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{{other uses}}{{redirect|Add||ADD (disambiguation)}}{{good article}}File:Addition01.svg|right|thumb|120px|3 + 2 = 5 with (apple]]s, a popular choice in textbooksFrom Enderton (p.138): " 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.")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 {{nowrap|1="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 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 {{num|1}} is the same as counting; addition of {{num|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, {{nowrap|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.

Notation and terminology

(File:PlusCM128.svg|right|120px|thumb|The plus sign)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 ("one plus one equals two") 2 + 2 = 4 ("two plus two equals four") 3 + 3 = 6 ("three plus three equals six") 5 + 4 + 2 = 11 (see "associativity" below) 3 + 3 + 3 + 3 = 12 (see "multiplication" below)
File:AdditionVertical.svg|right|thumb|Columnar addition – the numbers in the column are to be added, with the sum written below the underlineunderlineThere 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.Devine et al. p.263 For example,{{spaces|6}}3½ = 3 + ½ = 3.5.This notation can cause confusion since in most other contexts (wikt:juxtaposition|juxtaposition) denotes multiplication instead.Mazur, Joseph. Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers. Princeton University Press, 2014. p. 161
The sum of a series of related numbers can be expressed through capital sigma notation, which compactly denotes iteration. For example,
sum_{k=1}^5 k^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 55.
{{anchor|summand}}The numbers or the objects to be added in general addition are collectively referred to as the terms,Department of the Army (1961) Army Technical Manual TM 11-684: Principles and Applications of Mathematics for Communications-Electronics. Section 5.1 the {{vanchor|addend}}sBOOK, Shmerko, V. P., Yanushkevich [Ânuškevič], Svetlana N. [Svitlana N.], Lyshevski, S. E., 2009, Computer arithmetics for nanoelectronics, CRC Press, 80, or the summands;Hosch, W. L. (Ed.). (2010). The Britannica Guide to Numbers and Measurement. The Rosen Publishing Group. p.38this 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.BOOK, Decimal Computation, Hermann, Schmid, Hermann Schmid (computer scientist), 1974, 1, John Wiley & Sons, Binghamton, New York, USA, 0-471-76180-X, BOOK, Decimal Computation, Hermann, Schmid, Hermann Schmid (computer scientist), 1974, 1983, 1 (reprint), Robert E. Krieger Publishing Company, Malabar, Florida, USA, 0-89874-318-4, In fact, during the Renaissance, many authors did not consider the first addend an "addend" at all. Today, due to the commutative property of addition, "augend" is rarely used, and both terms are generally called addends.Schwartzman p.19All of the above terminology derives from Latin. "(wikt:addition|Addition)" and "(wikt:add|add)" are English words derived from the Latin verb addere, which is in turn a compound of ad "to" and dare "to give", from the Proto-Indo-European root {{PIE|*deh₃-}} "to give"; thus to add is to give to. Using the gerundive suffix -nd results in "addend", "thing to be added"."Addend" is not a Latin word; in Latin it must be further conjugated, as in numerus addendus "the number to be added". Likewise from augere "to increase", one gets "augend", "thing to be increased".(File:AdditionNombryng.svg|left|thumb|Redrawn illustration from The Art of Nombryng, one of the first English arithmetic texts, in the 15th century.Karpinski pp.56–57, reproduced on p.104)"Sum" and "summand" derive from the Latin noun summa "the highest, the top" and associated verb summare. This is appropriate not only because the sum of two positive numbers is greater than either, but because it was common for the ancient Greeks and Romans to add upward, contrary to the modern practice of adding downward, so that a sum was literally higher than the addends.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 "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.Addere and summare date back at least to Boethius, if not to earlier Roman writers such as Vitruvius and Frontinus; Boethius also used several other terms for the addition operation. The later Middle English terms "adden" and "adding" were popularized by Chaucer.Karpinski pp.150–153The plus sign "+" (Unicode:U+002B; ASCII: +) is an abbreviation of the Latin word et, meaning "and".BOOK, Cajori, Florian, A History of Mathematical Notations, Vol. 1, 1928, The Open Court Company, Publishers, Origin and meanings of the signs + and -, It appears in mathematical works dating back to at least 1489.{{OED|plus}}


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

(File:AdditionShapes.svg|right|200px|thumb)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.See Viro 2001 for an example of the sophistication involved in adding with sets of "fractional cardinality".One possible fix is to consider collections of objects that can be easily divided, such as pies or, still better, segmented rods.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." 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

(File:AdditionLineAlgebraic.svg|right|frame|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.)(File:AdditionLineUnary.svg|right|frame|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.Mosley, F. (2001). Using number lines with 5-8 year olds. Nelson Thornes. p.8
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 {{nowrap|a + b}} play asymmetric roles, and the operation {{nowrap|a + b}} is viewed as applying the unary operation +b to a.Li, Y., & Lappan, G. (2014). Mathematics curriculum in school education. Springer. p. 204 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.



