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Complex Plane — Definition, Graph & Examples

Complex Plane
Argand Plane

The coordinate plane used to graph complex numbers. The x-axis is called the real axis and the y-axis is called the imaginary axis. The complex number x + yi is graphed as the point (x, y).

 

Complex plane with real and imaginary axes. Point 3+2i plotted at coordinates (3,2), with gridlines marking values 2 and 3.

Key Formula

z=x+yi⟷(x, y)z = x + yi \quad \longleftrightarrow \quad (x,\, y)
Where:
  • zz = A complex number
  • xx = The real part of z, plotted on the horizontal (real) axis
  • yy = The imaginary part of z, plotted on the vertical (imaginary) axis
  • ii = The imaginary unit, where i² = −1

Worked Example

Problem: Plot the complex number z = 3 + 4i on the complex plane and find its distance from the origin (called the modulus).
Step 1: Identify the real and imaginary parts. The real part is 3 and the imaginary part is 4.
z=3+4i  ⟹  x=3,  y=4z = 3 + 4i \implies x = 3,\; y = 4
Step 2: Plot the point (3, 4) on the complex plane. Move 3 units right along the real axis and 4 units up along the imaginary axis.
z⟷(3, 4)z \longleftrightarrow (3,\, 4)
Step 3: Calculate the modulus (distance from the origin) using the Pythagorean theorem.
∣z∣=x2+y2=32+42=9+16=25|z| = \sqrt{x^2 + y^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25}
Step 4: Simplify to find the modulus.
∣z∣=5|z| = 5
Answer: The complex number 3 + 4i is plotted at the point (3, 4), and its modulus is 5.

Another Example

This example shows that the standard distance formula from coordinate geometry applies directly to the complex plane, reinforcing the connection between complex number arithmetic and geometry.

Problem: Two complex numbers are z₁ = 1 + 2i and z₂ = 4 + 6i. Find the distance between them on the complex plane.
Step 1: Write each complex number as a point in the plane.
z1⟷(1, 2),z2⟷(4, 6)z_1 \longleftrightarrow (1,\, 2), \quad z_2 \longleftrightarrow (4,\, 6)
Step 2: The distance between two points in the complex plane equals the modulus of their difference.
z2−z1=(4−1)+(6−2)i=3+4iz_2 - z_1 = (4 - 1) + (6 - 2)i = 3 + 4i
Step 3: Find the modulus of the difference using the distance formula.
∣z2−z1∣=32+42=9+16=25=5|z_2 - z_1| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
Answer: The distance between z₁ and z₂ on the complex plane is 5.

Frequently Asked Questions

What is the difference between the complex plane and the regular coordinate plane?
Both use two perpendicular axes and plot ordered pairs (x, y). The key difference is interpretation: on the regular coordinate plane both axes represent real numbers, while on the complex plane the horizontal axis represents the real part and the vertical axis represents the imaginary part of a complex number. Every point on the complex plane corresponds to exactly one complex number x + yi.
Why is the complex plane also called the Argand plane?
It is named after Jean-Robert Argand, a Swiss mathematician who published one of the first geometric representations of complex numbers in 1806. The diagram showing complex numbers as points in a plane is sometimes called an Argand diagram. The idea was developed independently by several mathematicians, but Argand's name became most widely associated with it.
How do you find the modulus and argument of a complex number on the complex plane?
The modulus is the distance from the origin to the point, calculated as ∣z∣=x2+y2|z| = \sqrt{x^2 + y^2}. The argument is the angle the line from the origin to the point makes with the positive real axis, found using θ=arctan⁡(y/x)\theta = \arctan(y/x) (adjusted for the correct quadrant). Together, these give the polar form z=∣z∣(cos⁡θ+isin⁡θ)z = |z|(\cos\theta + i\sin\theta).

Complex Plane vs. Coordinate Plane

Complex PlaneCoordinate Plane
AxesReal axis (horizontal) and imaginary axis (vertical)x-axis (horizontal) and y-axis (vertical), both real
Points representComplex numbers z = x + yiOrdered pairs of real numbers (x, y)
Distance from originModulus |z| = √(x² + y²)Distance d = √(x² + y²)
Multiplication meaningRotation and scaling (multiply moduli, add arguments)No direct geometric interpretation for multiplying two points
Common useVisualizing complex arithmetic, polar form, roots of unityGraphing equations, functions, and data in algebra and geometry

Why It Matters

You encounter the complex plane in precalculus and beyond whenever you work with complex numbers, polar form, or roots of polynomials. It turns abstract algebraic operations—like multiplying complex numbers—into concrete geometric transformations such as rotations and scalings. In physics and engineering, the complex plane is essential for analyzing electrical circuits (phasors), signal processing, and quantum mechanics.

Common Mistakes

Mistake: Reversing the axes by plotting the imaginary part on the horizontal axis and the real part on the vertical axis.
Correction: The real part always goes on the horizontal axis and the imaginary part on the vertical axis. Think of it as the real axis replacing the familiar x-axis.
Mistake: Plotting the complex number 3 + 4i at the point (3, 4i) instead of (3, 4).
Correction: When converting to a point, drop the 'i' and use only the numerical coefficient. The axes already tell you which component is imaginary, so the point is simply (3, 4).

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