Significance of "terms" of An Empty Sum
Since an empty sum by definition has no terms, it seems contradictory to talk about its terms; however in practice there almost always is an expression that describes the terms of a summation, even if the range of summation happens to be empty. Since this expression is never instantiated in an empty sum, its value is irrelevant; for instance the harmonic number
is perfectly well defined. However, the kind of values denoted by the summand is of importance for the value of the summation; for instance, an empty summation of elements of a vector space has as value the zero vector in that space, rather than the number 0. Even more important is the fact that the operation is summation; by contrast, the empty product—a product of no factors at all—has as value of one.
Read more about this topic: Empty Sum
Famous quotes containing the words significance of, significance, terms, empty and/or sum:
“It is necessary not to be Christian to appreciate the beauty and significance of the life of Christ.”
—Henry David Thoreau (1817–1862)
“To grasp the full significance of life is the actor’s duty, to interpret it is his problem, and to express it his dedication.”
—Marlon Brando (b. 1924)
“Certainly for us of the modern world, with its conflicting claims, its entangled interests, distracted by so many sorrows, so many preoccupations, so bewildering an experience, the problem of unity with ourselves in blitheness and repose, is far harder than it was for the Greek within the simple terms of antique life. Yet, not less than ever, the intellect demands completeness, centrality.”
—Walter Pater (1839–1894)
“Nor for my words shall ye forget your tears,
Or hope again for aught that I can say,
The idle singer of an empty day.”
—William Morris (1834–1896)
“What God abandoned, these defended,
And saved the sum of things for pay.”
—A.E. (Alfred Edward)