Visualizing Conjugates on the Complex Plane

We have established that a conjugate of a complex number is the same number with its imaginary part flipped. It is represented with an overline like $\overline{z}$ or $\text{conj}(z)$as Algebraically we have;

The blue point represents a complex number $z$ and you can move it around. The red point represents its conjugate $\overline{z}$

Some insights that you should be able to visually understand from this is that;

• Conjugate is formed by flipping the number on the x-axis.
• The magnitude of a complex number $|z|$and its conjugate $|\overline{z}|$is the same.
• Conjugate of a real number is the same as itself.

Addition of Conjugates is Real

We know previously that; $z + \overline{z} \in \mathbb{R}$ because the imaginary parts cancel out. Graphically we can see that $z + \overline{z}$ is a horizontal line meaning it has no imaginary component.

You can move the blue point around to see this in action.

Subtraction of Conjugates is Imaginary

We know previously that; $z - overline{z} \in \mathbb{I}$ because the real parts cancel out. Graphically we can see that $z - \overline{z}$ is a vertical line meaning it has no real component.

You can move the blue point around to see this in action.