Britannica Sources used in this answer

How are complex numbers represented on a coordinate plane?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — algebra This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

Complex numbers, which take the form $x + i y$ (where $x$ and $y$ are real numbers and $i$ is the square root of $- 1$ ), are represented on a coordinate plane known as an Argand diagram or the complex plane.12

In this system, the plane is defined by two perpendicular axes:
* Horizontal axis: Represents the real part of the complex number ( $x$ -values).13
* Vertical axis: Represents the pure imaginary part of the complex number ( $y$ -values).13

By using these axes, any complex number $x + i y$ can be plotted as a unique point $(x, y)$ on the plane. This geometric representation allows mathematicians to visualize complex numbers as points in a two-dimensional space, contrasting with real numbers, which are represented on a single-dimensional number line.453