Properties of Complex Numbers

Properties under addition (for any complex numbers $z_1,z_2,z_3$):

1. Closure: $z_1+z_2$ is again a complex number.
2. Commutative: $z_1+z_2=z_2+z_1$.
3. Associative: $(z_1+z_2)+z_3=z_1+(z_2+z_3)$.
4. Additive identity: there is a complex number $0=0+0i$ with $z+0=0+z=z$ for every $z$.
5. Additive inverse: for every $z$ there is $-z$ with $z+(-z)=(-z)+z=0$.

Properties under multiplication (for any complex numbers $z_1,z_2,z_3$):

1. Closure: $z_1z_2$ is again a complex number.
2. Commutative: $z_1z_2=z_2z_1$.
3. Associative: $(z_1z_2)z_3=z_1(z_2z_3)$.
4. Multiplicative identity: there is a complex number $1=1+0i$ with $z\cdot1=1\cdot z=z$ for every $z$.
5. Multiplicative inverse: for every nonzero $z$ there is $w$ with $zw=wz=1$; $w$ is denoted $z^{-1}$.

Distributive property (links the two operations): for any $z_1,z_2,z_3$, $z_1(z_2+z_3)=z_1z_2+z_1z_3$ and $(z_1+z_2)z_3=z_1z_3+z_2z_3$.

Proof of commutativity of addition. Let $z_1=x_1+iy_1$, $z_2=x_2+iy_2$, with $x_1,x_2,y_1,y_2\in\mathbb R$. Then

(since real-number addition is commutative) $=(x_2+iy_2)+(x_1+iy_1)=z_2+z_1$.

Proof of the multiplicative inverse. For nonzero $z=x+iy$, seek $z^{-1}=u+iv$ with $zz^{-1}=1$: $(x+iy)(u+iv)=1$, i.e. $(xu-yv)+i(xv+uy)=1+0i$. Equating real and imaginary parts, $xu-yv=1$ and $xv+uy=0$. Solving this pair of simultaneous equations for $u,v$ (valid since $z\ne0\Rightarrow x^2+y^2>0$) gives

$u=\frac{x}{x^2+y^2},\qquad v=\frac{-y}{x^2+y^2},\qquad\text{so}\qquad z^{-1}=\frac{x}{x^2+y^2}+i\,\frac{-y}{x^2+y^2}\quad(z^{-1}\text{ undefined when }z=0).$ …