# Complex plane

The complex plane (or the Argand plane) is a representation of the complex numbers established by the real axis (horizontal) and the imaginary axis (vertical). It can be thought of as a Cartesian plane, where you plot $$\text{Re}(z)$$ on the $$x$$-axis and $$\text{Im}(z)$$ on the $$y$$-axis.

Worked example: Plot $$z_1 = 3+2i$$ and $$z_2=3-2i$$ on the complex plane.
Solution: This is equivalent to plotting $$(3,2)$$ and $$(3,-2)$$ on a Cartesian plane.

Notice that $$z_2$$ is a reflection of $$z_1$$ on the imaginary axis, as they are conjugates.

## Problems

1. Plot the following complex numbers on the complex plane.
1. $$3+4i$$
2. $$-3-4i$$
3. $$i$$
4. $$1-i$$
5. $$\frac{1}{2}-\frac{3}{2}i$$
6. $$9i-1$$
2. Let $$z = 2+i$$. Sketch carefully on the same Argand diagram.
1. $$z$$
2. $$-z$$
3. $$\overline{z}$$
4. $$-2z$$
5. $$iz$$
3. What is the relationship between $$z$$, $$-z$$ and $$\overline{z}$$ on the complex plane?