Chapter 5.4, Problem 7E

Define $\otimes$ on $ℝ\times ℝ$ by setting $\left(a,b\right)\otimes \left(c,d\right)=\left(ac-bd,ad+bc\right).$

1. Show that $\left(ℝ\times ℝ,\otimes \right)$ is an algebraic system.
2. Show that the function h from the system $\left(ℂ,\cdot \right)$ to $\left(ℝ\times ℝ,\otimes \right)$ given by $h\left(a+bi\right)=\left(a,b\right)$ is a one-to-one function from the set of complex numbers that is onto $ℝ\times ℝ$ and is operation preserving.