(File:AdditionComm01.svg|right|113px|thumb|4 + 2 = 2 + 4 with blocks)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.


(File:AdditionAsc.svg|left|100px|thumb|2 + (1 + 3) = (2 + 1) + 3 with segmented rods)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 {{nowrap|1=(a + b) + c = a + (b + c)}}. For example, {{nowrap|1=(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.WEB,weblink Order of Operations Lessons, Algebra.Help, 5 March 2012,

Identity element

(File:AdditionZero.svg|right|70px|thumb|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 refined the concept; around the year 830, Mahavira wrote, "zero becomes the same as what is added to it", corresponding to the unary statement {{nowrap|1=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 {{nowrap|1=a + 0 = a}}.Kaplan pp.69–71


Within the context of integers, addition of one also plays a special role: for any integer a, the integer {{nowrap|(a + 1)}} is the least integer greater than a, also known as the successor of a.Hempel, C. G. (2001). The philosophy of Carl G. Hempel: studies in science, explanation, and rationality. p. 7 For instance, 3 is the successor of 2 and 7 is the successor of 6. Because of this succession, the value of {{nowrap|a + b}} can also be seen as the bth successor of a, making addition iterated succession. For examples, {{nowrap|6 + 2}} is 8, because 8 is the successor of 7, which is the successor of 6, making 8 the 2nd successor of 6.


To numerically add physical quantities with units, they must be expressed with common units.R. Fierro (2012) Mathematics for Elementary School Teachers. Cengage Learning. Sec 2.3 For example, adding 50 milliliters to 150 milliliters gives 200 milliliters. However, if a measure of 5 feet is extended by 2 inches, the sum is 62 inches, since 60 inches is synonymous with 5 feet. On the other hand, it is usually meaningless to try to add 3 meters and 4 square meters, since those units are incomparable; this sort of consideration is fundamental in dimensional analysis.

Performing addition

Innate ability

Studies on mathematical development starting around the 1980s have exploited the phenomenon of habituation: infants look longer at situations that are unexpected.Wynn p.5 A seminal experiment by Karen Wynn in 1992 involving Mickey Mouse dolls manipulated behind a screen demonstrated that five-month-old infants expect {{nowrap|1 + 1}} to be 2, and they are comparatively surprised when a physical situation seems to imply that {{nowrap|1 + 1}} is either 1 or 3. This finding has since been affirmed by a variety of laboratories using different methodologies.Wynn p.15 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.Wynn p.17Even 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 0 through 4, one chimpanzee was able to compute the sum of two numerals without further training.Wynn p.19 More recently, Asian elephants have demonstrated an ability to perform basic arithmetic.NEWS, The Guardian, Randerson, James,weblink Elephants have a head for figures, 21 August 2008, 29 March 2015,

Learning addition as children

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.F. Smith p.130 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 {{nowrap|1=6 + 6 = 12}} and then reason that {{nowrap|6 + 7}} is one more, or 13.BOOK, Carpenter, Thomas, Fennema, Elizabeth, Franke, Megan Loef, Levi, Linda, Empson, Susan, Children's mathematics: Cognitively guided instruction, Heinemann, 1999, Portsmouth, NH, 0-325-00137-5, 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.JOURNAL, Henry, Valerie J., Brown, Richard S., 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, 2008, 10.2307/30034895, Different nations introduce whole numbers and arithmetic at different ages, with many countries teaching addition in pre-school.Beckmann, S. (2014). The twenty-third ICMI study: primary mathematics study on whole numbers. International Journal of STEM Education, 1(1), 1-8.Chicago
However, throughout the world, addition is taught by the end of the first year of elementary school.Schmidt, W., Houang, R., & Cogan, L. (2002). A coherent curriculum. American educator, 26(2), 1-18.

Addition table

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 table}}

Decimal system

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:Fosnot and Dolk p. 99
  • Commutative property: Mentioned above, using the pattern a + b = b + a reduces the number of "addition facts" from 100 to 55.
  • One or two more: Adding 1 or 2 is a basic task, and it can be accomplished through counting on or, ultimately, intuition.
  • Zero: Since zero is the additive identity, adding zero is trivial. Nonetheless, in the teaching of arithmetic, some students are introduced to addition as a process that always increases the addends; word problems may help rationalize the "exception" of zero.
  • Doubles: Adding a number to itself is related to counting by two and to multiplication. Doubles facts form a backbone for many related facts, and students find them relatively easy to grasp.
  • Near-doubles: Sums such as 6 + 7 = 13 can be quickly derived from the doubles fact {{nowrap|1=6 + 6 = 12}} by adding one more, or from {{nowrap|1=7 + 7 = 14}} but subtracting one.
  • Five and ten: Sums of the form 5 + {{mvar|x}} and 10 + {{mvar|x}} are usually memorized early and can be used for deriving other facts. For example, {{nowrap|1=6 + 7 = 13}} can be derived from {{nowrap|1=5 + 7 = 12}} by adding one more.
  • Making ten: An advanced strategy uses 10 as an intermediate for sums involving 8 or 9; for example, {{nowrap|1=8 + 6 = 8 + 2 + 4 =}} {{nowrap|1=10 + 4 = 14}}.
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.


