Definition of Hyperintegers

When x varies over the hyperreal numbers, [x] is the greatest hyperinteger y such that y ≤ x. Because of the Transfer Principle, every hyperreal number x is between two hyperintegers [x] and [x] + 1,

[x] ≤ x < [x] + 1.

Also, sums, differences, and products of hyperintegers are again hyperintegers.

We are now going to use the hyperintegers. In sketching curves we divided a closed interval [a, b] into finitely many subintervals. For theoretical purposes in the calculus we often divide a closed interval into a finite or infinite number of equal subintervals. This is done as follows.