In the simplest sense, addition is a process of combining groups of things to form a new, larger group. Consider two baskets of apples: if apples are transferred from one basket into the other, a new bunch of apples is formed that is understood to be larger than either of the original two bunches. This process lead ancient peoples to create what is probably the earliest branch of mathematics, namely arithmetic. Eventually people abstracted from groups of objects (like bunches of apples) the concept of number that represents the quantity of objects in a particular group (i.e., five apples). In this process, the meaning of addition transformed from "combining groups of objects" to "combining the numbers associated with groups". Simply, the addition of numbers allowed for the counting of objects by increments greater than one. For instance, in order to add together two apples and three apples, all could be placed into a bunch and then counted individually from 1 to 5. However, arithmetic can more efficiently represent this procedure as "2 + 3 = 5". The concept of addition thus allowed people to determine the combined number of groups of things more readily, especially useful for larger quantities.
Addition is one of the four basic operations of arithmetic (the others being subtraction, multiplication, and division). Addition operates on the set of natural numbers {0, 1, 2, 3,...} such that for any two numbers added together, a (unique) third number is determined. For the expression "2 + 3" there is a unique natural number assigned, namely "5" (called the sum). For "2 + 3", the operation of addition is denoted by "+" (or "plus" sign), while the pair of natural numbers "2" and "3" are called the summands. Historically, the word addition is derived from the Latin "additio", used by Italian mathematician Leonardo Pisano Fibonacci (1170-1250). The "+" symbol appeared in print in 1489 under Johann Widman. However, Widman used the symbol to indicate excess, and not as a mathematical operation. Dutch mathematician Vander Hoeche is known to have used "+" as an algebraic operator in 1514.
The properties of addition in relation to the natural numbers can be expressed in the form of the following four laws that hold for any natural numbers "a" and "b":
"a + b" is a natural number, closure law.
"a + b = b + a", commutative law.
"a + (b + c) = (a + b) + c", associative law.
"a + 0 = a", identity law.
It is important to realize that the ancient peoples empirically determined these addition laws, like the number system itself. As the whole number system was expanded to include fractions, negative numbers, irrational numbers, etc., culminating in the construction of the real number system, the addition laws were found to still hold true. Thus, the four addition laws are just as valid when adding integers, fractions, or real numbers. With the advent of negative numbers, the inverse law of addition was added to these laws; so that for every real number "a" there exists a corresponding unique real number "-a" such that "a + (-a) = 0".
Particular rules have also been developed to aid the addition of particular types of numbers. For example, the addition of fractions requires a common denominator. To add the rational numbers "a / b" and "c / d", the unit fraction "d / d" is multiplied into the first term, and "b / b" into the second term, resulting in "ad / bd" and "cb / bd". With a common denominator "bd", the two can be added together "(ad / bd) + (cb / bd)" to result in "(ad + cb) / bd". As an example, "(1 / 2) + (2 / 5) = (1/ 2) ⋅ (5 / 5) + (2 / 5) ⋅ (2 / 2) = (5 / 10) + (4 / 10) = 9 / 10".
Addition can also be defined as an operation upon mathematical objects such as matrices, vectors, complex numbers, and functions. Some of the rules governing the operation of addition on these quantities are now laid out.
The addition of matrices is valid only if the matrices to be added are of the same order (i.e., contain the same number of rows and columns). Adding together the corresponding elements of each matrix performs addition on the matrices. The resulting matrix is of the same order as the summed matrices.
The addition of vectors is valid when corresponding components of vectors are added to form a resulting vector. Specifically, adding the vectors "a = axi + ayj" and "b = bxi + byj" results in the vector "r = a + b = (ax + bx)i + (ay + by)j", where i and j are unit vectors in the x-direction and y-direction (respectively) of the x-y plane, and ax, ay, bx, and by are scalar components of vectors a and b (respectively).
Complex numbers are expressed with a real part followed by an imaginary part. The operation of addition is defined as adding together the individual real parts and the individual imaginary parts. Adding the complex numbers "x = a + i(b)" and "y = c + i(d)" results in the sum "x + y = (a + c) + i(b + d)". For example, the sum of "2 + i(3)" and "1 + i(6)" is "(2 + 1) + i(3 + 6) = 3 + i(9)".
The addition of functions is accomplished by simply adding common terms. For example, the functions "f(x) = x2 + 3x - 4" and "g(x) = x3 - 2x2 + x + 3" are summed to "f(x) + g(x) = (x2 + 3x - 4) + (x3 - 2x2 + x + 3) = (x3 - x2 + 4x - 1)".
One of the triumphs of modern mathematics was the construction of a theory dealing with collections (or sets) of things, called set theory. Formulated in the late 1800s by German mathematician Georg Cantor (1845-1918), set theory has penetrated nearly every branch of mathematics. Early in the twentieth century it was shown that much of mathematics, including the natural numbers, were derivable from the concepts of set theory. Under set theory, the operation of addition on the natural numbers can be recast as the combining, or union, of sets, and the various addition laws can therefore be derived.
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