@inbook{10.4169/j.ctt6wpwc3.4, ISBN = {9780883853382}, URL = {http://www.jstor.org/stable/10.4169/j.ctt6wpwc3.4}, abstract = {We assume that the reader is familiar with the real number system R. We let${R^2} = \{ (x,y):x\varepsilon R,y\varepsilon R\}$(Figure 1.1). These are ordered pairs of real numbers.As we shall see, the complex numbers are nothing other than${R^2}$equipped with a special algebraic structure.The complex numbers C consist of${R^2}$equippedwith some binary algebraic operations. One defines$(x,y) + (x',y') = (x + x',y + y')$,$(x,y)\cdot(x',y') = (xx' - yy',xy' + yx')$.These operations of + and$\cdot$are commutative and associative.We denote (1; 0) by 1 and (0,1) byi. If$\alpha \varepsilon R$, then we identify$\alpha$with the complex number$(\alpha ,0)$Using this notation, we see that$\alpha \cdot (x,y) = (\alpha ,0) \cdot (x,y) = (\alpha x,\alpha y)$. (1.1.2.1)As if}, bookauthor = {Steven G. Krantz}, booktitle = {A Guide to Complex Variables}, edition = {1}, pages = {1--18}, publisher = {Mathematical Association of America}, title = {The Complex Plane}, volume = {32}, year = {2008} }