The Argand Plane

The Argand Plane

Every complex number $z=x+iy$ can be represented as a unique point in a plane, by plotting $x$ (the real part) along a horizontal axis and $y$ (the imaginary part) along a perpendicular vertical axis. This one-to-one correspondence between complex numbers and points in a plane is called the Argand plane (or complex plane), and the representation is called an Argand diagram.

• The horizontal axis is the real axis — every point on it has the form $x+0i$, i.e. it represents a real number.
• The vertical axis is the imaginary axis — every point on it has the form $0+iy$, i.e. it represents a purely imaginary number (or $0$ at the origin).
• The complex number $z=x+iy$ corresponds to the point $P(x,y)$: move a distance $x$ along the real axis and $y$ along the imaginary axis.

$z=1+2i$ is plotted at the point $(1,2)$, in the first quadrant. $z=-3+i$ is plotted at $(-3,1)$, in the second quadrant. The real number $z=4$ (that is, $4+0i$) is plotted at $(4,0)$, exactly on the real axis — confirming visually that every real number sits inside $\mathbb{C}$ as a point on this one axis.

Because each complex number corresponds to exactly one point, and each point to exactly one complex number, algebraic facts about complex numbers translate directly into geometric facts about points, and vice versa. Two facts used repeatedly later in the chapter:

• The conjugate $\bar z=x-iy$ of $z=x+iy$ is the point $(x,-y)$ — the reflection of $P$ in the real axis. This is why $z=\bar z$ exactly when $z$ is real (a point equals its own reflection only when it already lies on the mirror line, the real axis). …