addition

s, a popular choice in textbooks Addition, usually denoted with the plus sign , 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 or sum of those values combined. For example, the adjacent image shows two columns of apples, one with three apples and the other with two apples, totaling to five apples. This observation is expressed as , which is read as "three plus two equals five". 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, and elements of additive groups. Addition has several important properties. It is commutative, meaning that the order of the numbers being added does not matter, so 3 + 2 = 2 + 3. It is also associative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter. Repeated addition of is the same as counting (see Successor function). Addition of does not change a number.…

s, a popular choice in textbooks Addition, usually denoted with the plus sign , 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 or sum of those values combined. For example, the adjacent image shows two columns of apples, one with three apples and the other with two apples, totaling to five apples. This observation is expressed as , which is read as "three plus two equals five". 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, and elements of additive groups. Addition has several important properties. It is commutative, meaning that the order of the numbers being added does not matter, so 3 + 2 = 2 + 3. It is also associative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter. Repeated addition of is the same as counting (see Successor function). Addition of 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 perform. Addition of very small numbers is accessible to toddlers; the most basic task, , can be performed by infants as young as five months, and even by some members of other animal species. In primary education, students are taught to add numbers in the decimal system, beginning 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, and the result is expressed with an equals sign. For example, 1 + 2 = 3 reads "one plus two equals three". Nonetheless, there are some 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, with an example, 3\frac{1}{2}=3+\frac{1}{2}=3.5. This notation can cause confusion, since in most other contexts, juxtaposition denotes multiplication instead. The numbers or the objects to be added in general addition are collectively referred to as the , 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 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 , which is in turn a compound of "to" and "to give", from the Proto-Indo-European root "to give"; thus to add is to give to. Using the gerundive suffix -nd results in "addend", "thing to be added". Likewise from "to increase", one gets "augend", "thing to be increased". "Sum" and "summand" derive from the Latin noun "the highest" or "the top", used in Medieval Latin phrase ("top line") meaning the sum of a column of numerical quantities, following the ancient Greek and Roman practice of putting the sum at the top of a column. and 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. Addition is one of the four basic operations of arithmetic, with the other three being subtraction, multiplication, and division. This operation works by adding two or more terms. An arbitrary number of addition operations is called a summation. An infinite summation is a delicate procedure known as a series, and it can be expressed through capital sigma notation \sum , which compactly denotes iteration of the addition operation based on the given indexes. For example, \sum_{k=1}^5 k^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 55. 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. 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 operations are commutative too as in multiplication, but others are not as in subtraction and division. (a + b) + c = a + (b + c). For example, (1 + 2) + 3 = 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. 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, other 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 the same", corresponding to the unary statement a + 0 = a . === Single-digit addition === An ability to add a pair of single digits (numbers from 0 to 9) is a prerequisite for addition of arbitrary numbers in the decimal system. With 10 choices for each of the two digits to be added, this makes 100 single-digit "addition facts", which can be presented in an addition table. Learning to fluently and accurately compute single-digit additions is a major focus of early schooling in arithmetic. Sometimes students are encouraged to memorize the full addition table by rote, but pattern-based strategies are typically more enlightening and, for most people, more efficient: 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 by adding one more, or from but subtracting one. Five and ten: Sums of the form 5 + and 10 + are usually memorized early and can be used for deriving other facts. For example, can be derived from by adding one more. Making ten: An advanced strategy uses 10 as an intermediate for sums involving 8 or 9; for example, . As students gain experience with addition, they gradually memorize an increasing number of basic addition facts and become more fluent in