Additive Notation

Know this topic well? Help others and get FREE products!

Additive Notation

Addition is one of the four basic operations of arithmetic (the others being subtraction, multiplication, and division). In arithmetic, addition operates on the set of real numbers such that for any real numbers added together, another real number is uniquely determined. Additive notation is defined as the system of symbols used in the operation of addition to represent numbers and actions. For an expression like "2 + 3" the operation of addition is denoted by "+" (plus), while the numbers "2" and "3" are called summands. The familiar "+" sign has only been used to denote addition during the last four to five hundred years. Prior to that time other symbols or techniques were utilized. The "+" symbol appeared in print in 1489 under German mathematician Johann Widman (1462-1498) in his book Mercantile Arithmetic. However, Widman used the symbol to indicate excesses in business dealings, not as a mathematical operation. The "+" symbol is known to have been used as an algebraic operator by Dutch mathematician Vander Hoecke in 1514, but it is believed the symbol was used that way even earlier.

As described previously, addition on the real numbers is an operation that takes a pair of numbers and yields a third unique number; stated symbolically "a + b = c". Thus, addition is defined as a binary procedure (the sum of two values represents a third). For example, one could sum "a₁ + a₂ + a₃ + a₄ + a₅ + ... + a_n", where each subscripted "a" represents a different number and "n" terms are added. A compact notation for representing a long expression of added terms is the summation symbol, denoted by the Greek capital sigma (). The sigma notation was introduced to facilitate the writing of sums. Swiss mathematician Leonhard Euler (1707-1783) first used the summation symbol () in 1755. It was later used by French mathematician Joseph-Louis Lagrange (1736-1813) but was largely ignored during the eighteenth century. A typical summation could be of the form where "i" is the index of summation, "n" is the upper limit, and both symbols have only integer values. The summation can be written out explicitly as the sum of "n" terms. For example, to indicate the combined weights of 20 boxes one could write "total weight = a + b + c + ... + t", where each letter stands for the weight of a different box. One could also write "total weight = x_i = x₁+ x₂+ x₃ + ... + x₂₀", where each "x_i" represents an individual weight.

Most often in mathematics "" is used in the expression of a series. A series is a sequence of quantities, called terms, in which the relationship between consecutive terms is the same. Series can be used to find the value of constants like , and to construct tables of logarithms and trigonometric functions. When the upper bound of a summation is infinity, the result is called an infinite series.

Another additive notation involves the integral sign "". A series " f_k(x)", like the one above, can be extended to calculate an area under a curve by summing the areas of rectangles that together closely approximate the area under the curved region. An area is reached that is defined as its limit by reducing the size of the rectangles and increasing their numbers toward infinity (called the limit). This limit is denoted " f_k(x) x_k = f(x) dx"; where the curve is continuous on the closed interval [a, b], the area is contained under the curve, above lines of reference, and between the closed boundary, and the area is divided into n rectangles that approach infinity. This definite integral is thus the limiting value of a sum. It is used to evaluate the length of and the area under plane curves, the area of surfaces of revolution, the volume of solids of revolution, and for other problems.

The notation laid out in the preceding paragraphs use expressions whose values are numbers, particularly real numbers. The same notation can also be applied to the addition of complex numbers, vectors, matrices, etc. That is, addition is defined as an operation on these sets of mathematical objects, which means that a pair of elements of some set (e.g., the complex numbers) yields a third unique element of the set. The sigma and integral notation may be applied to these various mathematical objects; for example, within complex analysis and vector calculus.

In the nineteenth and twentieth centuries, an abstraction of algebra took place that possessed enormous implications for algebra and for mathematics in general. The use of additive notation within abstract algebra and set theory was, as a result, very important. Within abstract algebra, the definition of an operation is generalized to encompass sets possessing any type of element. So one finds the addition operator "+", used in regards to sets known as groups and fields, all of whose elements conform to certain rules under the operation of addition.

In the late nineteenth century, Georg Cantor formulated set theory to deal with collections (or sets) of things. Since then 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. Additive notation within set theory relies on the symbol "" for the union of sets. For sets A and B, the union of A and B is represented with the symbol "", where "A B" is defined as "the set of all elements x such that x is an element of either set A or B, or both". For instance, if A = {a, b, c} and B = {c, d, e}, then "A B = {a, b, c, d, e}".