Complex Numbers as a Metric Space

Distance in the Complex Plane

A metric gives a precise meaning to distance and makes it possible to discuss ideas such as neighborhoods, convergence, and completeness. For complex numbers, the modulus already measures distance from the origin. Applying it to the difference of two numbers therefore gives a natural distance between them. This viewpoint connects the algebra of complex numbers to the geometry of the plane and provides the setting for studying complex sequences in the next tutorial.

Write a complex number as \(z=x+iy\), where \(x,y\in\mathbb{R}\) and \(i^2=-1\). Its conjugate is \(\overline{z}=x-iy\), and its modulus is \(\lvert z\rvert=\sqrt{x^2+y^2}\). In particular, the modulus is nonnegative, and it is zero exactly when \(z=0\). We use the usual absolute value on the real numbers, and the usual algebraic operations on complex numbers.

For \(z,w\in\mathbb{C}\), define \(d(z,w)=\lvert z-w\rvert\). The central fact is that this formula really is a metric. Its triangle inequality follows from the corresponding inequality for the modulus, which we verify here to make clear how the geometry and algebra fit together.

Proof. The modulus is nonnegative, so \(d(z,w)\geq0\). Also, \(\lvert z-w\rvert=0\) if and only if \(z-w=0\), which holds exactly when \(z=w\). Since \(\lvert -(z-w)\rvert=\lvert z-w\rvert\), we have \(d(w,z)=d(z,w)\).

For the triangle inequality, first take any complex numbers \(u,v\). The identity \(\lvert a\rvert^2=a\overline{a}\) gives

For any complex number \(a=s+it\), \(\operatorname{Re}(a)=s\leq\sqrt{s^2+t^2}=\lvert a\rvert\). Thus \(\operatorname{Re}(u\overline{v})\leq\lvert u\overline{v}\rvert=\lvert u\rvert\lvert v\rvert\), where the last equality follows from multiplicativity of the modulus. Consequently,

Both sides are nonnegative, so taking square roots gives \(\lvert u+v\rvert\leq\lvert u\rvert+\lvert v\rvert\). Now set \(u=z-y\) and \(v=y-w\). Then \(u+v=z-w\), and therefore \(d(z,w)\leq d(z,y)+d(y,w)\). All the metric axioms hold. \(\square\)

Metric Balls Are Open Disks

In any metric space, the open ball of radius \(r>0\) centered at \(a\) is the set of points whose distance from \(a\) is less than \(r\). In \(\mathbb{C}\), this definition has a direct geometric interpretation: an open ball is an open disk in the plane. The boundary circle is excluded because the inequality defining the ball is strict.

To see the disk equation in coordinates, write \(a=\alpha+i\beta\) and \(z=x+iy\). Then \(z-a=(x-\alpha)+i(y-\beta)\), so \[ |z-a|=\sqrt{(x-\alpha)^2+(y-\beta)^2}. \] Thus membership in \(B(a,r)\) is equivalent to \((x-\alpha)^2+(y-\beta)^2<r^2\). This description also makes clear why the same distance measures horizontal and vertical displacement together rather than adding their absolute values.

The Complex Plane and Euclidean Distance

The coordinate map \(z=x+iy\mapsto(x,y)\) identifies \(\mathbb{C}\) with \(\mathbb{R}^2\). Under this identification, the complex metric is not merely similar to Euclidean distance; it is exactly the Euclidean metric. This means that metric questions about complex numbers can be interpreted geometrically in the ordinary plane, with no change in the distances.

Proof. Every complex number has a unique expression \(x+iy\) with real \(x,y\), so \(\Phi\) is both one-to-one and onto. Let \(z=x+iy\) and \(w=u+iv\). Their coordinate images are \((x,y)\) and \((u,v)\). The Euclidean distance between these images is \(\sqrt{(x-u)^2+(y-v)^2}\). On the other hand, \(z-w=(x-u)+i(y-v)\), and hence

The two distances are equal for every pair \(z,w\). Therefore \(\Phi\) preserves distance, as required. \(\square\)

An isometry preserves metric balls and distances between every pair of points. In particular, a ball in \(\mathbb{C}\) corresponds under \(\Phi\) to a Euclidean open disk with the same center coordinates and radius. Metric language in the complex plane can therefore be read either algebraically, using moduli, or geometrically, using the familiar plane.

Completeness of the Complex Metric

A metric space is complete if every Cauchy sequence in it converges to a point of the same space. A sequence \((z_n)\) is Cauchy when, for every \(\varepsilon>0\), all sufficiently late terms are within distance \(\varepsilon\) of one another. Completeness is an important guarantee that a process producing points closer and closer together does not converge only to a point missing from the space.

Proof. Let \((z_n)\) be a Cauchy sequence in \(\mathbb{C}\), and write \(z_n=x_n+iy_n\) with \(x_n,y_n\in\mathbb{R}\). For any \(n,m\),

Indeed, if \(z_n-z_m=(x_n-x_m)+i(y_n-y_m)\), then the square of its modulus is \((x_n-x_m)^2+(y_n-y_m)^2\), which is at least either squared coordinate difference. Since \((z_n)\) is Cauchy, these inequalities show that both real sequences \((x_n)\) and \((y_n)\) are Cauchy in \(\mathbb{R}\). By completeness of the real numbers, there are \(x,y\in\mathbb{R}\) such that \(x_n\to x\) and \(y_n\to y\).

Set \(z=x+iy\), which is a complex number. We show \(z_n\to z\). Given \(\varepsilon>0\), choose \(N\) large enough that for \(n\geq N\), \(\lvert x_n-x\rvert<\varepsilon/3\) and \(\lvert y_n-y\rvert<\varepsilon/3\). Then

Thus \(z_n\) converges to \(z\) in the complex metric. Since the original Cauchy sequence was arbitrary, \(\mathbb{C}\) is complete. \(\square\)

The proof depends on completeness of \(\mathbb{R}\), not on a separate assumption that complex numbers have some additional limiting property. The two real coordinate sequences determine the limit, and the modulus distance ensures that convergence of both coordinates gives convergence in \(\mathbb{C}\). Conversely, the coordinate inequalities show that complex convergence forces convergence of each coordinate.

Why the Metric Viewpoint Matters

The formula \(|z-w|\) unifies algebraic and geometric reasoning. It lets us express closeness without separately tracking real and imaginary parts, while the coordinate formula remains available whenever geometry makes a calculation easier. Open balls specify neighborhoods, and completeness ensures that Cauchy processes have limits in the complex plane. These are the basic metric ideas needed to study complex sequences systematically.

A common pitfall is to confuse distance from the origin, \(|z|\), with distance between arbitrary points. The distance from \(z\) to \(w\) is \(|z-w|\), not \(|z|-|w|\). For instance, if \(z=1+i\) and \(w=1-i\), then \(|z|=|w|=\sqrt{2}\), but \(|z-w|=|2i|=2\). Equal distances from the origin do not mean that two points coincide or are close to each other.