Addition Algorithms - Revision history

Khanh: /* Related operations */

2022-01-08T00:02:12Z

Related operations

Khanh: /* Performing addition */

2022-01-08T00:00:11Z

Performing addition

Khanh: /* Performing addition */

2022-01-07T23:58:06Z

Performing addition

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Khanh: /* Associativity */

2022-01-07T23:51:06Z

Associativity

Khanh: /* Properties */

2022-01-07T23:49:56Z

Properties

Khanh: /* Interpretations */

2022-01-07T23:45:01Z

Interpretations

Khanh: /* Notation and terminology */

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Notation and terminology

Khanh: /* Notation and terminology */

2022-01-07T23:38:44Z

Notation and terminology

Khanh at 23:35, 7 January 2022

2022-01-07T23:35:29Z

Khanh: /* Arithmetic */

2022-01-07T23:34:08Z

Arithmetic

← Older revision		Revision as of 00:02, 8 January 2022
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	===Ordering===		===Ordering===
−	~~[[File:XPlusOne.svg\|right\|thumb\|Log-log plot of {{mvar\|x}} + 1 and max ({{mvar\|x}}, 1) from {{mvar\|x}} = 0.001 to 1000]]~~
	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 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''.

@@ Line 90: / Line 90: @@
 ===Childhood learning===
-Typically, children first master counting. When given a problem that requires that two items and three items be combined, young children model the situation with physical objects, often fingers or a drawing, and then count the total. As they gain experience, they learn or discover the strategy of "counting-on": asked to find two plus three, children count three past two, saying "three, four, ''five''" (usually ticking off fingers), and arriving at five. This strategy seems almost universal; children can easily pick it up from peers or teachers.<ref>F. Smith p. 130</ref> 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. 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.
+Typically, children first master counting. When given a problem that requires that two items and three items be combined, young children model the situation with physical objects, often fingers or a drawing, and then count the total. As they gain experience, they learn or discover the strategy of "counting-on": asked to find two plus three, children count three past two, saying "three, four, ''five''" (usually ticking off fingers), and arriving at five. This strategy seems almost universal; children can easily pick it up from peers or teachers. Most discover it independently. With additional experience, children learn to add more quickly by exploiting the commutativity of addition by counting up from the larger number, in this case, starting with three and counting "four, ''five''." Eventually children begin to recall certain addition facts ("number bonds"), either through experience or rote memorization. Once some facts are committed to memory, children begin to derive unknown facts from known ones. For example, a child asked to add six and seven may know that 6 + 6 = 12 and then reason that 6 + 7 is one more, or 13. Such derived facts can be found very quickly and most elementary school students eventually rely on a mixture of memorized and derived facts to add fluently.
 Different nations introduce whole numbers and arithmetic at different ages, with many countries teaching addition in pre-school. However, throughout the world, addition is taught by the end of the first year of elementary school.
@@ Line 181: / Line 181: @@
 This is known as ''carrying''. When the result of an addition exceeds the value of a digit, the procedure is to "carry" the excess amount divided by the radix (that is, 10/10) to the left, adding it to the next positional value. This is correct since the next position has a weight that is higher by a factor equal to the radix. Carrying works the same way in binary:
-    {{brown|1 1 1 1 1    (carried digits)}}
+1 1 1 1    (carried digits)
 1 1 0 1
   +   1 0 1 1 1

@@ Line 67: / Line 67: @@
 Addition is associative, which means that when three or more numbers are added together, the order 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 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, (1 + 2) + 3 = 3 + 3 = 6 = 1 + 5 = 1 + (2 + 3).
+As an example, should the expression ''a'' + ''b'' + ''c'' be defined to mean (''a'' + ''b'') + ''c'' or ''a'' + (''b'' + ''c'')? Given that addition is associative, the choice of definition is irrelevant. For any three numbers ''a'', ''b'', and ''c'', it is true that (''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 root]]s, multiplication and division, but is given equal priority to subtraction.
+When addition is used together with other operations, the order of operations becomes important. In the standard order of operations, addition is a lower priority than exponentiation, nth roots, multiplication and division, but is given equal priority to subtraction.
 ===Identity element===

