The operations of addition and multiplication of complex numbers have the following properties.

• ACCN Additive Closure, Complex Numbers
  If $\alpha,\beta\in\complexes$, then $\alpha+\beta\in\complexes$.
  
  
• MCCN Multiplicative Closure, Complex Numbers
  If $\alpha,\beta\in\complexes$, then $\alpha\beta\in\complexes$.
  
  
• CACN Commutativity of Addition, Complex Numbers
  For any $\alpha,\,\beta\in\complexes$, $\alpha+\beta=\beta+\alpha$.
  
  
• CMCN Commutativity of Multiplication, Complex Numbers
  For any $\alpha,\,\beta\in\complexes$, $\alpha\beta=\beta\alpha$.
  
  
• AACN Additive Associativity, Complex Numbers
  For any $\alpha,\,\beta,\,\gamma\in\complexes$, $\alpha+\left(\beta+\gamma\right)=\left(\alpha+\beta\right)+\gamma$.
  
  
• MACN Multiplicative Associativity, Complex Numbers
  For any $\alpha,\,\beta,\,\gamma\in\complexes$, $\alpha\left(\beta\gamma\right)=\left(\alpha\beta\right)\gamma$.
  
  
• DCN Distributivity, Complex Numbers
  For any $\alpha,\,\beta,\,\gamma\in\complexes$, $\alpha(\beta+\gamma)=\alpha\beta+\alpha\gamma$.
  
  
• ZCN Zero, Complex Numbers
  There is a complex number $0=0+0i$ so that for any $\alpha\in\complexes$, $0+\alpha=\alpha$.
  
  
• OCN One, Complex Numbers
  There is a complex number $1=1+0i$ so that for any $\alpha\in\complexes$, $1\alpha=\alpha$.
  
  
• AICN Additive Inverse, Complex Numbers
  For every $\alpha\in\complexes$ there exists $-\alpha\in\complexes$ so that $\alpha+\left(-\alpha\right)=0$.
  
  
• MICN Multiplicative Inverse, Complex Numbers
  For every $\alpha\in\complexes$, $\alpha\neq 0$ there exists $\frac{1}{\alpha}\in\complexes$ so that $\alpha\left(\frac{1}{\alpha}\right)=1$.