The Argand plane

The complex number x + iy corresponds to the ordered pair (x, y), so it can be drawn as the point P(x, y) of the coordinate plane, and each point stands for exactly one complex number.

A plane in which every point is read as a complex number is called the complex plane or the Argand plane. For example 3 + 4i, −2 + 3i, 5 and 2i sit at (3, 4), (−2, 3), (5, 0) and (0, 2).

The x-axis carries the numbers a + i0 and is called the real axis; the y-axis carries the numbers 0 + ib and is called the imaginary axis.

|x + iy| = √(x² + y²) is the distance of P from the origin O. So all the numbers with |z| = 5, such as 5, 3 + 4i, −4 + 3i and −5i, lie on the circle of radius 5 about O.

The conjugate x − iy is the point Q(x, −y), the mirror image of P in the real axis. The negative −z = −x − iy is the point opposite P through O.

The distance between the points z₁ and z₂ is |z₁ − z₂|. The distance from 1 + i to 4 + 5i is |3 + 4i| = 5.

The signs of the parts fix the quadrant: −2 + 3i is in the second quadrant, −5 − 2i in the third and 1 − 2i in the fourth.