Algebraic Representation
If is a decimal number with no more than decimal digits, the Heaviside step function can be represented by means of the following algebraic expression:
where and are arbitrary integers that satisfy, and is a Kronecker delta function.
For instance, if is integer, the simplest choice is:, . On the other hand, if belongs to a set of decimal numbers with decimal digits, the simplest choice is:, .
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“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)