The Hyperreal Numbers: Infinitesimals and the Standard Part

The heart of the NSA spine. With ${}^{\ast }\mathbb{R}$ built (ultrapower.md) and transfer in hand, we organize its elements into infinitesimal, finite, and infinite, and introduce the operation that ties the hyperreals back to ordinary analysis: the standard part $\mathrm{st}$, which rounds a finite hyperreal to the nearest real. Standard part is the rigorous form of the physicist's "drop the higher-order terms."

References: Goldblatt, Lectures on the Hyperreals, ch. 5–6; Keisler, Elementary Calculus, ch. 1; Robinson, Non-standard Analysis, ch. 3.

1. Infinitesimal, finite, infinite

Let $x\in {}^{\ast }\mathbb{R}$.

• $x$ is infinitesimal if $|x|<r$ for every real $r>0$. (So $0$ is the only real infinitesimal.) Write $x\approx 0$.
• $x$ is finite (or limited) if $|x|<r$ for some real $r>0$.
• $x$ is infinite (or unlimited) if $|x|>r$ for every real $r>0$.

Every hyperreal is exactly one of finite-non-infinitesimal, infinitesimal (a special finite one), or infinite. From ultrapower.md §4.1: $\epsilon =[\frac{1}{n}]$ is a positive infinitesimal, $\omega =[n]$ is positive infinite, and $\omega =1/\epsilon$. Two hyperreals are infinitely close, $x\approx y$, if $x-y\approx 0$.

Arithmetic of orders of magnitude. Writing $\inf$/$\mathrm{fin}$/$>\mathrm{infinite}$ for the three classes: $$\inf \pm \inf =\inf ,>\mathrm{fin}\cdot \inf =\inf ,>\mathrm{fin}\pm \mathrm{fin}=\mathrm{fin},>\frac{1}{\text{positive inf}}=\text{infinite},>\frac{1}{\text{infinite}}=\inf .$$ The indeterminate combinations are exactly the classical indeterminate forms: $\inf \cdot \mathrm{infinite}$ ($0\cdot \infty$), $\mathrm{infinite}\pm \mathrm{infinite}$ ($\infty -\infty$), and $\inf /\inf$ ($0/0$) — the last is precisely the derivative quotient, whose value is the content of differentiation (infinitesimal-calculus.md).

2. ${}^{\ast }\mathbb{R}$ is non-Archimedean, and why $\mathbb{R}$ is not complete inside it

The existence of a nonzero infinitesimal $\epsilon$ says directly that ${}^{\ast }\mathbb{R}$ fails the Archimedean property over $\mathbb{R}$: no real multiple $n\epsilon$ ($n\in \mathbb{N}$) exceeds $1$. This does not contradict the transferred Archimedean sentence, which quantifies over ${}^{\ast }\mathbb{N}$ and merely says every hyperreal is below some hypernatural (transfer-principle.md §3).

The finite hyperreals form a subring $\mathcal{O}\subset {}^{\ast }\mathbb{R}$ and the infinitesimals form an ideal $\mathfrak{m}\subset \mathcal{O}$ (a maximal ideal: $\mathcal{O}/\mathfrak{m}\cong \mathbb{R}$, which is §4). Neither is internal — as transfer-principle.md §4 shows, "the set of infinitesimals" and "the set of finite hyperreals" are external, so completeness/least-upper-bound reasoning does not apply to them.

3. Halos (monads) and galaxies

Two equivalence relations organize ${}^{\ast }\mathbb{R}$:

• The halo (or monad) of $x$ is $\mathrm{hal}(x)=\{y:y\approx x\}$ — everything infinitely close to $x$. Halos are the "infinitesimal neighborhoods." $\mathrm{hal}(0)=\mathfrak{m}$ is the infinitesimals.
• The galaxy of $x$ is $\mathrm{gal}(x)=\{y:x-y\text{ finite}\}$ — everything a finite distance away. $\mathrm{gal}(0)=\mathcal{O}$ is the finite hyperreals.

Picture ${}^{\ast }\mathbb{R}$ as $\mathbb{R}$ with, around each real $r$, a cloud $\mathrm{hal}(r)$ of hyperreals infinitely close to it, and, beyond all the finite hyperreals, further galaxies of infinite numbers (and their halos). The reals are the shadows these clouds cast, made precise next.

