedekind cuts is a time-consuming case-by-case procedure similar to the addition of signed integers.^[65] 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, lima _n . Addition is defined term by term:

• Ascertain ${\displaystyle \lim _{north}a_{due north}+\lim _{north}b_{n}=\lim _{n}(a_{northward}+b_{n}).}$ ^[66]

This definition was get-go published by Georg Cantor, as well in 1872, although his formalism was slightly different.^[67] 1 must evidence 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 backdrop of rational numbers. Furthermore, the other arithmetics operations, including multiplication, have straightforward, coordinating definitions.^[68]

Complex numbers [edit]

Add-on of two complex numbers can be done geometrically by amalgam a parallelogram.

Complex numbers are added by adding the existent and imaginary parts of the summands.^{[69] [70]} That is to say:

${\displaystyle (a+bi)+(c+di)=(a+c)+(b+d)i.}$

Using the visualization of complex numbers in the circuitous plane, the addition has the post-obit geometric interpretation: the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point Ten obtained by building a parallelogram three of whose vertices are O, A and B. Equivalently, X is the indicate such that the triangles with vertices O, A, B, and X, B, A, are coinciding.

Generalizations [edit]

There are many binary operations that can be viewed as generalizations of the addition functioning on the existent numbers. The field of abstract algebra is centrally concerned with such generalized operations, and they also appear in fix theory and category theory.

Abstract algebra [edit]

Vectors [edit]

In linear algebra, a vector space is an algebraic construction that allows for adding whatever 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 airplane to the point (a,b) in the plane. The sum of two vectors is obtained by adding their individual coordinates:

${\displaystyle (a,b)+(c,d)=(a+c,b+d).}$

This addition operation is cardinal to classical mechanics, in which velocities, accelerations and forces are all represented past vectors.^[71]

Matrices [edit]

Matrix improver is defined for two matrices of the aforementioned dimensions. The sum of 2 m × n (pronounced "m past n") matrices A and B, denoted by A + B , is again an m × due north matrix computed by calculation respective elements:^{[72] [73]}

${\displaystyle {\begin{aligned}\mathbf {A} +\mathbf {B} &={\begin{bmatrix}a_{xi}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\\\stop{bmatrix}}+{\begin{bmatrix}b_{11}&b_{12}&\cdots &b_{1n}\\b_{21}&b_{22}&\cdots &b_{2n}\\\vdots &\vdots &\ddots &\vdots \\b_{m1}&b_{m2}&\cdots &b_{mn}\\\end{bmatrix}}\\&={\begin{bmatrix}a_{eleven}+b_{xi}&a_{12}+b_{12}&\cdots &a_{1n}+b_{1n}\\a_{21}+b_{21}&a_{22}+b_{22}&\cdots &a_{2n}+b_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}+b_{m1}&a_{m2}+b_{m2}&\cdots &a_{mn}+b_{mn}\\\stop{bmatrix}}\\\end{aligned}}}$

For instance:

${\displaystyle {\brainstorm{bmatrix}i&three\\1&0\\1&ii\end{bmatrix}}+{\begin{bmatrix}0&0\\7&5\\two&1\finish{bmatrix}}={\begin{bmatrix}ane+0&3+0\\1+seven&0+v\\i+2&2+1\end{bmatrix}}={\begin{bmatrix}1&3\\8&5\\three&3\end{bmatrix}}}$

Modular arithmetic [edit]

In modular arithmetics, the set of bachelor numbers is restricted to a finite subset of the integers, and add-on "wraps effectually" when reaching a sure value, called the modulus. For example, the set of integers modulo 12 has twelve elements; information technology inherits an addition operation from the integers that is cardinal to musical set up theory. The fix of integers modulo 2 has only ii elements; the addition performance it inherits is known in Boolean logic as the "sectional or" function. A similar "wrap around" operation arises in geometry, where the sum of 2 bending measures is oftentimes taken to exist their sum every bit real numbers modulo 2π. This amounts to an addition operation on the circumvolve, which in turn generalizes to addition operations on many-dimensional tori.

General theory [edit]

The general theory of abstruse algebra allows an "add-on" operation to be whatever associative and commutative functioning on a fix. Basic algebraic structures with such an addition operation include commutative monoids and abelian groups.