recalling them. Repeated practice also helps students recognize relationships among addition facts, allowing them to derive sums that they do not immediately recall. For example, a student who knows that 6 + 6 = 12 can use this fact to determine that 6 + 7 = 13. Over time, commonly used facts and strategies such as doubles, near-doubles, making ten, and counting on can become increasingly automatic. This development of fluency allows students to calculate single-digit sums more quickly and accurately, while reducing the amount of deliberate mental effort required for familiar additions. The standard algorithm for adding multi-digit numbers is to align the addends vertically and add the columns by using the above addition table, starting from the ones column on the right. If the result of a column exceeds nine, the extra digit is "carried" into the next column. For example, in the following image, the ones in the addition of is 9 + 7 = 16, and the digit 1 is the carry. 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. In scientific notation, numbers are written in the form x=a\times10^{b} , where a is the significand and 10^{b} is the exponential part. To add numbers in scientific notation, they should be expressed with the same exponent, so that the two significands can simply be added. For example: \begin{align} &2.34\times10^{-5} + 5.67\times10^{-6} \\ &\quad = 2.34\times10^{-5} + 0.567\times10^{-5} \\ &\quad = 2.907\times10^{-5}. \end{align} \begin{align} 0 + 0 &\to 0 \\ 0 + 1 &\to 1 \\ 1 + 0 &\to 1 \\ 1 + 1 &\to 0, \qquad \text{carry 1 since } 1 + 1 = 2 = 0 + 1 \times 2^1 \end{align} 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: \begin{align} 5 + 5 &\to 0, \qquad \text{carry 1 since } 5 + 5 = 10 = 0 + 1 \times 10^1 \\ 7 + 9 &\to 6, \qquad \text{carry 1 since } 7 + 9 = 16 = 6 + 1 \times 10^1 \end{align} 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. Although this definition can be useful for concrete problems, the number of cases to consider complicates proofs unnecessarily. So the following method is commonly used for defining integers. It is based on the remark that every integer is the difference of two natural integers and that two such differences, a - b and c - d are equal if and only if a + d = b + c . So, one can define formally the integers as the equivalence classes of ordered pairs of natural numbers under the equivalence relation (a,b) \sim (c,d) if and only if a + d = b + c . The equivalence class of (a,b) contains either (a-b,0) if a \ge b , or (0,b-a) if otherwise. Given that n is a natural number, then one can denote +n the equivalence class of (n,0) , and by -n the equivalence class of (0,n) . This allows identifying the natural number n with the equivalence class +n . The addition of ordered pairs is done component-wise: (a,b) + (c,d) = (a+c, b+d). A straightforward computation shows that the equivalence class of the result depends only on the equivalence classes of the summands, and thus that this defines an addition of equivalence classes, that is, integers. Another straightforward computation shows that this addition is the same as the above case definition. \frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}. As an example, the sum \frac 34 + \frac 18 = \frac{3 \,\times\, 8 \,+\, 4 \,\times\, 1}{4 \times 8} = \frac{24 \,+\, 4}{32} = \frac{28}{32} = \frac78 . Addition of fractions is much simpler when the denominators are the same; in this case, one can simply add the numerators while leaving the denominator the same: \frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}, so \frac 14 + \frac 24 = \frac{1\,+\,2}{4} = \frac 34 . The commutativity and associativity of rational addition are easy consequences of the laws of integer arithmetic. a+b = \{q+r \mid q\in a, r\in b\}. This definition was first published, in a slightly modified form, by Richard Dedekind in 1872¹. The commutativity and associativity of real addition are immediate; defining the real number 0 as the set of negative rationals, it is easily seen as the additive identity. Probably the trickiest part of this construction pertaining to addition is the definition of additive inverses. Unfortunately, dealing with the multiplication of Dedekind cuts is a time-consuming case-by-case process similar to the addition of signed integers. Another approach is the metric completion of the rational numbers. A real number is essentially defined to be the limit of a Cauchy sequence of rationals, \lim a_n . Addition is defined term by term: \lim_n a_n + \lim_n b_n = \lim_n (a_n + b_n). This definition was first published by Georg Cantor, also in 1872, although his formalism was slightly different. One must prove that this operation is well-defined, dealing with co-Cauchy sequences. Once that task is done, all the properties of real addition follow immediately from the properties of rational numbers. Furthermore, the other arithmetic operations, including multiplication, have straightforward, analogous definitions. (a+bi) + (c+di) = (a+c) + (b+d)i. Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point X obtained by building a parallelogram three of whose vertices are O, A and B. (a,b) + (c,d) = (a+c,b+d). This addition operation is central to classical mechanics, in which velocities, accelerations and forces are all represented by vectors. Matrix addition is defined for two matrices of the same dimensions. The sum of two m × n (pronounced "m by n") matrices A and B, denoted by , is again an matrix computed by adding corresponding elements: \begin{align} \mathbf{A}+\mathbf{B} &= \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} & \cdots & b_{1n} \\ b_{21} & b_{22} & \cdots & b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ b_{m1} & b_{m2} & \cdots & b_{mn} \\ \end{bmatrix}\\[8mu] &= \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} & \cdots & a_{1n} + b_{1n} \\ a_{21} + b_{21} & a_{22} + b_{22} & \cdots & a_{2n} + b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} + b_{m1} & a_{m2} + b_{m2} & \cdots & a_{mn} + b_{mn} \\ \end{bmatrix} \\ \end{align} For example: \begin{align} \begin{bmatrix} 1 & 3 \\ 1 & 0 \\ 1 & 2 \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 7 & 5 \\ 2 & 1 \end{bmatrix} &= \begin{bmatrix} 1+0 & 3+0 \\ 1+7 & 0+5 \\ 1+2 & 2+1 \end{bmatrix}\\[8mu] &= \begin{bmatrix} 1 & 3 \\ 8 & 5 \\ 3 & 3 \end{bmatrix} \end{align} In modular arithmetic, the set of available numbers is restricted to a finite subset of the integers, and addition "wraps around" when reaching a certain value, called the modulus. For example, the set of integers modulo 12 has twelve elements; it inherits an addition operation from the integers that is central to musical set theory. The set of integers modulo 2 has just two elements; the addition operation it inherits is known in Boolean logic as the "exclusive or" function. A similar "wrap around" operation arises in geometry, where the sum of two angle measures is often taken to be their sum as real numbers modulo 2π. This amounts to an addition operation on the circle, which in turn generalizes to the operations of higher-dimensional Lie groups. The general theory of abstract algebra allows an "addition" operation to be any associative and commutative operation on a set. Basic algebraic structures with such an addition operation include commutative monoids and abelian groups. Linear combinations combine multiplication and summation; they are sums in which each term has a multiplier, usually a real or complex number. Linear combinations are especially useful in contexts where straightforward addition would violate some normalization rule, such as mixing of strategies in game theory or superposition of states in quantum mechanics. e^{a+b} = e^a e^b. This identity allows multiplication to be carried out by consulting a table of logarithms and computing addition by hand; it also enables multiplication on a slide rule. The formula is still a good first-order approximation in the broad context of Lie groups, where it relates multiplication of infinitesimal group elements with addition of vectors in the associated Lie algebra. There are even more generalizations of multiplication than addition. In general, multiplication operations always distribute over addition; this requirement is formalized in the definition of a ring. In some contexts, integers, distributivity over addition, and the existence of a multiplicative identity are enough to determine the multiplication operation uniquely. The distributive property also provides information about the addition operation; by expanding the product (1+1)(a+b) in both ways, one concludes that addition is forced to be commutative. For this reason, ring addition is commutative in general. Division is an arithmetic operation remotely related to addition. Since a/b = ab^{-1} , division is right distributive over addition: (a+b)/c = a/c + b/c . However, division is not left distributive over addition, such as 1/(2+2) is not the same as 1/2 + 1/2 . of and from = 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, 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 approximation becomes exact in a kind of infinite limit; if either a or b is an infinite cardinal number, their cardinal sum is exactly equal to the greater of the two. Accordingly, there is no subtraction operation for infinite cardinals. Maximization is commutative and associative, like addition. Furthermore, since addition preserves the ordering of real numbers, addition distributes over "max" in the same way that multiplication distributes over addition: a + \max(b,c) = \max(a+b,a+c). For these reasons, in tropical geometry one replaces multiplication with addition and addition with maximization. In this context, addition is called "tropical multiplication", maximization is called "tropical addition", and the tropical "additive identity" is negative infinity. Some authors prefer to replace addition with minimization; then the additive identity is positive infinity. Tying these observations together, tropical addition is approximately related to regular addition through the logarithm: \log(a+b) \approx \max(\log a, \log b), which becomes more accurate as the base of the logarithm increases. The approximation can be made exact by extracting a constant h , named by analogy with the Planck constant from quantum mechanics, and taking the "classical limit" as h tends to zero: \max(a,b) = \lim_{h\to 0}h\log(e^{a/h}+e^{b/h}). In this sense, the maximum operation is a dequantized version of addition. == Further reading==

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