From a formula to a picture
You already know that a complex number z = x + i y is built from two real numbers, and in the last guide you learned to add, multiply, and conjugate them by pure algebra. Now we give that algebra a home. Take an ordinary flat plane, label the horizontal axis 'real' and the vertical axis 'imaginary', and plot z at the point with coordinates (x, y). That picture is the complex plane — the Argand plane — and from here on, every complex number is a place you can point to.
So the two pieces you met in guide 2 turn into the two coordinates of a point: the real part Re z = x is how far right you go, and the imaginary part Im z = y is how far up. The whole real number line you grew up with is just the horizontal axis of this plane, sitting inside it as the special points where y = 0. Complex numbers do not replace the reals; they surround them.
Point or arrow? Both
Here is the small idea that makes the complex plane powerful: a number z is both a point at (x, y) and a vector — an arrow drawn from the origin 0 out to that point. The point version is good for saying 'here is where z lives'. The arrow version is good for saying 'here is how z moves things'. Same z, two readings, and you switch freely between them depending on what you want to do.
The arrow reading instantly explains addition. From the last guide, z + w just adds real parts and adds imaginary parts. Geometrically that is exactly how you add vectors: slide the arrow for w so its tail sits at the tip of z, and z + w is the arrow to the new tip. Draw both ways around and you get a parallelogram — z + w is its far corner. This is why complex addition feels so natural: it is plain vector addition wearing an algebraic costume. Multiplication has its own geometry too, but that is a rotation-and-scaling story for the next guide.
Length and direction: modulus and argument
If z is an arrow, it has a length and a direction. The length is the modulus |z| = sqrt(x^2 + y^2) — exactly the Pythagorean distance from the origin out to the point, the same |z| you computed algebraically in guide 2, now revealed as a literal ruler measurement. The direction is the angle the arrow makes with the positive real axis, measured counterclockwise; this angle is the argument arg z.
Together, (length, direction) pin down z just as well as (x, y) do — they are polar coordinates, and rewriting z in terms of them is the polar form you will meet properly in the next guide. There is one honest catch with the angle. An arrow pointing in a given direction is unchanged if you spin it a full turn, so arg z is only defined up to adding any whole multiple of 2 pi. There is no single 'correct' argument; arg z is a whole family of angles that differ by full turns.
To get a single number you pick a window of width 2 pi and read the angle there; the usual choice is the principal argument Arg z, taken in the interval from -pi to pi. That choice is convenient but not free of consequences: as z crosses the negative real axis the principal value jumps abruptly from near pi to near -pi. That little jump is the first warning sign of the branch cuts that will haunt the complex logarithm later — geometry is honest about the seam even before the algebra forces us to name it.
Conjugation is a mirror
The conjugate z-bar = x - i y flips the sign of the imaginary part and leaves the real part alone. On the plane that is a single clean motion: reflect z across the real axis, like folding the page along the horizontal line. A point above the axis lands the same distance below it, and a point already on the axis — a plain real number — does not move at all.
This mirror picture makes two facts from guide 2 obvious instead of memorized. First, reflecting does not change an arrow's length, so |z-bar| = |z|. Second, the reflection sends the angle theta to -theta, so arg(z-bar) = -arg z. And the product z times z-bar, which you found algebraically equals x^2 + y^2, is now simply |z|^2 — the squared length — which is why dividing by |z|^2 is the trick that turns conjugation into reciprocals.
z = x + i y point above the real axis z-bar = x - i y its mirror image below |z-bar| = |z| reflection keeps length arg(z-bar) = -arg z reflection flips the angle z * z-bar = |z|^2 real, equal to the squared length
Triangles, lines, and circles
Once addition is arrow-sliding, one famous inequality becomes a picture you can almost feel. To go from 0 to z + w you can travel straight along the diagonal of the parallelogram, or take the bent path out along z and then along w. The bent path is never shorter, which is precisely the triangle inequality: |z + w| <= |z| + |w|. Equality holds only when z and w point the same way, so there is no bend to save. It is the ordinary 'a straight line is the shortest path' fact, written in complex notation.
Distances also give us a strikingly compact language for shapes, captured by the idea of lines and circles in the plane. A circle of radius r centered at a point z_0 is just every z whose distance to z_0 equals r — in symbols |z - z_0| = r, one tidy equation instead of the usual expanded x-and-y mess. A line through two points is the set equidistant in spirit from them; for instance |z - a| = |z - b| describes exactly the points equally far from a and b, which is the perpendicular bisector of the segment joining them.
- Want a region, not just a curve? Replace '=' with an inequality. |z - z_0| < r is the open disk inside the circle — every point strictly closer to z_0 than r.
- |z - z_0| > r is the outside, the unbounded region beyond the circle.
- Im z > 0 is the upper half-plane (everything above the real axis); Re z > 0 is the right half-plane. These half-planes and disks are the stages on which most of complex analysis is acted out.
When the plane is not quite enough
The plane is a beautiful arena, but it has a quiet gap: it has no place for 'infinitely far away'. As |z| grows without bound the arrow simply runs off the page, and different directions run off to different edges, so there is no single point you are heading toward. For many questions — what does the map z to 1/z do near 0, or how do lines and circles really relate — that missing destination is exactly what we need.
The fix is to add one extra point, written infinity, and agree that going far in any direction approaches that same single point. This gives the point at infinity, and the cleanest way to see it is to drape the flat plane over a sphere: every point of the plane corresponds to a point of the sphere, and the lone north pole stands in for infinity. That sphere is the Riemann sphere, and it is the subject of the last guide in this rung — for now, just hold the thought that the plane has a natural roof.