@@ Line 59: / Line 59: @@
 ===Commutativity===
 [[File:AdditionComm01.svg|right|113px|thumb|4 + 2 = 2 + 4 with blocks]]
-Addition is [[commutative]], meaning that one can change the order of the terms in a sum, but still get the same result. Symbolically, if ''a'' and ''b'' are any two numbers, then
+Addition is commutative, meaning that one can change the order of the terms in a sum, but still get the same result. Symbolically, if ''a'' and ''b'' are any two numbers, then
 :''a'' + ''b'' = ''b'' + ''a''.
-The fact that addition is commutative is known as the "commutative law of addition" or "commutative property of addition". Some other [[binary operation]]s are commutative, such as multiplication, but many others are not, such as subtraction and division.
+The fact that addition is commutative is known as the "commutative law of addition" or "commutative property of addition". Some other binary operations are commutative, such as multiplication, but many others are not, such as subtraction and division.
 ===Associativity===
 [[File:AdditionAsc.svg|left|100px|thumb|2 + (1 + 3) = (2 + 1) + 3 with segmented rods]]
-Addition is [[associativity|associative]], which means that when three or more numbers are added together, the [[order of operations]] does not change the result.
+Addition is associative, which means that when three or more numbers are added together, the order 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 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)}}.
+As an example, should the expression ''a'' + ''b'' + ''c'' be defined to mean (''a'' + ''b'') + ''c'' or ''a'' + (''b'' + ''c'')? Given that addition is associative, the choice of definition is irrelevant. For any three numbers ''a'', ''b'', and ''c'', it is true that {{nowrap|1=(''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 root]]s, multiplication and division, but is given equal priority to subtraction.<ref>{{cite book |title=Taschenbuch der Mathematik |author-first1=Ilja Nikolaevič<!-- Nikolajewitsch --> |author-last1=Bronstein<!-- 1903–1976 --> |author-first2=Konstantin Adolfovič<!-- Adolfowitsch --> |author-last2=Semendjajew<!-- 1908–1988 --> |editor-first1=Günter |editor-last1=Grosche |editor-first2=Viktor |editor-last2=Ziegler<!-- 1922–1980--> |editor-first3=Dorothea |editor-last3=Ziegler |others=Weiß, Jürgen<!-- lector --> |translator-first=Viktor |translator-last=Ziegler |volume=1 |date=1987 |edition=23 |orig-year=1945 |publisher=[[Verlag Harri Deutsch]] (and [[B.G. Teubner Verlagsgesellschaft]], Leipzig) |location=Thun and Frankfurt am Main |language=de |chapter=2.4.1.1. |pages=115–120 |isbn=978-3-87144-492-0 |title-link=Bronstein and Semendjajew}}</ref>
+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 root]]s, multiplication and division, but is given equal priority to subtraction.
 ===Identity element===
 [[File:AdditionZero.svg|right|70px|thumb|5 + 0 = 5 with bags of dots]]
-Adding [[0 (number)|zero]] to any number, does not change the number; this means that zero is the [[identity element]] for addition, and is also known as the [[additive identity]]. In symbols, for every {{math|''a''}}, one has
+Adding zero to any number, does not change the number; this means that zero is the identity element for addition, and is also known as the additive identity. In symbols, for every {{math|''a''}}, one has
 :{{math|1=''a'' + 0 = 0 + ''a'' = ''a''}}.
-This law was first identified in [[Brahmagupta]]'s ''[[Brahmasphutasiddhanta]]'' in 628&nbsp;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 (mathematician)|Mahavira]] wrote, "zero becomes the same as what is added to it", corresponding to the unary statement {{math|1=0 + ''a'' = ''a''}}. In the 12th&nbsp;century, [[Bhāskara II|Bhaskara]] wrote, "In the addition of cipher, or subtraction of it, the quantity, positive or negative, remains the same", corresponding to the unary statement {{math|1=''a'' + 0 = ''a''}}.<ref>Kaplan pp. 69–71</ref>
+This law was first identified in Brahmagupta's ''Brahmasphutasiddhanta'' in 628&nbsp;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 {{math|1=0 + ''a'' = ''a''}}. In the 12th&nbsp;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 {{math|1=''a'' + 0 = ''a''}}.
 ===Successor===
-Within the context of integers, addition of [[1 (number)|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 function|successor]] of ''a''.<ref>Hempel, C.G. (2001). The philosophy of Carl G. Hempel: studies in science, explanation, and rationality. p. 7</ref> 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 ''b''th successor of ''a'', making addition iterated succession. For example, {{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.
+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''. 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 ''b''th 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 6, making 8 the 2nd successor of 6.
 ===Units===
-To numerically add physical quantities with [[units of measurement|units]], they must be expressed with common units.<ref>R. Fierro (2012) ''Mathematics for Elementary School Teachers''. Cengage Learning. Sec 2.3</ref> For example, adding 50&nbsp;milliliters to 150&nbsp;milliliters gives 200&nbsp;milliliters. However, if a measure of 5&nbsp;feet is extended by 2&nbsp;inches, the sum is 62&nbsp;inches, since 60&nbsp;inches is synonymous with 5&nbsp;feet. On the other hand, it is usually meaningless to try to add 3&nbsp;meters and 4&nbsp;square meters, since those units are incomparable; this sort of consideration is fundamental in [[dimensional analysis]].
+To numerically add physical quantities with units, they must be expressed with common units. For example, adding 50&nbsp;milliliters to 150&nbsp;milliliters gives 200&nbsp;milliliters. However, if a measure of 5&nbsp;feet is extended by 2&nbsp;inches, the sum is 62&nbsp;inches, since 60&nbsp;inches is synonymous with 5&nbsp;feet. On the other hand, it is usually meaningless to try to add 3&nbsp;meters and 4&nbsp;square meters, since those units are incomparable; this sort of consideration is fundamental in dimensional analysis.
 ==Performing addition==

