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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 numberFor every nonzero hypersmall number applies that it can be inverted and the result is the number εω = 1 / .ε |

• | Hyperlarge numberFor the hyperlarge number ω applies that |ω| > m for all m ∈ N. If ω is positive we can computeWe also have ( ω + 1)·(ω − 1) = ω^{2} − 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≃ 0The hypersmall number εis infinitely close to zero.

δ≈ 0The hypersmall number δis approximatively equal to zero, but is nonzero.

ω~ ∞The hyperlarge number ωis positive and has the same order of magnitude as infinity +∞. The difference betweenωand ∞ isnothypersmall.

−ω~ −∞The hyperlarge number − ωis negative and has the same order of magnitude as infinity −∞. The difference between −ωand −∞ isnothypersmall.

## HistoryThe German-American mathematician Abraham Robinson defined hyperreal numbers in the early 1960. |