(related to Proposition: Existence of Inverse Complex Numbers With Respect to Addition)
Let \(x\) be a complex number represented by the ordered pair of real numbers \((a,b)\). Because there exist inverse real number swith respect to the addition of real numbers \(-a\) and \(-b\) , we can use them to define a new complex number
\[-x:=(-a,-b).\]
From the definition of adding complex numbers, it follows
\[\begin{array}{rcll} x+(-x)&=&(a,b)+(-a,-b)&\text{by definition of complex numbers}\\ &=&(a+(-a),b+(-b))&\text{by definition of adding complex numbers}\\ &=&(0,0)&\text{by existence of inverse real numbers with respect to addition}\\ &=&0&\text{by existence of complex zero} \end{array}\]
where \(0\) denotes the complex zero, as required.