Set up theory and category theory [edit]

A far-reaching generalization of add-on of natural numbers is the addition of ordinal numbers and central numbers in set theory. These give ii unlike generalizations of addition of natural numbers to the transfinite. Unlike most improver operations, addition of ordinal numbers is not commutative.^[74] Addition of primal numbers, withal, is a commutative operation closely related to the disjoint union performance.

In category theory, disjoint spousal relationship is seen every bit a particular case of the coproduct operation,^[75] and general coproducts are perhaps the virtually abstract of all the generalizations of improver. Some coproducts, such as direct sum and wedge sum, are named to evoke their connection with improver.

[edit]

Addition, along with subtraction, multiplication and division, is considered i of the basic operations and is used in elementary arithmetic.

Arithmetic [edit]

Subtraction can be thought of equally a kind of add-on—that is, the addition of an additive inverse. Subtraction is itself a sort of inverse to addition, in that calculation x and subtracting ten are inverse functions.

Given a set with an addition operation, one cannot e'er ascertain a corresponding subtraction performance on that prepare; the set of natural numbers is a elementary instance. On the other hand, a subtraction functioning uniquely determines an addition performance, an additive inverse operation, and an condiment identity; for this reason, an condiment group can be described as a set that is closed under subtraction.^[76]

Multiplication can be thought of as repeated addition. If a unmarried term 10 appears in a sum n times, and then the sum is the production of n and x. If north is not a natural number, the product may nevertheless brand sense; for case, multiplication by −i yields the additive inverse of a number.

In the real and complex numbers, addition and multiplication can be interchanged by the exponential office:^[77]

${\displaystyle e^{a+b}=e^{a}e^{b}.}$

This identity allows multiplication to exist carried out by consulting a tabular array of logarithms and calculating add-on by manus; information technology as well enables multiplication on a slide dominion. The formula is notwithstanding a adept get-go-gild approximation in the broad context of Lie groups, where it relates multiplication of infinitesimal group elements with add-on of vectors in the associated Lie algebra.^[78]

There are even more generalizations of multiplication than addition.^[79] In full general, multiplication operations e'er distribute over addition; this requirement is formalized in the definition of a ring. In some contexts, such as the integers, distributivity over improver and the existence of a multiplicative identity is enough to uniquely determine the multiplication operation. The distributive belongings also provides information about addition; by expanding the product (ane + ane)(a + b) in both ways, one concludes that addition is forced to be commutative. For this reason, ring addition is commutative in general.^[80]

Division is an arithmetic functioning remotely related to addition. Since a/b = a(b ⁻¹), partitioning is right distributive over improver: (a + b) / c = a/c + b/c .^[81] Still, partition is not left distributive over add-on; i / (2 + 2) is not the aforementioned as 1/2 + 1/ii.

Ordering [edit]

The maximum operation "max (a, b)" is a binary performance similar to addition. In fact, if ii nonnegative numbers a and b are of different orders of magnitude, and then their sum is approximately equal to their maximum. This approximation is extremely useful in the applications of mathematics, for instance in truncating Taylor serial. Notwithstanding, it presents a perpetual difficulty in numerical assay, 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 fifty-fifty 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 primal number, their cardinal sum is exactly equal to the greater of the 2.^[83] Appropriately, at that place is no subtraction performance for infinite cardinals.^[84]

Maximization is commutative and associative, like add-on. Furthermore, since addition preserves the ordering of existent numbers, add-on distributes over "max" in the same way that multiplication distributes over addition:

${\displaystyle a+\max(b,c)=\max(a+b,a+c).}$

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 add-on", and the tropical "additive identity" is negative infinity.^[85] Some authors adopt to supplant addition with minimization; then the additive identity is positive infinity.^[86]

Tying these observations together, tropical addition is approximately related to regular improver through the logarithm:

${\displaystyle \log(a+b)\approx \max(\log a,\log b),}$

which becomes more than accurate as the base of operations of the logarithm increases.^[87] The approximation tin be fabricated exact by extracting a constant h, named by illustration with Planck'southward abiding from breakthrough mechanics,^[88] and taking the "classical limit" every bit h tends to goose egg:

${\displaystyle \max(a,b)=\lim _{h\to 0}h\log(e^{a/h}+e^{b/h}).}$

In this sense, the maximum performance is a dequantized version of improver.^[89]

Other ways to add [edit]

Incrementation, too known as the successor functioning, is the addition of 1 to a number.

