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Home Real and Hyperreal Numbers Standard Parts Theorem 2:
 
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Theorem 2:

THEOREM 2

Assume a ≈ b. Then

(i) If a is infinitesimal, so is b.

(ii) If a is finite, so is b.

(iii) If a is infinite, so is b.

The real numbers are sometimes called "standard" numbers, while the hyperreal numbers that are not real are called "nonstandard" numbers. For this reason, the real number that is infinitely close to b is called the "standard part" of b. An infinite number cannot have a standard part, because it can't be infinitely close to a finite number (Theorem 2). Our third principle (stated next) on hyperreal numbers is that every finite number has a standard part.
Home Real and Hyperreal Numbers Standard Parts Theorem 2:

Last Update: 2006-11-05