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 {{nowrap|27 + 59}}
+ 59
7 + 9 = 16, and the digit 1 is the carry.Some authors think that "carry" may be inappropriate for education; Van de Walle (p.211) calls it "obsolete and conceptually misleading", preferring the word "trade". However, "carry" remains the standard term. An alternate strategy starts adding from the most significant digit on the left; this route makes carrying a little clumsier, but it is faster at getting a rough estimate of the sum. There are many alternative methods.

Addition of decimal fractions

Decimal fractions can be added by a simple modification of the above process.Rebecca Wingard-Nelson (2014) Decimals and Fractions: It's Easy Enslow Publishers, Inc. One aligns two decimal fractions above each other, with the decimal point in the same location. If necessary, one can add trailing zeros to a shorter decimal to make it the same length as the longer decimal. Finally, one performs the same addition process as above, except the decimal point is placed in the answer, exactly where it was placed in the summands.As an example, 45.1 + 4.34 can be solved as follows:
4 5 . 1 0
+ 0 4 . 3 4
4 9 . 4 4

Scientific notation

In scientific notation, numbers are written in the form x=atimes10^{b}, where a is the significand and 10^{b} is the exponential part. Addition requires two numbers in scientific notation to be represented using the same exponential part, so that the two significands can simply be added.For example:
2.34times10^{-5} + 5.67times10^{-6} = 2.34times10^{-5} + 0.567times10^{-5} = 2.907times10^{-5}

Addition in other bases

Addition in other bases is very similar to decimal addition. As an example, one can consider addition in binary.Dale R. Patrick, Stephen W. Fardo, Vigyan Chandra (2008) Electronic Digital System Fundamentals The Fairmont Press, Inc. p. 155 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.P.E. Bates Bothman (1837) The common school arithmetic. Henry Benton. p. 31 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:
{{brown|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, {{nowrap|1=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: {{nowrap|1=1 + 0 + 1 = 102}} again; the 1 is carried, and 0 is written at the bottom. The third column: {{nowrap|1=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).


File:Opampsumming2.svg|right|frame|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.Truitt and Rogers pp.1;44–49 and pp.2;77–78Addition 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.File:BabbageDifferenceEngine.jpg|left|thumb|Part of Charles Babbage's Difference EngineDifference EngineThe 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.BOOK, Ifrah, Georges, 2001, The Universal History of Computing: From the Abacus to the Quantum Computer, John Wiley & Sons, Inc., New York, NY, 978-0471396710, p.11Blaise Pascal invented the mechanical calculator in 1642;Jean Marguin, p. 48 (1994) ; Quoting René Taton (1963) it was the first operational adding machine. It made use of a gravity-assisted carry mechanism. It was the only operational mechanical calculator in the 17th centurySee Competing designs in Pascal's calculator article and the earliest automatic, digital computer. Pascal's calculator was limited by its carry mechanism, which forced its wheels to only turn one way so it could add. To subtract, the operator had to use the Pascal's calculator's complement, which required as many steps as an addition. Giovanni Poleni followed Pascal, building the second functional mechanical calculator in 1709, a calculating clock made of wood that, once setup, could multiply two numbers automatically.File:Full-adder.svg|thumb|"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 {{nowrap|999 + 1}}, but one bypasses the group of 9s and skips to the answer.Flynn and Overman pp.2, 8In 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.Flynn and Overman pp.1–9Yeo, 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 Unlike addition on paper, addition on a computer often changes the addends. On the ancient abacus and adding board, both addends are destroyed, leaving only the sum. The influence of the abacus on mathematical thinking was strong enough that early Latin texts often claimed that in the process of adding "a number to a number", both numbers vanish.Karpinski pp.102–103 In modern times, the ADD instruction of a microprocessor often replaces the augend with the sum but preserves the addend.The identity of the augend and addend varies with architecture. For ADD in x86 see Horowitz and Hill p.679; for ADD in 68k see p.767. In a high-level programming language, evaluating {{nowrap|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 {{nowrap|1=a = a + b}}. Some languages such as C or C++ allow this to be abbreviated as {{nowrap|1=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

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