Summation describes the add-on of arbitrarily many numbers, usually more just two. It includes the thought of the sum of a unmarried number, which is itself, and the empty sum, which is zero.^[xc] An space summation is a delicate procedure known equally a series.^[91]

Counting a finite set up is equivalent to summing 1 over the ready.

Integration is a kind of "summation" over a continuum, or more than 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, normally a real or complex number. Linear combinations are particularly useful in contexts where straightforward addition would violate some normalization rule, such every bit mixing of strategies in game theory or superposition of states in quantum mechanics.^[92]

Convolution is used to add together ii independent random variables divers past distribution functions. Its usual definition combines integration, subtraction, and multiplication.^[93] In general, convolution is useful every bit a kind of domain-side improver; by contrast, vector addition is a kind of range-side add-on.

Encounter also [edit]

• Lunar arithmetic
• Mental arithmetic
• Parallel add-on (mathematics)
• Verbal arithmetics (also known as cryptarithms), puzzles involving addition

Notes [edit]

1. ^ "Addend" is not a Latin discussion; in Latin information technology must exist further conjugated, as in numerus addendus "the number to be added".
2. ^ Some authors think that "deport" may be inappropriate for education; Van de Walle (p. 211) calls information technology "obsolete and conceptually misleading", preferring the word "merchandise". Still, "acquit" remains the standard term.

Footnotes [edit]

