Let $z_1$ and $z_2$ $\in \Bbb C$. Prove that $z_1z_2$ = $z_2z_1$.
Proof : For two complex numbers to be equal, they must have the same real and imaginary parts. First let's define $z_1$ and $z_2$ $$z_1= a+bi$$$$z_2= x+yi$$ Where a,b,x,y $\in \Bbb R$ $$z_1z_2=(a+bi)(x+yi)$$ $$ax+ayi+bxi-by$$ Thus $Re(z_1z_2)=ax-by$ and $Im(z_1z_2)=(ay+bx)$. $$z_2z_1$$ $$(x+yi)(a+bi)$$ $$ax+bxi+ayi+byi^2$$ $$ax+(bx+ay)i-by$$ $$ax-by+(bx+ay)i$$ Now $Re(z_2z_1)=ax-by$ and $Im(z_2z_1)=(ay+bx)$. This shows that the complex numbers, i.e the products are equal. Correct?