The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined

Fix two objects $-\mathrm{\infty }$ and $+\mathrm{\infty }$, distinct from one another and neither of them a real number (The real numbers), and set

$\bar{\mathbb{R}}:=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}\mathrm{.}$

This is a new object, introduced here explicitly with its own order and its own partial arithmetic. It is not an enlargement of the field $\mathbb{R}$, and no operation of $\mathbb{R}$ (Complete ordered field (least-upper-bound property)) is redefined by anything below.

The order. For $a,b\in \bar{\mathbb{R}}$ declare

$a\le b:⟺a=-\mathrm{\infty }\text{ or }b=+\mathrm{\infty }\text{ or }(a,b\in \mathbb{R}\text{ and }a\le b\text{ in }\mathbb{R}),$

with $\mathbb{R}$ ordered as in Order on the reals, and write $a<b$ for "$a\le b$ and $a\mathrm{\ne }b$" as usual (Partial order and partially ordered set).

$(\bar{\mathbb{R}},\le )$ is a totally ordered set, and the inclusion of $\mathbb{R}$ preserves and reflects the order. All four checks are immediate from the displayed clauses.

• Reflexive. For $a=\pm \mathrm{\infty }$ one of the first two clauses applies; for $a\in \mathbb{R}$ the third does, since $a\le a$ in $\mathbb{R}$.
• Antisymmetric. Suppose $a\le b$ and $b\le a$. If $a=-\mathrm{\infty }$ then $b\le a$ forces $b=-\mathrm{\infty }$, since the clause $a=+\mathrm{\infty }$ fails and $b,a$ are not both real. Symmetrically $b=-\mathrm{\infty }$ forces $a=-\mathrm{\infty }$, and $a=+\mathrm{\infty }$ or $b=+\mathrm{\infty }$ forces the other to be $+\mathrm{\infty }$. In the one remaining situation $a$ and $b$ are both real and antisymmetry is that of $\mathbb{R}$.
• Transitive. Let $a\le b\le c$. If $a=-\mathrm{\infty }$ or $c=+\mathrm{\infty }$ the conclusion is one of the first two clauses. Otherwise $a\mathrm{\ne }-\mathrm{\infty }$ forces, in $a\le b$, either $b=+\mathrm{\infty }$ or $a,b\in \mathbb{R}$; and $c\mathrm{\ne }+\mathrm{\infty }$ forces, in $b\le c$, either $b=-\mathrm{\infty }$ or $b,c\in \mathbb{R}$. The value $b=+\mathrm{\infty }$ is incompatible with the second alternative pair, so $b$ is real, hence so are $a$ and $c$, and transitivity is that of $\mathbb{R}$.
• Total. If $a=-\mathrm{\infty }$ or $b=+\mathrm{\infty }$ then $a\le b$; if $b=-\mathrm{\infty }$ or $a=+\mathrm{\infty }$ then $b\le a$; otherwise both are real and the order of $\mathbb{R}$ is total.
• Preserved and reflected. For $a,b\in \mathbb{R}$ the first two clauses fail, so $a\le b$ in $\bar{\mathbb{R}}$ says exactly $a\le b$ in $\mathbb{R}$.

In particular $-\mathrm{\infty }$ is the least and $+\mathrm{\infty }$ the greatest element of $\bar{\mathbb{R}}$, and $-\mathrm{\infty }<x<+\mathrm{\infty }$ for every $x\in \mathbb{R}$.

Reflection. Extend negation by

$-(+\mathrm{\infty }):=-\mathrm{\infty },-(-\mathrm{\infty }):=+\mathrm{\infty },$

keeping the field negative on $\mathbb{R}$. The resulting map $\nu :\bar{\mathbb{R}}\to \bar{\mathbb{R}}$, $\nu (a)=-a$, satisfies $\nu (\nu (a))=a$ and

$a\le b⟺-b\le -a(a,b\in \bar{\mathbb{R}})\mathrm{.}$

For $a$ and $b$ real this is the elementwise order reversal in $\mathbb{R}$: translation invariance (Order is preserved by adding a constant and by adding inequalities) applied with the constant $-a-b$ turns $a<b$ into $-b<-a$ and, applied with the constant $a+b$, turns it back, while $a=b$ holds exactly when $-a=-b$. In every other case both sides are decided by the first two clauses of the order: $a=-\mathrm{\infty }$ makes both sides true, as does $b=+\mathrm{\infty }$, and if $a\mathrm{\ne }-\mathrm{\infty }$, $b\mathrm{\ne }+\mathrm{\infty }$ and $a,b$ are not both real then one of $a=+\mathrm{\infty }$, $b=-\mathrm{\infty }$ holds and both sides are false.