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 ==Interpretations==
-Addition is used to model many physical processes. Even for the simple case of adding [[natural number]]s, there are many possible interpretations and even more visual representations.
+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===
@@ Line 44: / Line 44: @@
 * When two or more disjoint collections are combined into a single collection, the number of objects in the single collection is the sum of the numbers of objects in the original collections.
-This interpretation is easy to visualize, with little danger of ambiguity. It is also useful in higher mathematics (for the rigorous definition it inspires, see {{Section link||Natural numbers}} below). However, it is not obvious how one should extend this version of addition to include fractional numbers or negative numbers.<ref>See Viro 2001 for an example of the sophistication involved in adding with sets of "fractional cardinality".</ref>
+This interpretation is easy to visualize, with little danger of ambiguity. It is also useful in higher mathematics (for the rigorous definition it inspires). However, it is not obvious how one should extend this version of addition to include fractional numbers or negative numbers.
-One possible fix is to consider collections of objects that can be easily divided, such as pies or, still better, segmented rods.<ref>''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."</ref> Rather than solely combining collections of segments, rods can be joined end-to-end, which illustrates another conception of addition: adding not the rods but the lengths of the rods.
+One possible fix is to consider collections of objects that can be easily divided, such as pies or, still better, segmented rods. Rather than solely combining collections of segments, rods can be joined end-to-end, which illustrates another conception of addition: adding not the rods but the lengths of the rods.
 ===Extending a length===
@@ Line 52: / Line 52: @@
 [[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.<ref>Mosley, F. (2001). ''Using number lines with 5–8 year olds''. Nelson Thornes. p. 8</ref>
+* When an original length is extended by a given amount, the final length is the sum of the original length and the length of the extension.
-The sum ''a'' + ''b'' can be interpreted as a [[binary operation]] 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''.<ref>Li, Y., & [[Glenda Lappan|Lappan, G.]] (2014). ''Mathematics curriculum in school education''. Springer. p. 204</ref> 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''.
+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''. 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==

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 [[File:AdditionNombryng.svg|left|thumb|Redrawn illustration from ''The Art of Nombryng'', one of the first English arithmetic texts, in the 15th century.]]
-"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 Greece|ancient Greeks]] and [[Ancient Rome|Romans]] to add upward, contrary to the modern practice of adding downward, so that a sum was literally higher than the addends.
+"Sum" and "summand" derive from the Latin noun ''summa'' "the highest, the top" and associated verb ''summare''. This is appropriate not only because the sum of two positive numbers is greater than either, but because it was common for the ancient Greeks and Romans to add upward, contrary to the modern practice of adding downward, so that a sum was literally higher than the addends.
-''Addere'' and ''summare'' date back at least to [[Anicius Manlius Severinus Boethius|Boethius]], if not to earlier Roman writers such as [[Vitruvius]] and [[Sextus Julius Frontinus|Frontinus]]; Boethius also used several other terms for the addition operation. The later [[Middle English]] terms "adden" and "adding" were popularized by [[Geoffrey Chaucer|Chaucer]].
+''Addere'' and ''summare'' date back at least to Boethius, if not to earlier Roman writers such as Vitruvius and Frontinus; Boethius also used several other terms for the addition operation. The later Middle English terms "adden" and "adding" were popularized by Chaucer.
-The [[plus sign]] "+" ([[Unicode]]:U+002B; [[ASCII]]: <code>&amp;#43;</code>) is an abbreviation of the Latin word ''et'', meaning "and". It appears in mathematical works dating back to at least 1489.
+The plus sign "+" (Unicode:U+002B; ASCII: <code>&amp;#43;</code>) is an abbreviation of the Latin word ''et'', meaning "and". It appears in mathematical works dating back to at least 1489.
 ==Interpretations==