1. ^ From Enderton (p. 138): "...select ii sets One thousand and L with bill of fare Thou = ii and bill of fare 50 = iii. Sets of fingers are handy; sets of apples are preferred by textbooks."
2. ^ "Addition". www.mathsisfun.com . Retrieved 2020-08-25 .
3. ^ Devine et al. p. 263
4. ^ Mazur, Joseph. Enlightening Symbols: A Brusk History of Mathematical Notation and Its Hidden Powers. Princeton Academy Press, 2014. p. 161
5. ^ Section of the Army (1961) Army Technical Transmission TM xi-684: Principles and Applications of Mathematics for Communications-Electronics. Department 5.1
6. ^ ^{a b} Shmerko, 5.P.; Yanushkevich [Ânuškevič], Svetlana Due north. [Svitlana N.]; Lyshevski, South.E. (2009). Calculator arithmetic for nanoelectronics. CRC Press. p. 80.
7. ^ ^{a b} Schmid, Hermann (1974). Decimal Ciphering (1st ed.). Binghamton, NY: John Wiley & Sons. ISBN0-471-76180-10. and Schmid, Hermann (1983) [1974]. Decimal Computation (reprint of 1st ed.). Malabar, FL: Robert E. Krieger Publishing Company. ISBN978-0-89874-318-0.
8. ^ ^{a b} Weisstein, Eric W. "Addition". mathworld.wolfram.com . Retrieved 2020-08-25 .
9. ^ Hosch, W.50. (Ed.). (2010). The Britannica Guide to Numbers and Measurement. The Rosen Publishing Group. p. 38
10. ^ ^{a b} Schwartzman p. 19
11. ^ Karpinski pp. 56–57, reproduced on p. 104
12. ^ Schwartzman (p. 212) attributes adding upwards to the Greeks and Romans, saying it was nigh as common as adding downward. On the other hand, Karpinski (p. 103) writes that 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.
13. ^ Karpinski pp. 150–153
14. ^ Cajori, Florian (1928). "Origin and meanings of the signs + and -". A History of Mathematical Notations, Vol. ane. The Open up Courtroom Visitor, Publishers.
15. ^ "plus". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
16. ^ Run into Viro 2001 for an example of the composure involved in adding with sets of "fractional cardinality".
17. ^ Adding it upwardly (p. 73) compares adding measuring rods to calculation 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 split them into parts, and it seriously changes their nature."
18. ^ Mosley, F. (2001). Using number lines with 5–8 twelvemonth olds. Nelson Thornes. p. 8
19. ^ Li, Y., & Lappan, Thousand. (2014). Mathematics curriculum in school education. Springer. p. 204
20. ^ Bronstein, Ilja Nikolaevič; Semendjajew, Konstantin Adolfovič (1987) [1945]. "2.4.1.ane.". In Grosche, Günter; Ziegler, Viktor; Ziegler, Dorothea (eds.). Taschenbuch der Mathematik (in German). Vol. ane. Translated past Ziegler, Viktor. Weiß, Jürgen (23 ed.). Thun and Frankfurt am Main: Verlag Harri Deutsch (and B.G. Teubner Verlagsgesellschaft, Leipzig). pp. 115–120. ISBN978-3-87144-492-0.
21. ^ Kaplan pp. 69–71
22. ^ Hempel, C.G. (2001). The philosophy of Carl G. Hempel: studies in science, caption, and rationality. p. 7
23. ^ R. Fierro (2012) Mathematics for Elementary Schoolhouse Teachers. Cengage Learning. Sec two.3
24. ^ Moebs, William; et al. (2022). "1.4 Dimensional Analysis". University Physics Volume 1. OpenStax. ISBN978-one-947172-xx-3.
25. ^ Wynn p. five
26. ^ Wynn p. 15
27. ^ Wynn p. 17
28. ^ Wynn p. xix
29. ^ Randerson, James (21 August 2008). "Elephants have a head for figures". The Guardian. Archived from the original on 2 April 2015. Retrieved 29 March 2015.
30. ^ F. Smith p. 130
31. ^ Carpenter, Thomas; Fennema, Elizabeth; Franke, Megan Loef; Levi, Linda; Empson, Susan (1999). Children's mathematics: Cognitively guided educational activity . Portsmouth, NH: Heinemann. ISBN978-0-325-00137-i.
32. ^ ^{a b} Henry, Valerie J.; Brown, Richard S. (2008). "Beginning-grade basic facts: An investigation into teaching and learning of an accelerated, high-demand memorization standard". Journal for Research in Mathematics Educational activity. 39 (2): 153–183. doi:10.2307/30034895. JSTOR 30034895.
33. ^ Beckmann, S. (2014). The twenty-third ICMI study: primary mathematics written report on whole numbers. International Journal of Stem Teaching, i(1), 1-8. Chicago
34. ^ Schmidt, W., Houang, R., & Cogan, L. (2002). "A coherent curriculum". American Educator, 26(2), 1–xviii.
35. ^ ^{a b c d e f g} Fosnot and Dolk p. 99
36. ^ Rebecca Wingard-Nelson (2014) Decimals and Fractions: It's Easy Enslow Publishers, Inc.
37. ^ Dale R. Patrick, Stephen W. Fardo, Vigyan Chandra (2008) Electronic Digital System Fundamentals The Fairmont Printing, Inc. p. 155
