62-440: Sum most commonly means the total of two or more numbers added together; see addition . Sum can also refer to: Addition 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
124-419: 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
186-420: 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 0 + a = a . In the 12th century, Bhaskara wrote, "In the addition of cipher, or subtraction of it, the quantity, positive or negative, remains
248-423: 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 . 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 The fact that addition
310-412: A box; the youngest responded well for small numbers, while older subjects were able to compute sums up to 5. Even some nonhuman animals show a limited ability to add, particularly primates . In a 1995 experiment imitating Wynn's 1992 result (but using eggplants instead of dolls), rhesus macaque and cottontop tamarin monkeys performed similarly to human infants. More dramatically, after being taught
372-407: 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 the order of the operands does not matter, and it is associative , meaning that when one adds more than two numbers, the order in which addition
434-461: 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
496-486: A noun phrase and then a prepositional phrase often led by to or for . For example: "The players gave their teammates high fives." "The players gave high fives to their teammates." When two noun phrases follow a transitive verb, the first is an indirect object, that which is receiving something, and the second is a direct object, that being acted upon. Indirect objects can be noun phrases or prepositional phrases. Double transitive verbs (sometimes called Vc verbs after
558-457: A screen demonstrated that five-month-old infants expect 1 + 1 to be 2, and they are comparatively surprised when a physical situation seems to imply that 1 + 1 is either 1 or 3. This finding has since been affirmed by a variety of laboratories using different methodologies. 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
620-511: A simple modification of the above process. One aligns two decimal fractions above each other, with the decimal point in the same location. If necessary, one can add trailing zeros to a shorter decimal to make it the same length as the longer decimal. Finally, one performs the same addition process as above, except the decimal point is placed in the answer, exactly where it was placed in the summands. As an example, 45.1 + 4.34 can be solved as follows: In scientific notation , numbers are written in
682-463: A state of being ( be , exist , stand ). In the usual description of English , the basic form, with or without the particle to , is the infinitive . In many languages , verbs are inflected (modified in form) to encode tense , aspect , mood , and voice . A verb may also agree with the person , gender or number of some of its arguments , such as its subject , or object . In English, three tenses exist: present , to indicate that an action
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#1732766227416744-458: A transitive verb can often drop its object and become intransitive; or an intransitive verb can take an object and become transitive. For example, in English the verb move has no grammatical object in he moves (though in this case, the subject itself may be an implied object, also expressible explicitly as in he moves himself ); but in he moves the car , the subject and object are distinct and
806-406: Is "understood", even though no symbol appears: The sum of a series of related numbers can be expressed through capital sigma notation , which compactly denotes iteration . For example, The numbers or the objects to be added in general addition are collectively referred to as the terms , the addends or the summands ; this terminology carries over to the summation of multiple terms. This
868-401: Is being carried out; past , to indicate that an action has been done; future , to indicate that an action will be done, expressed with the auxiliary verb will or shall . For example: Every language discovered so far makes a some form of noun -verb distinction, possibly because of the graph-like nature of communicated meaning by humans, i.e. nouns being the "entities" and verbs being
930-402: 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, such as subtraction and division, are not. 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
992-579: 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 . Studies on mathematical development starting around the 1980s have exploited the phenomenon of habituation : infants look longer at situations that are unexpected. A seminal experiment by Karen Wynn in 1992 involving Mickey Mouse dolls manipulated behind
1054-449: 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. Children are often presented with
1116-445: Is one that does not have a direct object. Intransitive verbs may be followed by an adverb (a word that addresses how, where, when, and how often) or end a sentence. For example: "The woman spoke softly." "The athlete ran faster than the official." "The boy wept ." A transitive verb is followed by a noun or noun phrase . These noun phrases are not called predicate nouns, but are instead called direct objects because they refer to
1178-564: Is performed does not matter. Repeated addition of 1 is the same as counting (see Successor function ). Addition of 0 does not change a number. Addition also obeys rules concerning related operations such as subtraction and multiplication. Performing addition is one of the simplest numerical tasks to do. Addition of very small numbers is accessible to toddlers; the most basic task, 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
1240-497: Is the ripple carry adder, which follows the standard multi-digit algorithm. One slight improvement is the carry skip design, again following human intuition; one does not perform all the carries in computing 999 + 1 , but one bypasses the group of 9s and skips to the answer. Verb A verb (from Latin verbum 'word') is part of speech that in syntax generally conveys an action ( bring , read , walk , run , learn ), an occurrence ( happen , become ), or
1302-429: Is the use of auxiliary verbs or inflections to convey whether the action or state is before, simultaneous with, or after some reference point. The reference point could be the time of utterance , in which case the verb expresses absolute tense , or it could be a past, present, or future time of reference previously established in the sentence, in which case the verb expresses relative tense . Aspect expresses how
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#17327662274161364-511: Is to be distinguished from factors , which are multiplied . 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
