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Home Real and Hyperreal Numbers Infinitesimal, Finite and Infinite Numbers Hyperreals
 
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Hyperreals

In the next theorem we list facts about the ordering of the hyperreals.

THEOREM 1

(i) Every hyperreal number which is between two infinitesimals is infinitesimal.

(ii) Every hyperreal number which is between two finite hyperreal numbers is finite.

(iii) Every hyperreal number which is greater than some positive infinite number is positive infinite.

(iv) Every hyperreal number which is less than some negative infinite number is negative infinite.

All the proofs are easy. We prove (iii), which is especially useful. Assume H is positive infinite and H < K. Then for any real number r, r < H < K. Therefore, r < K and K is positive infinite.

Example 6

Our last example concerns square roots.

Example 7

Home Real and Hyperreal Numbers Infinitesimal, Finite and Infinite Numbers Hyperreals

Last Update: 2006-11-15