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How to Graph Complex Numbers

To graph complex numbers, you simply combine the ideas of the real-number coordinate plane and the Gauss or Argand coordinate plane to create the complex coordinate plane. In other words, given a complex number A+Bi, you take the real portion of the complex number (A) to represent the x-coordinate, and you take the imaginary portion (B) to represent the y-coordinate.

In the Gauss or Argand coordinate plane, pure real numbers in the form a + 0i exist completely on the real axis (the horizontal axis), and pure imaginary numbers in the form 0 + Bi exist completely on the imaginary axis (the vertical axis). Figure a shows the graph of a real number, and Figure b shows that of an imaginary number.

Comparing the graphs of a real and an imaginary number.
Comparing the graphs of a real and an imaginary number.

Although you graph complex numbers much like any point in the real-number coordinate plane, complex numbers aren't real! The x-coordinate is the only real part of a complex number, so you call the x-axis the real axis and the y-axis the imaginary axis when graphing in the complex coordinate plane.

Graphing complex numbers gives you a way to visualize them, but a graphed complex number doesn't have the same physical significance as a real-number coordinate pair. For an (x, y) coordinate, the position of the point on the plane is represented by two numbers. In the complex plane, the value of a single complex number is represented by the position of the point, so each complex number A + Bi can be expressed as the ordered pair (A, B).

Complex numbers plotted on the complex coordinate plane.
Complex numbers plotted on the complex coordinate plane.

You can see several examples of graphed complex numbers in this figure:

  • Point A. The real part is 2 and the imaginary part is 3, so the complex coordinate is (2, 3) where 2 is on the real (or horizontal) axis and 3 is on the imaginary (or vertical) axis. This point is 2 + 3i.

  • Point B. The real part is –1 and the imaginary part is –4; you can draw the point on the complex plane as (–1, –4). This point is –1 – 4i.

  • Point C. The real part is 1/2 and the imaginary part is –3, so the complex coordinate is (1/2, –3). This point is 1/2 – 3i.

  • Point D. The real part is –2 and the imaginary part is 1, which means that on the complex plane, the point is (–2, 1). This coordinate is –2 + i.

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