Sums and Differences
Sums and differences are the results obtained by applying the operations of addition and subtraction, respectively, to particular mathematical objects. For the sake of clarity, the discussion is divided into two parts, with sums examined first, followed by differences.
A sum is defined as the result obtained by the addition of two or more quantities or expressions. Students first encounter sums in arithmetic, and early on memorize the sums of various whole numbers. Beyond the whole numbers of elementary school arithmetic, one can consider the sums of integers, rational numbers, real numbers and complex numbers; and (following the applicable rules) the sums of vectors, matrices, tensors, and other higher mathematical objects.
The process of determining a sum is called summation (or addition). The word sum originally came from Latin for "summa" (highest) and later became the word "summe" in Middle English. An early use of the mathematical meaning of the word sum came in the form of "some" within Nicolas Chuquet's 1484 document Triparty en la Science des Nombres . A sum is represented in its simplest form by a "+" sign between two variables; that is, "a + b" where "+" is read plus.
Besides the "+" sign, the operation of summation (or addition) can also be represented by the Greek capital letter sigma (), being especially useful when the addition of many terms is necessary. The sigma notation, which corresponds to the English letter S, was introduced to facilitate the writing of these sums. Swiss mathematician Leonhard Euler (1707-1783) first used the summation symbol () in 1755. A summation using sigma notation is of the form: " ai = a1 + a2 + a3 + ... + an - 1 + an", where "i" is called the index of summation and "n" is called the upper limit, where both "i" and "n" possess only integer values, and where n i.
A particular type of sigma summation that is of great importance in mathematics is called a series. A series is a sequence of quantities, called terms, in which the relationship between consecutive terms is the same. There are different types of series. An arithmetic series is a series in which the difference between successive terms is a constant, commonly called the common difference. Another type of series is called a geometric series and is one for which each term equals the previous term multiplied by a constant, commonly called the common ratio. Series can be used to find the value of constants like pi () and Euler's number "e", 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. Important examples of infinite series are the binomial series, Taylor's series, and Maclaurin's series. Because an infinite series means that an infinite number of terms are to be summed together, in practice an approximation is made by truncating the number of terms to be summed. In other words, in the case of 'real-world' problems that are encountered in physics and engineering, the upper limit is changed from infinity to some integer value "n".
A last example of sums is defined by integral calculus. The definite integral can be thought of as the limiting value of a sum. In practice, the definite integral 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 is used to find solutions to many other problems in physics, chemistry, and engineering.
Difference is defined in arithmetic and mathematics as the result obtained by subtracting one quantity or expression from another. The process of determining a difference is subtraction. The difference is represented in its most general form by two letter symbols with a "-" sign in between, that is "a - b" where "-" is read "minus". The word difference originally came from the Latin word "differentia" (meaning difference or diversity).
The term difference is often used to mean sequences of differences. Perhaps this concept of sequences of differences is best developed through use of an example. Take, for instance, the set of positive integers {1, 2, 3, 4, 5, 6,...}. Squaring each term produces the sequence {1, 4, 9, 16, 25, 36,...}; called the "original sequence". Starting from the original sequence, the "sequence of first differences" is the sequence: {3, 5, 7, 9, 11,...}. In a sequence of first differences each term is the difference between two successive terms in the original sequence; in this case, "3 = 4 - 1", "5 = 9 - 4", "7 = 16 - 9", and so on. Continuing along this line of reasoning, one may consider differences between consecutive terms of the sequence {3, 5, 7, 9, 11,...} (remembering that this sequence is called a "first differences" sequence because it was derived from differences of the original sequence {1, 4, 9, 16, 25, 36,...}). Differences of the sequence {3, 5, 7, 9, 11,...} are called "second differences", and the resulting "second differences" sequence is {2, 2, 2, 2, 2, 2,...}. Sequences of third, fourth, or nth sequences are possible depending upon the original sequence. The study of these sequences of differences is called the calculus of finite differences. In a practical sense, certain calculating machines, particularly difference engines, are based on the principle of differences.
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