1426-403: 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. A second interpretation of addition comes from extending an initial length by a given length: The sum
1488-406: Is very similar to decimal addition. As an example, one can consider addition in binary. Adding two single-digit binary numbers is relatively simple, using a form of carrying: 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
1550-413: 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. To numerically add physical quantities with units , they must be expressed with common units. For example, adding 50 milliliters to 150 milliliters gives 200 milliliters. However, if a measure of 5 feet
1612-464: 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. 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, There are also situations where addition
1674-427: 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 . Addition is also fundamental to the operation of digital computers , where
1736-415: The "links" between them. In languages where the verb is inflected, it often agrees with its primary argument (the subject) in person, number or gender. With the exception of the verb to be , English shows distinctive agreements only in the third person singular, present tense form of verbs, which are marked by adding "-s" ( walk s ) or "-es" ( fish es ). The rest of the persons are not distinguished in
1798-539: 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". "Sum" and "summand" derive from the Latin noun summa "the highest,
1860-780: The action or state occurs through time. Important examples include: Aspect can either be lexical , in which case the aspect is embedded in the verb's meaning (as in "the sun shines", where "shines" is lexically stative), or it can be grammatically expressed, as in "I am running." Modality expresses the speaker's attitude toward the action or state given by the verb, especially with regard to degree of necessity, obligation, or permission ("You must go", "You should go", "You may go"), determination or willingness ("I will do this no matter what"), degree of probability ("It must be raining by now", "It may be raining", "It might be raining"), or ability ("I can speak French"). All languages can express modality with adverbs , but some also use verbal forms as in
1922-493: The addition operation. The later Middle English terms "adden" and "adding" were popularized by Chaucer . The plus sign "+" ( Unicode :U+002B; ASCII : + ) is an abbreviation of the Latin word et , meaning "and". It appears in mathematical works dating back to at least 1489. 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. Possibly
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1984-476: The addition table of pairs of numbers from 0 to 9 to memorize. 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: As students grow older, they commit more facts to memory, and learn to derive other facts rapidly and fluently. Many students never commit all
2046-463: The adjacent image shows two columns of three apples and two apples each, totaling at 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,
2108-418: The carry bits used. Starting in the rightmost column, 1 + 1 = 10 2 . The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. The second column from the right is added: 1 + 0 + 1 = 10 2 again; the 1 is carried, and 0 is written at the bottom. The third column: 1 + 1 + 1 = 11 2 . This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives
2170-458: 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
2232-414: The destination takes the active suffix -i (> mangai- ) in the intransitive form, and as a transitive verb the stem is not suffixed. The TAM ending -nu is the general today past attainative perfective, found with all numbers in the perfective except the singular active, where -ma is found. Depending on the language, verbs may express grammatical tense , aspect , or modality . Grammatical tense
2294-463: The efficiency of addition, in particular the carry mechanism, is an important limitation to overall performance. The abacus , also called a counting frame, is a calculating tool that was in use centuries before the adoption of the written modern numeral system and is still widely used by merchants, traders and clerks in Asia , Africa , and elsewhere; it dates back to at least 2700–2300 BC, when it
2356-429: 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
2418-423: 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 27 + 59 7 + 9 = 16, and the digit 1 is the carry. An alternate strategy starts adding from
2480-432: The final answer 100100 2 (36 10 ). 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
2542-472: The form x = a × 10 b {\displaystyle x=a\times 10^{b}} , where a {\displaystyle a} is the significand and 10 b {\displaystyle 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: Addition in other bases
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2604-413: The given examples. If the verbal expression of modality involves the use of an auxiliary verb, that auxiliary is called a modal verb . If the verbal expression of modality involves inflection, we have the special case of mood ; moods include the indicative (as in "I am there"), the subjunctive (as in "I wish I were there"), and the imperative ("Be there!"). The voice of a verb expresses whether
2666-440: The hardest she has ever completed." Copular verbs ( a.k.a. linking verbs) include be , seem , become , appear , look , and remain . For example: "Her daughter was a writing tutor." "The singers were very nervous." "His mother looked worried." "Josh remained a reliable friend." These verbs precede nouns or adjectives in a sentence, which become predicate nouns and predicate adjectives. Copulae are thought to 'link'
2728-483: The meanings of the Arabic numerals 0 through 4, one chimpanzee was able to compute the sum of two numerals without further training. More recently, Asian elephants have demonstrated an ability to perform basic arithmetic. 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
2790-408: The most basic interpretation of addition lies in combining sets : 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. One possible fix
2852-462: The most significant digit on the left; this route makes carrying a little clumsier, but it is faster at getting a rough estimate of the sum. There are many alternative methods. Since the end of the 20th century, some US programs, including TERC, decided to remove the traditional transfer method from their curriculum. This decision was criticized, which is why some states and counties did not support this experiment. Decimal fractions can be added by
