The plane of complex numbers spanned by the vectors 1 and , where is the imaginary number. Every complex number corresponds to a unique point in the complex plane. The line in the plane with is the real line. The complex plane is sometimes called the Argand plane or Gauss plane, and a plot of complex numbers in the plane is sometimes called an Argand diagram.

The complex plane together with the point at infinity is known as the Riemann sphere or extended complex plane and denoted or . However, the notation is also used to denote the punctured plane .

The Riemann sphere , also called the extended complex plane. The notation is sometimes also used (Krantz 1999, p. 82).

The notation is also used to denote the punctured plane .

The axis in the complex plane corresponding to zero real part, .

The term "real line" has a number of different meanings in mathematics.

Most commonly, "real line" is used to mean real axis, i.e., a line with a fixed scale so that every real number corresponds to a unique point on the line. The generalization of the real line to two dimensions is called the complex plane.

The term "real line" is also used to distinguish an ordinary line from a so-called imaginary line which can arise in algebraic geometry.