---
tag: 1ZBY4W4E81
kind: concept
name: Standard part function
msc: ["26E35", "03H05"]
sources:
  - url: https://en.wikipedia.org/wiki/Standard_part_function
same_as:
  wikidata: Q4439341
  wikipedia: Standard part function
---

## Summary

The standard part function sends each finite [[hyperreal number|421QMTFG9T]] $x$ to the unique [[real number|08J2M1JYAG]] $\operatorname{st}(x)$ that is infinitely close to it.
It is the bridge back from the hyperreals to the reals in [[nonstandard analysis|WM62GKCBWN]].
Derivatives, integrals and limits are defined as standard parts of hyperreal expressions.

## Explanations

### L1 human

A finite hyperreal number is a real number plus an infinitely small error.
The standard part throws away the error and keeps the real number.
For example, if $\varepsilon$ is a positive infinitesimal, then $\operatorname{st}(3 + \varepsilon) = 3$ and $\operatorname{st}(\varepsilon^2 - 5\varepsilon) = 0$.
An infinite hyperreal such as $1/\varepsilon$ has no standard part, because no real number is close to it.