Partial addition. For $a,b\in \bar{\mathbb{R}}$ the sum $a+b$ is defined by

• $a+b$ = the field sum, when $a,b\in \mathbb{R}$;
• $a+b:=+\mathrm{\infty }$ when $a=+\mathrm{\infty }$ and $b\mathrm{\ne }-\mathrm{\infty }$, or $b=+\mathrm{\infty }$ and $a\mathrm{\ne }-\mathrm{\infty }$;
• $a+b:=-\mathrm{\infty }$ when $a=-\mathrm{\infty }$ and $b\mathrm{\ne }+\mathrm{\infty }$, or $b=-\mathrm{\infty }$ and $a\mathrm{\ne }+\mathrm{\infty }$;

and the two sums $(+\mathrm{\infty })+(-\mathrm{\infty })$ and $(-\mathrm{\infty })+(+\mathrm{\infty })$ are left undefined. Addition is commutative where defined, and

$-(a+b)=(-a)+(-b),$

each side being defined exactly when the other is: the excluded pairs $\{+\mathrm{\infty },-\mathrm{\infty }\}$ are exchanged by $\nu$, and the three clauses above are exchanged accordingly.

Partial multiplication. For $a,b\in \bar{\mathbb{R}}$ the product $ab$ is defined by

• $ab$ = the field product, when $a,b\in \mathbb{R}$;
• $ab:=+\mathrm{\infty }$ when one of $a,b$ is $\pm \mathrm{\infty }$, the other is $\mathrm{\ne }0$, and both are $>0$ or both are $<0$;
• $ab:=-\mathrm{\infty }$ when one of $a,b$ is $\pm \mathrm{\infty }$, the other is $\mathrm{\ne }0$, and one is $>0$ and the other $<0$;

and every product with one factor $0$ and the other $\pm \mathrm{\infty }$ is left undefined. The comparisons $>0$ and $<0$ here are taken in the order above, under which $+\mathrm{\infty }>0>-\mathrm{\infty }$.

Nothing else is defined. There is no subtraction, no division, no exponentiation and no absolute value on $\bar{\mathbb{R}}$ in this library; where such an expression is wanted it is written out in the two defined operations, and where a case falls in the undefined list the statement carries an explicit hypothesis saying so.

The real numbers

Complete ordered field (least-upper-bound property)

Order on the reals

Partial order and partially ordered set

Order is preserved by adding a constant and by adding inequalities

• $\bar{\mathbb{R}}$ is not a field, and not an ordered field. It has no additive inverse for $+\mathrm{\infty }$: $(+\mathrm{\infty })+b$ is $+\mathrm{\infty }$ whenever it is defined and is never $0$. So none of the field axioms (Complete ordered field (least-upper-bound property)) is available here, and no algebraic manipulation valid in $\mathbb{R}$ may be transported to $\bar{\mathbb{R}}$ without a separate justification.

• Why the excluded cases are excluded. The three defined clauses of each operation are exactly the cases in which the value is forced by the limiting behaviour of the sequences involved, and the excluded cases are exactly the ones in which it is not. For the product this is proved on the companion page: Null times divergent has no rule: $x_k=1\mathrm{/}k$ with $y_k=ck$ gives product limit $c$, and with $y_k=k^2$ gives divergence ↗ exhibits a null sequence and two sequences diverging to $+\mathrm{\infty }$ whose products behave differently, so no value assigned to $0\cdot (+\mathrm{\infty })$ could be compatible with products of limits. The same phenomenon rules out a value for $(+\mathrm{\infty })+(-\mathrm{\infty })$: with $a_k=k$ and $b_k=-k$ the sum is constantly $0$, while with $b_k=-2k$ it diverges to $-\mathrm{\infty }$. Leaving them undefined is not squeamishness, it is the only option that keeps every later statement about limits true without a side condition hidden inside the arithmetic.

• This is the separate introduction that Conventions: $\sup \mathrm{\varnothing }$, unbounded sets, and the extended reals points to. That remark refuses the conventions $\sup S=+\mathrm{\infty }$ and $\inf \mathrm{\varnothing }=+\mathrm{\infty }$ inside $\mathbb{R}$, and records that the extended real line is introduced explicitly here, with its own order and its own partial arithmetic kept separate from $\mathbb{R}$ rather than quietly extending it. This is that introduction. The suprema and infima of Complete ordered field (least-upper-bound property), Greatest lower bound (infimum) and the whole suprema page remain real numbers with their nonempty and bounded hypotheses intact; what is new is a separate supremum operation, taken in $\bar{\mathbb{R}}$ and named as such, supplied by Every subset of $\bar{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\bar{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$.

• The symbols were already in circulation, and this definition does not change what they meant. Divergence to $+\mathrm{\infty }$ and to $-\mathrm{\infty }$ defines the single phrase "$x_k\to +\mathrm{\infty }$" as an abbreviation for a condition on $(x_k)$, and says in as many words that it does not define an object named $+\mathrm{\infty }$. That reading is still correct: nothing in Divergence to $+\mathrm{\infty }$ and to $-\mathrm{\infty }$ is restated or reinterpreted here, and Convergence in $\bar{\mathbb{R}}$ and the extended subsequential limit set: $L\in \bar{\mathbb{R}}$ is an extended subsequential limit when some subsequence converges to $L$, or diverges to $L=\pm \mathrm{\infty }$ is where the two are related, by a definition that quotes the old one rather than replacing it. Likewise the interval notation $(a,\mathrm{\infty })$ of Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length is notation for a condition on one side, not an endpoint, and stays that way.

• Why the order is defined by three clauses rather than by a picture. The clauses are what the verifications above actually use, and they make the two facts that later proofs lean on immediate: every element is $\le +\mathrm{\infty }$ and every element is $\ge -\mathrm{\infty }$, with no case analysis at the point of use.