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 Some authors call the first addend the ''augend''. In fact, during the Renaissance, many authors did not consider the first addend an "addend" at all. Today, due to the commutative property of addition, "augend" is rarely used, and both terms are generally called addends.
-All of the above terminology derives from Latin. "Addition" and "add" are English words derived from the Latin verb ''addere'', which is in turn a compound of ''ad'' "to" and ''dare'' "to give", from the Proto-Indo-European root {{PIE|''*deh₃-''}} "to give"; thus to ''add'' is to ''give to''. Using the gerundive suffix ''-nd'' results in "addend", "thing to be added". Likewise from ''augere'' "to increase", one gets "augend", "thing to be increased".
+All of the above terminology derives from Latin. "Addition" and "add" are English words derived from the Latin verb ''addere'', which is in turn a compound of ''ad'' "to" and ''dare'' "to give", from the Proto-Indo-European root ''*deh₃-'' "to give"; thus to ''add'' is to ''give to''. Using the gerundive suffix ''-nd'' results in "addend", "thing to be added". Likewise from ''augere'' "to increase", one gets "augend", "thing to be increased".
 [[File:AdditionNombryng.svg|left|thumb|Redrawn illustration from ''The Art of Nombryng'', one of the first English arithmetic texts, in the 15th century.]]
-"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 Greece|ancient Greeks]] and [[Ancient Rome|Romans]] to add upward, contrary to the modern practice of adding downward, so that a sum was literally higher than the addends.<ref>Schwartzman (p. 212) attributes adding upwards to the [[Ancient Greece|Greeks]] and [[Ancient Rome|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.</ref>
+"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 Greece|ancient Greeks]] and [[Ancient Rome|Romans]] to add upward, contrary to the modern practice of adding downward, so that a sum was literally higher than the addends.
-''Addere'' and ''summare'' date back at least to [[Anicius Manlius Severinus Boethius|Boethius]], if not to earlier Roman writers such as [[Vitruvius]] and [[Sextus Julius Frontinus|Frontinus]]; Boethius also used several other terms for the addition operation. The later [[Middle English]] terms "adden" and "adding" were popularized by [[Geoffrey Chaucer|Chaucer]].<ref>Karpinski pp. 150–153</ref>
+''Addere'' and ''summare'' date back at least to [[Anicius Manlius Severinus Boethius|Boethius]], if not to earlier Roman writers such as [[Vitruvius]] and [[Sextus Julius Frontinus|Frontinus]]; Boethius also used several other terms for the addition operation. The later [[Middle English]] terms "adden" and "adding" were popularized by [[Geoffrey Chaucer|Chaucer]].
-The [[plus sign]] "+" ([[Unicode]]:U+002B; [[ASCII]]: <code>&amp;#43;</code>) is an abbreviation of the Latin word ''et'', meaning "and".<ref>{{cite book |last=Cajori |first=Florian |title=A History of Mathematical Notations, Vol. 1 |url=https://archive.org/details/in.ernet.dli.2015.200372 |year=1928 |publisher=The Open Court Company, Publishers |chapter=Origin and meanings of the signs + and -}}</ref> It appears in mathematical works dating back to at least 1489.<ref name="OED">{{OED|plus}}</ref>
+The [[plus sign]] "+" ([[Unicode]]:U+002B; [[ASCII]]: <code>&amp;#43;</code>) is an abbreviation of the Latin word ''et'', meaning "and". It appears in mathematical works dating back to at least 1489.
 ==Interpretations==

@@ Line 1: / Line 1: @@
 [[File:Addition01.svg|right|thumb|120px|3 + 2 = 5 with apples, a popular choice in textbooks]]
-'''Addition''' (usually signified by the plus symbol {{char|+}}) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or ''sum'' of those values combined. The example in the adjacent image shows a combination of three apples and two apples, making a total of five apples. This observation is equivalent to the mathematical expression "3 + 2 = 5" (that is, "3 ''plus'' 2 is equal to 5").
+'''Addition''' (usually signified by the plus symbol +) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or ''sum'' of those values combined. The example in the adjacent image shows a combination of three apples and two apples, making a total of five apples. This observation is equivalent to the mathematical expression "3 + 2 = 5" (that is, "3 ''plus'' 2 is equal to 5").
 Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers and complex numbers. Addition belongs to arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects such as vectors, matrices, subspaces and subgroups.
-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 {{num|0}} does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication.
+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.

@@ Line 234: / Line 234: @@
 ===Arithmetic===
-Subtraction can be thought of as a kind of addition—that is, the addition of an additive inverse. Subtraction is itself a sort of inverse to addition, in that adding {{mvar|x}} and subtracting {{mvar|x}} are [[inverse function]]s.
+Subtraction can be thought of as a kind of addition—that is, the addition of an additive inverse. Subtraction is itself a sort of inverse to addition, in that adding {{mvar|x}} and subtracting {{mvar|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.
-Multiplication can be thought of as repeated addition. If a single term {{mvar|x}} appears in a sum ''n'' times, then the sum is the product of ''n'' and {{mvar|x}}. If ''n'' is not a natural number, the product may still make sense; for example, multiplication by {{num|−1}} yields the additive inverse of a number.
+Multiplication can be thought of as repeated addition. If a single term {{mvar|x}} appears in a sum ''n'' times, then the sum is the product of ''n'' and {{mvar|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.
 [[File:Csl.JPG|thumb|A circular slide rule]]