Definitions

Starting from a familiar example. Take the ordinary addition and multiplication of natural numbers, $m+n$ and $m\times n$ for $m,n\in\mathbb N=\{1,2,3,\ldots\}$. Both operations share two features: (1) exactly two elements of $\mathbb N$ are processed at a time, and (2) the resulting element is again in $\mathbb N$. Any operation on a non-empty set with these two features is called a binary operation (or binary composition) in abstract algebra.

Definition 12.1. An operation $*$ defined on a non-empty set $S$ is a binary operation on $S$ if:

1. $*$ is defined for every ordered pair $(a,b)\in S\times S$, and
2. it assigns a unique element $a*b\in S$ to every such ordered pair. In other words, $*$ is a rule (a function/mapping) with input in the Cartesian product $S\times S$ and output in $S$:

   $*:S\times S\to S;\qquad *(a,b)=a*b\in S,$

   where $a*b$ is a single, unambiguous element. Because the output $a*b$ is required to lie in $S$ itself and never outside it, we say $*$ is closed on $S$, or $S$ is closed with respect to $*$ -- the closure property. Every binary operation automatically satisfies closure, by definition. Definition 12.2. A non-empty set on which one or more binary operations are defined is called an algebraic structure. An equivalent way to state Definition 12.1: $\forall\,a,b\in S,\ a*b$ is unique and $a*b\in S$. The symbol $*$ is generic -- depending on the set it may stand for $+,\times,-,\div$, matrix addition, matrix multiplication, and so on. Why not every "obvious" operation is binary -- and why number systems keep expanding. $+$ and $\times$ are binary on $\mathbb N$, but $-$ is not: for $(3,4)\in\mathbb N\times\mathbb N$, $3-4=-1\notin\mathbb N$. So $\mathbb N$ is extended to $\mathbb Z$, on which $-$ is binary; $(\mathbb Z,+,\times,-)$ is an algebraic structure. The same pattern repeats:

• $\div$ is not binary on $\mathbb Z$: for $(1,2)\in\mathbb Z\times\mathbb Z$, $1\div2=\tfrac12\notin\mathbb Z$ -- forcing the extension $\mathbb Z\to\mathbb Q$.
• Division by $0$ is never defined, so $\div$ is binary on $\mathbb Q\setminus\{0\}$, not on all of $\mathbb Q$.
• Needing roots of equations such as $x^2-2=0$ (irrational) and $x^2+1=0$ (imaginary) forces $\mathbb Q\to\mathbb R\to\mathbb C$. $\mathbb C$ is the biggest of these systems, properly containing $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R$ as subsets.

Table 12.1 summarises which of $+,-,\times,\div$ are binary on each number system: