Negative infinite and infinitesimal quantities are just the negatives of positive infinite and infinitesimal quantities. The numbers infinitely close to 0 are the infinitesimals. A positive infinitesimal is a positive quantity mathA/math that violates it the other way, by mathnA/math being less than math1/math for any positive integer mathn/math. About each real number c is a portion of the hyperreal line composed of the numbers infinitely close to c (shown under an infinitesimal microscope for c = 0 and c = 100). The set R of real numbers is scattered among the finite numbers. The circles represent "infinitesimal microscopes" which are powerful enough to show an infinitely small portion of the hyperreal line. Figure 1.4.3 shows a drawing of the hyperreal line. Hyperreal numbers which are not infinite numbers are called finite numbers. On the other hand, -1/ε will be an infinite negative number, i.e., a number less than every real number. 1/ε will be an infinite positive number, that is, it will be greater than any real number. If ε is positive infinitesimal, then -ε will be a negative infinitesimal. For example, if Δx is infinitesimal then x 0 + Δx is infinitely close to x 0. If a and b are hyperreal numbers whose difference a - b is infinitesimal, we say that a is infinitely close to b. and the Greek letters ε (epsilon) and δ (delta) will be used for infinitesimals. The infinitesimals in R* are of three kinds: positive, negative, and the real number 0. Every real number is a member of R*, but R* has other elements too. The set of all hyperreal numbers is denoted by R*. First we shall give an intuitive picture of the hyperreal numbers and show how they can be used to find the slope of a curve.
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