( 2 2 + 2 2 i ) 2 = ( − 2 2 − 2 2 i ) 2 = i {\displaystyle \left({\frac {\sqrt {2}}{2}}+{\frac {\sqrt {2}}{2}}i\right)^{2}=\left(-{\frac {\sqrt {2}}{2}}-{\frac {\sqrt {2}}{2}}i\right)^{2}=i}
V {\displaystyle V} is a set, + {\displaystyle +} an addition on V {\displaystyle V} , ∗ {\displaystyle *} a scalor multiplication on V {\displaystyle V} , a b := a ∗ b {\displaystyle ab:=a*b} , and F {\displaystyle \mathbf {F} } a field with multiplication identity 1 {\displaystyle 1} , then
( V , + , ∗ ) i s a v e c t o r s p a c e o v e r F {\displaystyle (V,+,*)\ is\ a\ vector\ space\ over\ \mathbf {F} }
:⇔ ∀ u ∈ V , v ∈ V u + v = v + u ∧ ∀ u ∈ V , v ∈ V , w ∈ V ( u + v ) + w = u + ( v + w ) ∧ ∀ a ∈ F , b ∈ F , v ∈ V ( a b ) v = a ( b v ) {\displaystyle :\Leftrightarrow {\underset {u\in V,\,v\in V}{\forall }}u+v=v+u\land {\underset {u\in V,\,v\in V,\,w\in V}{\forall }}(u+v)+w=u+(v+w)\land {\underset {a\in \mathbf {F} ,\,b\in \mathbf {F} ,\,v\in V}{\forall }}(ab)v=a(bv)}
∧ ∃ 0 ∈ V ∀ v ∈ V ( v + 0 = v ∧ ∃ w ∈ V v + w = 0 ) ∧ ∀ v ∈ V 1 v = v ∧ ∀ a ∈ F , u ∈ V , v ∈ V a ( u + v ) = a u + a v ∧ ∀ a ∈ F , b ∈ F , v ∈ V ( a + b ) v = a v + b v {\displaystyle \land {\underset {0\in V}{\exists }}\ {\underset {v\in V}{\forall }}\left(v+0=v\land {\underset {w\in V}{\exists }}v+w=0\right)\land {\underset {v\in V}{\forall }}1v=v\land {\underset {a\in \mathbf {F} ,\,u\in V,\,v\in V}{\forall }}a(u+v)=au+av\land {\underset {a\in \mathbf {F} ,\,b\in \mathbf {F} ,\,v\in V}{\forall }}(a+b)v=av+bv}
is a set, 
an addition on , 
a scalor multiplication on , 
, and 
a field with multiplication identity 
, then