38. ^ P.E. Bates Bothman (1837) The common school arithmetics. Henry Benton. p. 31
39. ^ Truitt and Rogers pp. i;44–49 and pp. 2;77–78
40. ^ Ifrah, Georges (2001). The Universal History of Calculating: From the Abacus to the Breakthrough Figurer. New York: John Wiley & Sons, Inc. ISBN978-0-471-39671-0. p. 11
41. ^ Jean Marguin, p. 48 (1994) ; Quoting René Taton (1963)
42. ^ See Competing designs in Pascal's calculator commodity
43. ^ Flynn and Overman pp. 2, 8
44. ^ Flynn and Overman pp. 1–nine
45. ^ 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. i. Springer, 2010. p. 194
46. ^ Karpinski pp. 102–103
47. ^ The identity of the augend and addend varies with architecture. For Add together in x86 come across Horowitz and Hill p. 679; for ADD in 68k see p. 767.
48. ^ Joshua Bloch, "Extra, Extra – Read All About It: Nearly All Binary Searches and Mergesorts are Broken" Archived 2016-04-01 at the Wayback Machine. Official Google Inquiry Web log, June two, 2006.
49. ^ Neumann, Peter One thousand. "The Risks Digest Volume 4: Issue 45". The Risks Digest. Archived from the original on 2014-12-28. Retrieved 2015-03-30 .
50. ^ Enderton capacity 4 and 5, for example, follow this development.
51. ^ According to a survey of the nations with highest TIMSS mathematics exam scores; meet Schmidt, W., Houang, R., & Cogan, L. (2002). A coherent curriculum. American educator, 26(2), p. iv.
52. ^ Baez (p. 37) explains the historical development, in "stark dissimilarity" with the set theory presentation: "Apparently, half an apple is easier to understand than a negative apple tree!"
53. ^ Begle p. 49, Johnson p. 120, Devine et al. p. 75
54. ^ Enderton p. 79
55. ^ For a version that applies to whatsoever poset with the descending concatenation condition, see Bergman p. 100.
56. ^ Enderton (p. 79) observes, "Only nosotros want i binary operation +, not all these piddling i-identify functions."
57. ^ Ferreirós p. 223
58. ^ M. Smith p. 234, Sparks and Rees p. 66
59. ^ Enderton p. 92
60. ^ Schyrlet Cameron, and Carolyn Craig (2013)Adding and Subtracting Fractions, Grades v–viii Mark Twain, Inc.
61. ^ 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.
62. ^ Enderton p. 114
63. ^ Ferreirós p. 135; meet section 6 of Stetigkeit und irrationale Zahlen Archived 2005-10-31 at the Wayback Automobile.
64. ^ The intuitive approach, inverting every element of a cut and taking its complement, works only for irrational numbers; meet Enderton p. 117 for details.
65. ^ Schubert, E. Thomas, Phillip J. Windley, and James Alves-Foss. "Higher Social club Logic Theorem Proving and Its Applications: Proceedings of the 8th International Workshop, volume 971 of." Lecture Notes in Computer Science (1995).
66. ^ Textbook constructions are commonly not so cavalier with the "lim" symbol; see Burrill (p. 138) for a more careful, fatigued-out development of addition with Cauchy sequences.
67. ^ Ferreirós p. 128
68. ^ Burrill p. 140
69. ^ Conway, John B. (1986), Functions of One Complex Variable I, Springer, ISBN978-0-387-90328-half-dozen
70. ^ Joshi, Kapil D (1989), Foundations of Discrete Mathematics, New York: John Wiley & Sons, ISBN978-0-470-21152-6
71. ^ Gbur, p. i
72. ^ Lipschutz, Southward., & Lipson, M. (2001). Schaum's outline of theory and problems of linear algebra. Erlangga.
73. ^ Riley, M.F.; Hobson, M.P.; Bence, S.J. (2010). Mathematical methods for physics and engineering . Cambridge University Printing. ISBN978-0-521-86153-3.
74. ^ Cheng, pp. 124–132
75. ^ Riehl, p. 100
76. ^ The set still must be nonempty. Dummit and Foote (p. 48) hash out this benchmark written multiplicatively.
77. ^ Rudin p. 178
78. ^ Lee p. 526, Proposition twenty.9
79. ^ Linderholm (p. 49) observes, "By multiplication, properly speaking, a mathematician may hateful practically anything. By addition he may mean a great variety of things, but not so great a variety as he will mean by 'multiplication'."
80. ^ 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.
81. ^ For an instance of left and right distributivity, see Loday, peculiarly p. xv.
82. ^ Compare Viro Figure 1 (p. ii)
83. ^ Enderton calls this statement the "Absorption Law of Central Arithmetic"; it depends on the comparability of cardinals and therefore on the Axiom of Choice.
84. ^ Enderton p. 164
85. ^ Mikhalkin p. ane
86. ^ Akian et al. p. 4
87. ^ Mikhalkin p. two
88. ^ Litvinov et al. p. three
89. ^ Viro p. iv
90. ^ Martin p. 49
91. ^ Stewart p. 8
92. ^ Rieffel and Polak, p. 16
93. ^ Gbur, p. 300

