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65-411: 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 two columns of three apples and two apples each, totaling at five apples. This observation

130-613: 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. 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

195-552: 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. More recently, Asian elephants have demonstrated an ability to perform basic arithmetic. Typically, children first master counting . When given

260-417: 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 . Studies on mathematical development starting around the 1980s have exploited

325-422: A move that puts the opponent into check , while a double plus ++ is sometimes used to denote double check . Combinations of the plus and minus signs are used to evaluate a move (+/−, +/=, =/+, −/+). In linguistics, a superscript plus sometimes replaces the asterisk , which denotes unattested linguistic reconstruction . In botanical names , a plus sign denotes graft-chimaera . In Catholicism,

390-521: A pattern to be matched. For example, x+ means "one or more of the letter x". This is the Kleene plus notation. There is no concept of negative zero in mathematics, but in computing −0 may have a separate representation from zero. In the IEEE floating-point standard , 1 / −0 is negative infinity ( − ∞ {\displaystyle -\infty } ) whereas 1 / 0

455-503: A plus or minus to indicate the presence or absence of the Rh factor . For example, A+ means type A blood with the Rh factor present, while B− means type B blood with the Rh factor absent. In music, augmented chords are symbolized with a plus sign, although this practice is not universal (as there are other methods for spelling those chords). For example, "C+" is read "C augmented chord". Sometimes

520-682: A positive or negative charge of 1 (e.g., NH + 4   ). If the charge is greater than 1, a number indicating the charge is written before the sign (as in SO 2− 4   ). A plus sign prefixed to a telephone number is used to indicate the form used for International Direct Dialing . Its precise usage varies by technology and national standards. In the International Phonetic Alphabet , subscripted plus and minus signs are used as diacritics to indicate advanced or retracted articulations of speech sounds. The minus sign

585-603: 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

650-510: 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

715-429: 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 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

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780-447: 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 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

845-414: Is positive infinity ( ∞ {\displaystyle \infty } ). + is also used to denote added lines in diff output in the context format or the unified format . In physics, the use of plus and minus signs for different electrical charges was introduced by Georg Christoph Lichtenberg . In chemistry, superscripted plus and minus signs are used to indicate an ion with

910-493: Is a simplification of the Latin : et (comparable to the evolution of the ampersand & ). The − may be derived from a macron ◌̄ written over ⟨m⟩ when used to indicate subtraction; or it may come from a shorthand version of the letter ⟨m⟩ itself. In his 1489 treatise, Johannes Widmann referred to the symbols − and + as minus and mer (Modern German mehr ; "more"): "[...]

975-415: Is also used as tone letter in the orthographies of Dan , Krumen , Karaboro , Mwan , Wan , Yaouré , Wè , Nyabwa and Godié . The Unicode character used for the tone letter (U+02D7) is different from the mathematical minus sign. The plus sign sometimes represents / ɨ / in the orthography of Huichol . In the algebraic notation used to record games of chess , the plus sign + is used to denote

1040-403: Is also used in chemistry and physics . For more, see § Other uses . The minus sign ( − ) has three main uses in mathematics: In many contexts, it does not matter whether the second or the third of these usages is intended: −5 is the same number. When it is important to distinguish them, a raised minus sign ( ¯ ) is sometimes used for negative constants, as in elementary education ,

1105-458: 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 the most basic interpretation of addition lies in combining sets : This interpretation

1170-576: 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 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

1235-576: 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

1300-522: Is generally called "minus five degrees".) Further, a few textbooks in the United States encourage − x to be read as "the opposite of x " or "the additive inverse of x "—to avoid giving the impression that − x is necessarily negative (since x itself may already be negative). In mathematics and most programming languages, the rules for the order of operations mean that −5 is equal to −25 : Exponentiation binds more strongly than

1365-445: 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

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1430-424: 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 the decimal system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from

1495-413: Is the only character that resembles a minus sign or a dash so it is also used for these. The name hyphen-minus derives from the original ASCII standard, where it was called hyphen–(minus) . The character is referred to as a hyphen , a minus sign , or a dash according to the context where it is being used. A Jewish tradition that dates from at least the 19th century is to write plus using

1560-410: 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. Plus and minus signs#Plus sign The forms ⟨+⟩ and ⟨−⟩ are used in many countries around

1625-400: 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. 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

1690-405: 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

1755-475: 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 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

1820-478: 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 *deh₃- "to give"; thus to add is to give to . Using

1885-451: 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, 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

1950-400: The negative numbers ( +5 versus −5 ). The plus sign can also indicate many other operations, depending on the mathematical system under consideration. Many algebraic structures , such as vector spaces and matrix rings , have some operation which is called, or is equivalent to, addition. It is though conventional to use the plus sign to only denote commutative operations . The symbol

2015-419: 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 ( a + b ) + c = a + ( b + c ) . For example, (1 + 2) + 3 = 3 + 3 = 6 = 1 + 5 = 1 + (2 + 3) . When addition

Addition - Misplaced Pages Continue

2080-426: 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

2145-476: The ASCII hyphen-minus character, - . In APL a raised minus sign (here written using Unicode U+00AF MACRON) is used to denote a negative number, as in ¯3 . While in J a negative number is denoted by an underscore , as in _5 . In C and some other computer programming languages, two plus signs indicate the increment operator and two minus signs a decrement; the position of the operator before or after

2210-475: 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

2275-543: 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 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

2340-418: 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

2405-458: 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

2470-463: 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

2535-421: 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

2600-432: 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

2665-412: The first is made thus + and betokeneth more: the other is thus made − and betokeneth lesse." The plus sign ( + ) is a binary operator that indicates addition , as in 2 + 3 = 5 . It can also serve as a unary operator that leaves its operand unchanged ( + x means the same as x ). This notation may be used when it is desired to emphasize the positiveness of a number, especially in contrast with

