Proof of The Modulus Metric on the Complex Numbers

By The Distance on the Complex Numbers §distance, $d_{\mathbb{C}}$ is a map $\mathbb{C}\times\mathbb{C}\to\mathbb{R}$ with $d_{\mathbb{C}}(z,w)=|z-w|$, where $|\cdot|$ is the modulus. $\mathbb{C}$ is a field by The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers §complex, with negatives and differences as in Negatives, Differences, Reciprocals and Quotients §negative. Let $z,w,u\in\mathbb{C}$; we verify the conditions of Metric Space §metric.

Nonnegative. $0\le|z-w|=d_{\mathbb{C}}(z,w)$ by Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums §modulus, which is Metric Space §nonnegative.

Zero. By Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums §modulus, $|z-w|=0$ if and only if $z-w=0$. If $z=w$, then $z-w=w+(-w)=0$ by Negatives, Differences, Reciprocals and Quotients §negative. Conversely, if $z-w=0$, then by associativity of $+$ and $(-w)+w=0$ (Negatives, Differences, Reciprocals and Quotients §negative, with commutativity)

$$z=z+\bigl((-w)+w\bigr)=\bigl(z+(-w)\bigr)+w=0+w=w .$$

Hence $d_{\mathbb{C}}(z,w)=0$ if and only if $z=w$, which is Metric Space §zero.

Symmetric. $w-z=-(z-w)$ by Rules of Arithmetic in a Commutative Ring: Zero, Signs and Squares, and No Zero Divisors in a Field §signs, and $|-(z-w)|=|z-w|$ by Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums §modulus; so $d_{\mathbb{C}}(w,z)=d_{\mathbb{C}}(z,w)$, which is Metric Space §symmetric.

Triangle. By associativity and commutativity of $+$ and $(-w)+w=0$ (Negatives, Differences, Reciprocals and Quotients §negative), $(z-w)+(w-u)=z+\bigl((-w)+w\bigr)+(-u)=z-u$. Hence Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums §triangle gives

$$d_{\mathbb{C}}(z,u)=|(z-w)+(w-u)|\le|z-w|+|w-u|=d_{\mathbb{C}}(z,w)+d_{\mathbb{C}}(w,u),$$

which is Metric Space §triangle.

As $z,w,u$ were arbitrary, $d_{\mathbb{C}}$ is a metric on $\mathbb{C}$ by Metric Space §metric.