References [edit]

History

• Ferreirós, José (1999). Labyrinth of Thought: A History of Set Theory and Its Role in Mod Mathematics . Birkhäuser. ISBN978-0-8176-5749-9.
• Karpinski, Louis (1925). The History of Arithmetic. Rand McNally. LCC QA21.K3.
• Schwartzman, Steven (1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English . MAA. ISBN978-0-88385-511-9.
• Williams, Michael (1985). A History of Computing Technology . Prentice-Hall. ISBN978-0-13-389917-7.

Elementary mathematics

• Sparks, F.; Rees C. (1979). A Survey of Basic Mathematics. McGraw-Hill. ISBN978-0-07-059902-iv.

Didactics

• Begle, Edward (1975). The Mathematics of the Elementary School. McGraw-Loma. ISBN978-0-07-004325-1.
• California Country Board of Didactics mathematics content standards Adopted December 1997, accessed December 2005.
• Devine, D.; Olson, J.; Olson, M. (1991). Unproblematic Mathematics for Teachers (2e ed.). Wiley. ISBN978-0-471-85947-5.
• National Research Quango (2001). Adding It Upward: Helping Children Learn Mathematics. National University Printing. doi:10.17226/9822. ISBN978-0-309-06995-3.
• Van de Walle, John (2004). Elementary and Middle School Mathematics: Teaching developmentally (5e ed.). Pearson. ISBN978-0-205-38689-five.

Cognitive science

• Fosnot, Catherine T.; Dolk, Maarten (2001). Young Mathematicians at Piece of work: Amalgam Number Sense, Improver, and Subtraction. Heinemann. ISBN978-0-325-00353-5.
• Wynn, Karen (1998). "Numerical competence in infants". The Evolution of Mathematical Skills. Taylor & Francis. ISBN0-86377-816-X.

Mathematical exposition

• Bogomolny, Alexander (1996). "Improver". Interactive Mathematics Miscellany and Puzzles (cut-the-knot.org). Archived from the original on April 26, 2006. Retrieved 3 Feb 2006.
• Cheng, Eugenia (2017). Beyond Infinity: An Expedition to the Outer Limits of Mathematics. Bones Books. ISBN978-1-541-64413-7.
• Dunham, William (1994). The Mathematical Universe . Wiley. ISBN978-0-471-53656-7.
• Johnson, Paul (1975). From Sticks and Stones: Personal Adventures in Mathematics. Science Research Assembly. ISBN978-0-574-19115-1.
• Linderholm, Carl (1971). Mathematics Made Hard. Wolfe. ISBN978-0-7234-0415-6.
• Smith, Frank (2002). The Glass Wall: Why Mathematics Can Seem Difficult . Teachers Higher Press. ISBN978-0-8077-4242-half-dozen.
• Smith, Karl (1980). The Nature of Modern Mathematics (tertiary ed.). Wadsworth. ISBN978-0-8185-0352-viii.

Advanced mathematics

• Bergman, George (2005). An Invitation to General Algebra and Universal Constructions (2.iii ed.). General Printing. ISBN978-0-9655211-4-seven.
• Burrill, Claude (1967). Foundations of Existent Numbers. McGraw-Hill. LCC QA248.B95.
• Dummit, D.; Foote, R. (1999). Abstruse Algebra (2 ed.). Wiley. ISBN978-0-471-36857-one.
• Gbur, Greg (2011). Mathematical Methods for Optical Physics and Engineering science. Cambridge University Press. ISBN978-0-511-91510-9. OCLC 704518582.
• Enderton, Herbert (1977). Elements of Set Theory. Bookish Press. ISBN978-0-12-238440-0.
• Lee, John (2003). Introduction to Smooth Manifolds. Springer. ISBN978-0-387-95448-6.
• Martin, John (2003). Introduction to Languages and the Theory of Computation (3 ed.). McGraw-Hill. ISBN978-0-07-232200-ii.
• Riehl, Emily (2016). Category Theory in Context. Dover. ISBN978-0-486-80903-eight.
• Rudin, Walter (1976). Principles of Mathematical Analysis (iii ed.). McGraw-Hill. ISBN978-0-07-054235-eight.
• Stewart, James (1999). Calculus: Early Transcendentals (4 ed.). Brooks/Cole. ISBN978-0-534-36298-0.

