Triangle Inequality with Complex Numbers

I was wondering how to prove the triangle inequality with complex numbers:

Verify that the function $d(z_1, z_2)$ is a distance funtion on $\mathbb{C}$ and also on any subdomain on $\mathbb{C}$.

I finished all but the inequality, which is what I need assistance with. I don't know how to prove the distance function inequality:

$d(z_1, z_3) \le d(z_1, z_2) + d(z_2, z_3) $

Solution 1:

The more formal proof goes as so:

Let us consider $|z_1+z_2|^2 = (z_1 +z_2)(\overline{z_1}+\overline{z_2})$

Multiplying out,

$(z_1 +z_2)(\overline{z_1}+\overline{z_2}) = z_1\overline{z_1}+z_1\overline{z_2}+\overline{z_1\overline{z_2}}+z_2\overline{z_2}$

$=|z_1|^2 + 2Re(z_1\overline{z_2})+|z_2|^2$, and it is here we note that $2Re(z_1\overline{z_2})\leq 2|z_1||z_2|$, so substituting in,

$$|z_1|^2 + 2Re(z_1\overline{z_2})+|z_2|^2 \leq |z_1|^2 + 2|z_1||z_2|+|z_2|^2 = (|z_1|+|z_2|)^2$$

So, we have shown $|z_1+z_2|^2 \leq (|z_1|+|z_2|)^2 \implies |z_1+z_2| \leq |z_1|+|z_2|$

Also, the very last step is justified since we know that the modulus is always greater than or equal to 0.

Solution 2:

Without loss of generality, we only need to show that $|z_2-z_1|\le |z_2|+|z_1|$. We will use the notation that $z_1=|z_1|e^{i\theta_1}$ and $z_2=|z_2|e^{i\theta_2}$ and $z^{*}$ will denote the complex conjugate of $z$.

So, we write

$$\begin{align} |z_2-z_1|^2&=(z_2-z_1)(z_2-z_1)^*\\\\ &=|z_2|^2+|z_1|^2-2\text{Re}\{z_1z_2^*\}\\\\ &=|z_2|^2+|z_1|^2-2|z_2||z_1|\cos (\theta_1-\theta_2)\\\\ &=(|z_2|+|z_1|)^2-2|z_2||z_1|(1+\cos (\theta_1-\theta_2))\\\\ &\le(|z_2|+|z_1|)^2 \end{align}$$

and we are done! Just let $z_2 \to z_3-z_2$ and $z_1 \to z_1-z_2$.

Solution 3:

Note that we identify $\mathbb C$ with the plane $\mathbb R^2$. If you realize that complex addition in $\mathbb C$ is the same thing as vector addition in $\mathbb R^2$, and the absolute value in $\mathbb C$ is the same thing as the norm $\|\vec{v}\|$ in $\mathbb R^2$, then the triangle inequality in $\mathbb C$ is just exactly the same as the triangle inequality in $\mathbb R^2$.

Solution 4:

Using just the algebraic form. Set $u=z_1-z_2$ and $v=z_2-z_3$, so you have to prove that $$ |u+v|\le |u|+|v| $$ This is equivalent to $$ |u+v|^2\le (|u|+|v|)^2 $$ Since $|z|^2=z\bar{z}$, this becomes $$ u\bar{u}+u\bar{v}+\bar{u}v+v\bar{v}\le u\bar{u}+2|u|\,|v|+v\bar{v} $$ and, removing the alike terms, we get the equivalent inequality $$ u\bar{v}+\bar{u}v\le 2|u|\,|v| $$ If the left-hand side is negative, we are done. If it is nonnegative, the inequality is equivalent to $$ (u\bar{v}+\bar{u}v)^2\le 4|u|^2\,|v|^2 $$ that becomes $$ u^2\bar{v}^2+2u\bar{u}v\bar{v}+\bar{u}^2v^2\le 4u\bar{u}v\bar{v} $$ Transporting the terms to the left-hand side, we obtain $$ (u\bar{v}-\bar{u}v)^2\le0 $$ which is true, because $z=u\bar{v}-\bar{u}v$ is purely imaginary, since $z=-\bar{z}$.