---

Addition

Define

… as the binary operation

\[\begin{gather*}
  \forall a, b \in F, \\[1ex]
  a + b \in F
\end{gather*}
\]

where	\(a, b\)	are the terms,
	\(a + b\)	is the sum, and
	\(F\)	is the field with \(a\) and \(b\).

Constrain

• Additive identity

  … is \(0\), so

  \[\begin{gather*}
    \forall a \in F, \\[1ex]
    a + 0 = a.
  \end{gather*}
  \]

  More generally,

  \[\begin{gather*}
    \forall a \in F, \> \exists! b \in F : \\[1ex]
    a + b = a,
  \end{gather*}
  \]

  where the identity is

  \[\begin{equation*}
    b = 0.
  \end{equation*}
  \]

• Additive inverse

  … is \(-a\), so

  \[\begin{gather*}
    \forall a \in F, \\[1ex]
    a + (-a) = 0.
  \end{gather*}
  \]

  More generally,

  \[\begin{gather*}
    \forall a \in F, \> \exists! b \in F : \\[1ex]
    a + b = 0, \\[1ex]
  \end{gather*}
  \]

  where the inverse is \(b = -a\).

• Commutative property

  … holds, so

  \[\begin{gather*}
    \forall a, b \in F, \\[1ex]
    a + b = b + a.
  \end{gather*}
  \]

• Associative property

  … holds, so

  \[\begin{gather*}
    \forall a, b, c \in F, \\[1ex]
    (a + b) + c = a + (b + c).
  \end{gather*}
  \]