The field of complex numbers ℂ is the set of all ordered pairs (a, b)where a and b are real numbers and where addition and multiplication are defined by: $$\begin{gathered} (a,b) + (c,d) = (a + c{\text{, }}b + d) \hfill \\ (a,b)(c,d) = (ac - bd,{\text{ }}bc + ad). \hfill \\ \end{gathered}$$ We will write a for the complex number (a, 0). In fact, the mapping a ↦ (a, 0) defines a field isomorphism of ℝ into ℂ, hence we may consider ℝ as a subset of ℂ. If we put ι = (0,1), then (a, b) = a + bι. For z = z +ιb we put Re z = a and Im z = b. Real numbers are associated with points on the x-axis and called the real axis. Purely imaginary numbers are associated with points on the y-axis and called the imaginary axis.