Mathematical research

• Akian, Marianne; Bapat, Ravindra; Gaubert, Stephane (2005). "Min-plus methods in eigenvalue perturbation theory and generalised Lidskii-Vishik-Ljusternik theorem". INRIA Reports. arXiv:math.SP/0402090. Bibcode:2004math......2090A.
• Baez, J.; Dolan, J. (2001). Mathematics Unlimited – 2001 and Across. From Finite Sets to Feynman Diagrams. p. 29. arXiv:math.QA/0004133. ISBN3-540-66913-2.
• Litvinov, Grigory; Maslov, Victor; Sobolevskii, Andreii (1999). Idempotent mathematics and interval assay. Reliable Computing, Kluwer.
• Loday, Jean-Louis (2002). "Arithmetree". Journal of Algebra. 258: 275. arXiv:math/0112034. doi:ten.1016/S0021-8693(02)00510-0.
• Mikhalkin, Grigory (2006). Sanz-Solé, Marta (ed.). Proceedings of the International Congress of Mathematicians (ICM), Madrid, Espana, August 22–30, 2006. Volume 2: Invited lectures. Tropical Geometry and its Applications. Zürich: European Mathematical Society. pp. 827–852. arXiv:math.AG/0601041. ISBN978-iii-03719-022-vii. Zbl 1103.14034.
• Viro, Oleg (2001). Cascuberta, Carles; Miró-Roig, Rosa Maria; Verdera, Joan; Xambó-Descamps, Sebastià (eds.). European Congress of Mathematics: Barcelona, July 10–fourteen, 2000, Volume I. Dequantization of Real Algebraic Geometry on Logarithmic Newspaper. Progress in Mathematics. Vol. 201. Basel: Birkhäuser. pp. 135–146. arXiv:math/0005163. Bibcode:2000math......5163V. ISBN978-three-7643-6417-five. Zbl 1024.14026.

Computing

• Flynn, Yard.; Oberman, S. (2001). Advanced Computer Arithmetics Blueprint. Wiley. ISBN978-0-471-41209-0.
• Horowitz, P.; Hill, Westward. (2001). The Art of Electronics (2 ed.). Cambridge UP. ISBN978-0-521-37095-0.
• Jackson, Albert (1960). Analog Computation. McGraw-Hill. LCC QA76.4 J3.
• Rieffel, Eleanor Thousand.; Polak, Wolfgang H. (four March 2011). Breakthrough Computing: A Gentle Introduction. MIT Press. ISBN978-0-262-01506-6.
• Truitt, T.; Rogers, A. (1960). Basics of Analog Computers. John F. Rider. LCC QA76.4 T7.
• Marguin, Jean (1994). Histoire des Instruments et Machines à Calculer, Trois Siècles de Mécanique Pensante 1642–1942 (in French). Hermann. ISBN978-two-7056-6166-3.
• Taton, René (1963). Le Calcul Mécanique. Que Sais-Je ? n° 367 (in French). Presses universitaires de France. pp. xx–28.

Farther reading [edit]

• Baroody, Arthur; Tiilikainen, Sirpa (2003). The Evolution of Arithmetics Concepts and Skills. 2 perspectives on improver development. Routledge. p. 75. ISBN0-8058-3155-X.
• Davison, David Yard.; Landau, Marsha South.; McCracken, Leah; Thompson, Linda (1999). Mathematics: Explorations & Applications (TE ed.). Prentice Hall. ISBN978-0-13-435817-8.
• Bunt, Lucas N.H.; Jones, Phillip S.; Bedient, Jack D. (1976). The Historical roots of Elementary Mathematics . Prentice-Hall. ISBN978-0-xiii-389015-0.
• Poonen, Bjorn (2010). "Addition". Girls' Angle Bulletin. iii (three–5). ISSN 2151-5743.
• Weaver, J. Fred (1982). "Improver and Subtraction: A Cerebral Perspective". Improver and Subtraction: A Cerebral Perspective. Interpretations of Number Operations and Symbolic Representations of Addition and Subtraction. Taylor & Francis. p. 60. ISBN0-89859-171-half-dozen.

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Source: https://en.wikipedia.org/wiki/Addition

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