Integer powers $a^m$

Let $a\in \mathbb{R}$, where $\mathbb{R}$ is the ambient ordered field (Ordered field, Field).

Natural exponents. By the recursion theorem (The recursion theorem) applied to the set $\mathbb{R}$, the starting element $1$ and the function $f(x)=x\cdot a$, there is a unique function $\mathbb{N}\to \mathbb{R}$, written $n\mapsto a^n$, with

$a^0=1,a^{n+1}=a^n\cdot a(n\in \mathbb{N})\mathrm{.}$

Thus $a^1=a$, $a^2=a\cdot a$, and so on. Note that this is defined for every $a$, including $a=0$.

Negative exponents. If $a\mathrm{\ne }0$ and $n\in \mathbb{N}$ with $n\ge 1$, set

$a^{-n}:=(a^n{)}^{-1}\mathrm{.}$

Why that is legitimate. The right-hand side presupposes that $a^n$ is invertible, that is, that $a^n\mathrm{\ne }0$. This is a proof obligation and not an observation, and it is discharged by claim 2 of Laws of integer exponents: for $a\mathrm{\ne }0$ in a field, $a^n\mathrm{\ne }0$ for every $n\in \mathbb{N}$, proved there by induction on $n$ from the fact that a field has no zero divisors (A field has no zero divisors: $ab=0\Rightarrow a=0$ or $b=0$). That lemma is a statement about the operation introduced here, so it depends on this definition and is recorded in this item's justified_by rather than in its deps (SCHEMA §3). Given $a^n\mathrm{\ne }0$, the value $(a^n{)}^{-1}$ is a single well-determined element, because multiplicative inverses in a field are unique (Identities and inverses in a field are unique).

Integer exponents. Every integer $m$ (The integers as equivalence classes of pairs of naturals) is either $n$ or $-n$ for a unique natural $n$, where $n$ in these integer expressions denotes its image under the embedding $\iota :\mathbb{N}\to \mathbb{Z}$ (The naturals embed in the integers, Arithmetic on the integers). This too is a citation and not a slogan: the order on $\mathbb{Z}$ is total (The integers form a totally ordered ring), so $m\ge 0$ or $m<0$; the image of $\iota$ is exactly the set of nonnegative integers, and each of them is $n$ for a unique natural $n$ (The naturals embed in the integers); and if $m<0$ then $-m>0$, by compatibility of the order with addition (The integers form a totally ordered ring), so $-m=n$ and $m=-n$, with $n$ unique because $\iota$ is injective. The two clauses above therefore define $a^m$ for every $m\in \mathbb{Z}$ whenever $a\mathrm{\ne }0$, and for every $m\in \mathbb{N}$ for arbitrary $a$. The two exponent domains are disjoint: the natural clause handles $m\ge 0$, including $m=0$, while the negative clause requires $n\ge 1$ and therefore handles only $m<0$.

Ordered field

Field

The recursion theorem

Laws of integer exponents ↗

A field has no zero divisors: $ab=0\Rightarrow a=0$ or $b=0$

Identities and inverses in a field are unique

The integers as equivalence classes of pairs of naturals

The naturals embed in the integers

Arithmetic on the integers

The integers form a totally ordered ring

• The convention $0^0=1$ is adopted here, and it is not a matter of taste but of agreement with the recursion above: $a^0=1$ is the starting value for every $a$, exactly as the empty product is $1$ (Finite sums and finite products, by recursion). This is the convention that makes the empty product, the binomial theorem, and polynomial notation ${\sum }_k{c}_k{x}^k$ work at $x=0$ without an exception. The competing convention "$0^0$ undefined" belongs to contexts where $x^y$ is studied as a function of two real variables and one wants continuity, which is unavailable and irrelevant here: the exponent in $a^m$ is an integer, never a real.
• $0^n=0$ for every $n\ge 1$, since $0^n=0^{n-1}\cdot 0=0$, a product with a zero factor (Multiplication by zero: $0\cdot a=0$); and $0^{-n}$ is not defined for $n\ge 1$, since $0$ has no inverse.
• The exponent is an integer and stays an integer. Rational exponents are a separate construction resting on the existence of roots (Existence and uniqueness of $n$-th roots: a unique $a^{1\mathrm{/}n}\ge 0$ with $(a^{1\mathrm{/}n}{)}^n=a$, Rational powers $a^r$ of a positive base), and real exponents are not available at this point in the reading order; they are introduced later in Real powers for positive bases, with the zero-base positive-exponent convention ↗ (see Why real exponents are deferred on the rational-powers page).
• The laws $a^{m+n}=a^m{a}^n$, $(a^m{)}^n=a^{mn}$ and $(ab{)}^n=a^n{b}^n$ are proved, not assumed, in Laws of integer exponents ↗; the order behaviour of $a\mapsto a^n$ is Monotonicity of $x\mapsto x^n$ and of $n\mapsto a^n$.