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Every real number has a place on a standard one-dimensional number line — but where does a number like 3+2i3 + 2i live?
Because a complex number has two independent pieces — a real part and an imaginary part — it cannot fit on a single line and needs two dimensions to exist visually.
What does a 2D plane for numbers look like?

The Complex Plane

In 1806, amateur mathematician Jean-Robert Argand published a geometric interpretation of complex numbers to give −1\sqrt{-1} a concrete physical meaning.
An Argand diagram is a geometric plane where the horizontal axis represents the real axis (labeled Re\text{Re}) and the vertical axis represents the imaginary axis (labeled Im\text{Im}).
How do we locate a specific complex number on this coordinate grid?

Plotting Complex Numbers

Any complex number z=a+biz = a + bi is plotted as the Cartesian coordinate pair (a,b)(a, b).
For example, z1=3+2iz_1 = 3 + 2i sits at (3,2)(3, 2), while purely real numbers sit directly on the real axis (z2=−4z_2 = -4 at (−4,0)(-4, 0)) and purely imaginary numbers sit on the imaginary axis (z3=−3iz_3 = -3i at (0,−3)(0, -3)).
A common mistake is writing the vertical coordinate as bibi instead of bb. The axis itself is already designated as the imaginary direction, so the vertical coordinate is purely the real coefficient bb.