4. The standard part (shadow)

Standard Part Theorem. Every finite hyperreal $x$ is infinitely close to a unique real number, called its standard part (or shadow) $\mathrm{st}(x)={}^{\circ }x$: $$x\text{ finite}⟹\exists !r\in \mathbb{R}:x\approx r.$$

Proof. Let $A=\{s\in \mathbb{R}:s<x\}$. Since $x$ is finite, $A$ is nonempty and bounded above (by any real exceeding $|x|$), so by the completeness of $\mathbb{R}$ (real-numbers.md) $r:=\sup A$ exists. We claim $x\approx r$. If not, $|x-r|>t$ for some real $t>0$. If $x>r+t$, then $r+t\in A$ contradicts $r=\sup A$; if $x<r-t$, then $r-t$ is a smaller upper bound of $A$, again a contradiction. So $x\approx r$. Uniqueness: two reals both infinitely close to $x$ are infinitely close to each other, hence equal (the only real infinitesimal is $0$). $\square$

The map $\mathrm{st}:\mathcal{O}\to \mathbb{R}$ is exactly the quotient $\mathcal{O}\twoheadrightarrow \mathcal{O}/\mathfrak{m}\cong \mathbb{R}$ of §2: it is a ring homomorphism onto $\mathbb{R}$ with kernel the infinitesimals.

4.1 Rules for the standard part

For finite $x,y$ (and only these), $\mathrm{st}$ respects all the algebra and the weak order:

$$\mathrm{st}(x\pm y)=\mathrm{st}(x)\pm \mathrm{st}(y),\mathrm{st}(xy)=\mathrm{st}(x)\mathrm{st}(y),\mathrm{st}(x/y)=\mathrm{st}(x)/\mathrm{st}(y)(\mathrm{st}(y)\ne 0),$$

and $x\le y\Rightarrow \mathrm{st}(x)\le \mathrm{st}(y)$. Two caveats that mirror classical analysis:

• Strict order is not preserved. $x<y$ only gives $\mathrm{st}(x)\le \mathrm{st}(y)$: e.g. $0<\epsilon$ but $\mathrm{st}(0)=\mathrm{st}(\epsilon )=0$. (Compare: strict inequalities can degrade to $\le$ under limits.)
• Finiteness is required. $\mathrm{st}$ is undefined on infinite hyperreals; applying it there is the nonstandard face of the classical indeterminate forms of §1.

The physicist's "drop $dx$" made rigorous. When a physicist forms $\frac{f(x+dx)-f(x)}{dx}$ and then "sets $dx=0$ at the end," the honest operation is: compute the quotient as an exact hyperreal (with $dx$ a nonzero infinitesimal, so division is legal), then apply $\mathrm{st}$. Berkeley's "ghost of a departed quantity" is resolved because dividing and taking the shadow are two different, individually well-defined steps (motivation-history.md §3).

5. Worked micro-examples

• $\mathrm{st}(3+2\epsilon )=3$ for any infinitesimal $\epsilon$.
• $\mathrm{st}\left(\frac{(2+\epsilon {)}^2-4}{\epsilon }\right)=\mathrm{st}(4+\epsilon )=4$ — this is $\frac{d}{dx}{x}^2$ at $x=2$, computed with no limit, just infinitesimal algebra + shadow (infinitesimal-calculus.md).
• $\frac{1}{\omega }\approx 0$ for infinite $\omega$; $\mathrm{st}(1/\omega )=0$, recovering "$1/\infty =0$" honestly.

Where this page is used

• The standard part is the return map from ${}^{\ast }\mathbb{R}$ to $\mathbb{R}$ used in every definition of infinitesimal-calculus.md and nonstandard-integration.md.
• Halos underlie the nonstandard treatment of limits and compactness in sequences-topology.md.
• The external status of $\mathfrak{m}$, $\mathcal{O}$, $\mathbb{R}$ is spelled out in internal-sets.md.

Next: Internal and Hyperfinite Sets — the sets transfer can touch, and the hyperfinite sums integration needs.