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Hyperreal number
In non-standard analysis, a hyperreal number is used to handle infinite and infinitesimal quantities.
Explanation
The set of hyperreal numbers is denoted by the symbol *R. This set is an extension of R and is called the ∗-transform of R.
| • | Hypersmall number For every nonzero hypersmall number ε applies that it can be inverted and the result is the number ω = 1 / ε. |
| • | Hyperlarge number For the hyperlarge number ω applies that |ω| > m for all m ∈ N. If ω is positive we can compute m < √ω < ω / 2 < ω − 1 < ω < ω + 1 < 2ω < ω2We also have (ω + 1)·(ω − 1) = ω2 − 1 or (ω + 1) + (ω − 1) = 2ω. This is certainly not true for infinity ∞, which is not regarded a number at all. |
Examples
The various hyperreal numbers have special properties.
ε ≃ 0 The hypersmall number ε is infinitely close to zero.
δ ≈ 0 The hypersmall number δ is approximatively equal to zero, but is nonzero.
ω ~ ∞ The hyperlarge number ω is positive and has the same order of magnitude as infinity +∞. The difference between ω and ∞ is not hypersmall.
−ω ~ −∞ The hyperlarge number −ω is negative and has the same order of magnitude as infinity −∞. The difference between −ω and −∞ is not hypersmall.
HistoryThe German-American mathematician Abraham Robinson defined hyperreal numbers in the early 1960. |