Addition - Misplaced Pages Continue

2730-472: 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

2795-437: 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 the addition operation. The later Middle English terms "adden" and "adding" were popularized by Chaucer . The plus sign "+" ( Unicode :U+002B; ASCII : + )

2860-460: 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

2925-479: 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 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

2990-439: 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 . Addition is commutative , meaning that one can change

3055-412: The operation of subtraction. The same convention is also used in some computer languages. For example, subtracting −5 from 3 might be read as "positive three take away negative 5", and be shown as which can be read as: or even as When placed after a number, a plus sign can indicate an open range of numbers. For example, "18+" is commonly used as shorthand for "ages 18 and up". In US grading systems,

3120-480: 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

3185-468: 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 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,

3250-409: 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 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

3315-496: The plus is written as a superscript . As well as the normal mathematical usage, plus and minus signs may be used for a number of other purposes in computing. Plus and minus signs are often used in tree view on a computer screen—to show if a folder is collapsed or not. In some programming languages, concatenation of strings is written "a" + "b" , and results in "ab" . In most programming languages, subtraction and negation are indicated with

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3380-618: The plus sign before a last name denotes a Bishop , and a double plus is used to denote an Archbishop. Variants of the symbols have unique codepoints in Unicode: There is a commercial minus sign , ⁒ , which is used in Germany and Scandinavia. The symbol ÷ is used to denote subtraction in Scandinavia . The hyphen-minus symbol ( - ) is the form of hyphen most commonly used in digital documents . On most keyboards, it

3445-450: The plus sign indicates a grade one level higher and the minus sign a grade lower. For example, B− ("B minus") is one grade lower than B . In some occasions, this is extended to two plus or minus signs (e.g., A++ being two grades higher than A ). A common trend in branding, particularly with streaming video services, has been the use of the plus sign at the end of brand names, e.g. Google+ , Disney+ , Paramount+ and Apple TV+ . Since

3510-408: The programming language APL , and some early graphing calculators. All three uses can be referred to as "minus" in everyday speech, though the binary operator is sometimes read as "take away". In American English nowadays, −5 (for example) is generally referred to as "negative five" though speakers born before 1950 often refer to it as "minus five". (Temperatures tend to follow the older usage; −5°

3575-546: 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

3640-700: The reverse sign indicating subtraction: Nicole Oresme 's manuscripts from the 14th century show what may be one of the earliest uses of + as a sign for plus. In early 15th century Europe, the letters "P" and "M" were generally used. The symbols (P with overline, p̄ , for più (more), i.e., plus, and M with overline, m̄ , for meno (less), i.e., minus) appeared for the first time in Luca Pacioli 's mathematics compendium , Summa de arithmetica, geometria, proportioni et proportionalità , first printed and published in Venice in 1494. The + sign

3705-404: 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 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

3770-405: The same as what is added to it", corresponding to the unary statement 0 + a = a . In the 12th century, Bhaskara wrote, "In the addition of cipher, or subtraction of it, the quantity, positive or negative, remains the same", corresponding to the unary statement a + 0 = a . Within the context of integers, addition of one also plays a special role: for any integer a , the integer (

3835-593: The symbol ﬩ . This practice was adopted into Israeli schools and is still commonplace today in elementary schools (including secular schools) but in fewer secondary schools . It is also used occasionally in books by religious authors, but most books for adults use the international symbol + . The reason for this practice is that it avoids the writing of a symbol + that looks like a Christian cross . Unicode has this symbol at position U+FB29 ﬩ HEBREW LETTER ALTERNATIVE PLUS SIGN . Gerundive Too Many Requests If you report this error to

3900-456: The unary minus, which binds more strongly than multiplication or division. However, in some programming languages ( Microsoft Excel in particular), unary operators bind strongest, so in those cases −5^2 is 25, but 0−5^2 is −25. Similar to the plus sign, the minus sign is also used in chemistry and physics . For more, see § Other uses below. Some elementary teachers use raised minus signs before numbers to disambiguate them from

3965-401: The variable indicates whether the new or old value is read from it. For example, if x equals 6, then y = x++ increments x to 7 but sets y to 6, whereas y = ++x would set both x and y to 7. By extension, ++ is sometimes used in computing terminology to signify an improvement, as in the name of the language C++ . In regular expressions , + is often used to indicate "1 or more" in

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4030-656: The word "plus" can mean an advantage, or an additional amount of something, such "+" signs imply that a product offers extra features or benefits. Positive and negative are sometimes abbreviated as +ve and −ve . In mathematics the one-sided limit x → a means x approaches a from the right (i.e., right-sided limit), and x → a means x approaches a from the left (i.e., left-sided limit). For example, 1/ x → + ∞ {\displaystyle \infty } as x → 0 but 1/ x → − ∞ {\displaystyle \infty } as x → 0 . Blood types are often qualified with

4095-496: The world. Other designs include ⟨ ﬩ ⟩ for plus and ⟨ ⁒ ⟩ for minus. Though the signs now seem as familiar as the alphabet or the Hindu–Arabic numerals , they are not of great antiquity. The Egyptian hieroglyphic sign for addition, for example, resembled a pair of legs walking in the direction in which the text was written ( Egyptian could be written either from right to left or left to right), with

4160-485: 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,

4225-506: Was − ist das ist minus [...] und das + das ist mer das zu addirst" . They were not used for addition and subtraction in the treatise, but were used to indicate surplus and deficit; usage in the modern sense is attested in a 1518 book by Henricus Grammateus . Robert Recorde , the designer of the equals sign , introduced plus and minus to Britain in 1557 in The Whetstone of Witte : "There be other 2 signes in often use of which

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