Provider: JSTOR http://www.jstor.org Database: JSTOR Content: text/plain; charset="us-ascii" TY - CHAP TI - The Complex Plane A2 - Krantz, Steven G. AB -

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

EP - 18 ET - 1 PB - Mathematical Association of America PY - 2008 SN - 9780883853382 SP - 1 T2 - A Guide to Complex Variables UR - http://www.jstor.org/stable/10.4169/j.ctt6wpwc3.4 VL - 32 Y2 - 2021/09/26/ ER -