If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

The complex plane

Learn what the complex plane is and how it is used to represent complex numbers.
The Imaginary unit, or i, is the number with the following equivalent properties:
  • i2=1
  • 1=i
A complex number is any number that can be written as a+bi, where i is the imaginary unit and a and b are real numbers.
a is called the real part of the number, and b is called the imaginary part of the number.

The complex plane

Just like we can use the number line to visualize the set of real numbers, we can use the complex plane to visualize the set of complex numbers.
A coordinate plane where the x-axis is labeled the real axis and the y-axis is labeled the imaginary-axis. Both axes are scaled by one.
The complex plane consists of two number lines that intersect in a right angle at the point (0,0).
The horizontal number line (what we know as the x-axis on a Cartesian plane) is the real axis.
The vertical number line (the y-axis on a Cartesian plane) is the imaginary axis.

Plotting a complex number

Every complex number can be represented by a point in the complex plane.
For example, consider the number 35i. This number, also expressed as 3+(5)i, has a real part of 3 and an imaginary part of 5.
The location of this number on the complex plane is the point that corresponds to 3 on the real axis and 5 on the imaginary axis.
A coordinate plane where the x-axis is labeled the real axis and the y-axis is labeled the imaginary-axis. Both axes are scaled by one. A point is at (three, negative five i). A vertical dashed line extends from the three on the real axis to the point and a horizontal line extends from negative five on the imaginary-axis to the point.
So the number 3+(5)i is associated with the point (3,5). In general, the complex number a+bi corresponds to the point (a,b) in the complex plane.

Check your understanding

Problem 1
Plot the complex number 4+7i.

Problem 2
Plot the complex number 6i+1.

Problem 3
Plot the complex number i3.

Problem 4
Plot the complex number 4i.

Problem 5
Plot the complex number 7.

Connections to the real number line

In Pythagoras's days, the existence of irrational numbers was a surprising discovery! They wondered how something like 2 could exist without an accurate complete decimal expansion.
The real number line, however, helps rectify this dilemma. Why? Because 2 has a specific location on the real number line, showing that it is indeed a real number. (If you take the diagonal of a unit square and place one end on 0, the other end corresponds to the number 2.)
Likewise, every complex number does indeed exist because it corresponds to an exact location on the complex plane! Perhaps by being able to visualize these numbers, we can understand that calling these numbers "imaginary" was an unfortunate misnomer.
Complex numbers exist and are very much a part of mathematics. The real number line is simply the real axis on the complex plane, but there is so much beyond that single line!

Want to join the conversation?