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Elementary Calculus: An Infinitesimal approach (the subtitle is sometimes given as An approach using infinitesimals) is a textbook by H. Jerome Keisler. The subtitle alludes to the infinitesimal numbers of Abraham Robinson's non-standard analysis.



In his textbook, Keisler pioneered the pedagogical technique of an infinite-magnification microscope, so as to represent graphically, distinct hypperreal numbers infinitely close to each other. A typical pair of infinitely close hyperreals is represented in the figure.

When one examines a curve, say the graph of ƒ, under a magnifying glass, its curvature decreases proportionally to the magnification power of the lens. Similarly, an infinite-magnification microscope will transform an infinitesimal arc of a graph of ƒ, into a straight line, up to an infinitesimal error (only visible by applying a higher-magnification "microscope"). The derivative of ƒ is then the (standard part of the) slope of that line. Thus the microscope is a useful device in explaining the derivative.

Examples of a real statement

To provide a freshman-level explanation of the transfer principle, Keisler first gives a few examples of real statements to which the principle applies:

  • Closure law for addition: for any x and y, the sum x + y is defined.
  • Commutative law for addition: x + y = y + x.
  • A rule for order: if 0 < x < y then 0 < 1/y < 1/x.
  • Division by zero is never allowed: x/0 is undefined.
  • An algebraic identity: (xy)2 = x2 − 2xy + y2.
  • A trigonometric identity: sin2x + cos2x = 1.
  • A rule for logarithms: If x > 0 and y > 0, then log10(xy) = log10x + log10y.

Transfer principle

Every real statement that holds for one or more particular real functions holds for the hyperreal natural extensions of these functions.

See also




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