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...