2914-525: The norm. In the objective, the verb takes an object but no subject; the nonreferent subject in some uses may be marked in the verb by an incorporated dummy pronoun similar to that used with the English weather verbs. Impersonal verbs in null subject languages take neither subject nor object, as is true of other verbs, but again the verb may show incorporated dummy pronouns despite the lack of subject and object phrases. Verbs are often flexible with regard to valency. In non-valency marking languages such as English,
2976-422: The object that is being acted upon. For example: "My friend read the newspaper." "The teenager earned a speeding ticket." A way to identify a transitive verb is to invert the sentence, making it passive. For example: "The newspaper was read by my friend." "A speeding ticket was earned by the teenager." Ditransitive verbs (sometimes called Vg verbs after the verb give ) precede either two noun phrases or
3038-480: The operator had to use the Pascal's calculator's complement , which required as many steps as an addition. Giovanni Poleni followed Pascal, building the second functional mechanical calculator in 1709, a calculating clock made of wood that, once setup, could multiply two numbers automatically. Adders execute integer addition in electronic digital computers, usually using binary arithmetic . The simplest architecture
3100-586: The predicate adjective or noun to the subject. They can also be followed by an adverb of place, which is sometimes referred to as a predicate adverb. For example: "My house is down the street." The main copular verb be is manifested in eight forms be , is , am , are , was , were , been , and being in English. The number of arguments that a verb takes is called its valency or valence . Verbs can be classified according to their valency: Weather verbs often appear to be impersonal (subjectless, or avalent) in null-subject languages like Spanish , where
3162-547: The radix (10), the digit to the left is incremented: 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: In this example, two numerals are being added together: 01101 2 (13 10 ) and 10111 2 (23 10 ). The top row shows
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#17327662274163224-626: The relationship those words have with the verb itself. Classified by the number of their valency arguments, usually four basic types are distinguished: intransitives, transitives, ditransitives and double transitive verbs. Some verbs have special grammatical uses and hence complements, such as copular verbs (i.e., be ); the verb do used for do -support in questioning and negation; and tense or aspect auxiliaries, e.g., be , have or can . In addition, verbs can be non-finite (not inflected for person, number, tense, etc.), such special forms as infinitives , participles or gerunds . An intransitive verb
3286-407: The same", corresponding to the unary statement a + 0 = a . 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
3348-593: The standard order of operations, addition is a lower priority than exponentiation , nth roots , multiplication and division, but is given equal priority to subtraction. 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 a , one has This law was first identified in Brahmagupta 's Brahmasphutasiddhanta in 628 AD, although he wrote it as three separate laws, depending on whether
3410-564: The subject of the verb is performing the action of the verb or whether the action is being performed on the subject. The two most common voices are the active voice (as in "I saw the car") and the passive voice (as in "The car was seen by me" or simply "The car was seen"). Most languages have a number of verbal nouns that describe the action of the verb. In the Indo-European languages, verbal adjectives are generally called participles . English has an active participle, also called
3472-426: The subject—it is a strictly dependent-marking language . On the other hand, Basque , Georgian , and some other languages, have polypersonal agreement : the verb agrees with the subject, the direct object, and even the secondary object if present, a greater degree of head-marking than is found in most European languages. Verbs vary by type, and each type is determined by the kinds of words that accompany it and
3534-488: 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 at the top of the addends. Addere and summare date back at least to Boethius , if not to earlier Roman writers such as Vitruvius and Frontinus ; Boethius also used several other terms for
3596-467: The verb consider ) are followed by a noun phrase that serves as a direct object and then a second noun phrase, adjective, or infinitive phrase. The second element (noun phrase, adjective, or infinitive) is called a complement, which completes a clause that would not otherwise have the same meaning. For example: "The young couple considers the neighbors wealthy people." "Some students perceive adults quite inaccurately." "Sarah deemed her project to be
3658-511: The verb llueve means "It rains". In English, French and German, they require a dummy pronoun and therefore formally have a valency of 1. As verbs in Spanish incorporate the subject as a TAM suffix, Spanish is not actually a null-subject language, unlike Mandarin (see above). Such verbs in Spanish also have a valency of 1. Intransitive and transitive verbs are the most common, but the impersonal and objective verbs are somewhat different from
3720-534: The verb ( I walk , you walk , they walk , etc.). Latin and the Romance languages inflect verbs for tense–aspect–mood (abbreviated 'TAM'), and they agree in person and number (but not in gender, as for example in Polish ) with the subject. Japanese , like many languages with SOV word order, inflects verbs for tense-aspect-mood, as well as other categories such as negation, but shows absolutely no agreement with
3782-724: The verb has a different valency. Some verbs in English have historically derived forms that show change of valency in some causative verbs, such as fall-fell-fallen : fell-felled-felled ; rise-rose-risen : raise-raised-raised ; cost-cost-cost : cost-costed-costed . In valency marking languages, valency change is shown by inflecting the verb in order to change the valency. In Kalaw Lagaw Ya of Australia, for example, verbs distinguish valency by argument agreement suffixes and TAM endings: Verb structure: manga-i-[number]-TAM "arrive+active+singular/dual/plural+TAM" Verb structure: manga-Ø-[number]-TAM "arrive+attainative+singular/dual/plural+TAM" The verb stem manga- 'to take/come/arrive' at
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#17327662274163844-488: Was used in Sumer . Blaise Pascal invented the mechanical calculator in 1642; it was the first operational adding machine . It made use of a gravity-assisted carry mechanism. It was the only operational mechanical calculator in the 17th century and the earliest automatic, digital computer. Pascal's calculator was limited by its carry mechanism, which forced its wheels to only turn one way so it could add. To subtract,
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