Triangle Inequality

Real Numbers
Let $x, y \in \R$ be real numbers.

Let $\left\vert{x}\right\vert$ be the absolute value of $x$.

Then:
 * $\left\vert{x + y}\right\vert \le \left\vert{x}\right\vert + \left\vert{y}\right\vert$

Complex Numbers
Let $z_1, z_2 \in \C$ be complex numbers.

Let $\left\vert{z}\right\vert$ be the modulus of $z$.

Then:
 * $\left\vert{z_1 + z_2}\right\vert \le \left\vert{z_1}\right\vert + \left\vert{z_2}\right\vert$

Vectors in $\R^n$
Let $\mathbf{x}$,$\mathbf{y}$ be vectors in $\R^n$.

Let $\left\Vert{\cdot}\right\Vert$ denote vector length.

Then:


 * $\left \Vert {\mathbf{x} + \mathbf{y} }\right \Vert \le \left \Vert {\mathbf{x}}\right \Vert + \left \Vert { \mathbf{y} }\right \Vert$

If the two vectors are scalar multiples where said scalar is strictly positive, an equality holds:


 * $\exists \lambda \in \R, \lambda > 0: \mathbf x = \lambda \mathbf y \iff \left \Vert {\mathbf x + \mathbf y } \right \Vert = \left \Vert { \mathbf x } \right \Vert + \left \Vert { \mathbf y } \right \Vert$

Proof 1 for Real Numbers
Then by Order of Squares in Totally Ordered Ring:
 * $\left\vert{x + y}\right\vert \le \left\vert{x}\right\vert + \left\vert{y}\right\vert$

Proof 2 for Real Numbers
This can be seen to be a special case of Minkowski's Inequality, with $n = 1$.

Proof 3 for Real Numbers
From Real Numbers form Ordered Integral Domain, we can directly apply Sum of Absolute Values, which is applicable on all ordered integral domains, of which $\R$ is one.

Note that this result can not directly be used for the complex numbers $\C$ as they do not form an ordered integral domain.

Proof For Vectors
Let $\mathbf{x,y} \in \R^n$

We have:

From the Cauchy-Schwarz Inequality:

To prove that the equality holds if the vectors are scalar multiples of each other:

Sufficient Condition
Assume:


 * $\exists \lambda \in \R, \lambda > 0: \mathbf v = \lambda \mathbf w$

Then:

Proof for Complex Numbers
Let $z_1 = a_1 + i a_2, z_2 = b_1 + i b_2$.

Then from the definition of the modulus, the above equation translates into:
 * $\left({\left({a_1 + b_1}\right)^2 + \left({a_2 + b_2}\right)^2}\right)^{\frac 1 2} \le \left({a_1^2 + a_2^2}\right)^{\frac 1 2} + \left({b_1^2 + b_2^2}\right)^{\frac 1 2}$

This is a special case of Minkowski's Inequality, with $n = 2$.

Note
There is also a geometric interpretation of the triangle inequality. It is in fact a special case of this algebraic triangle equality in the Euclidean metric space.