Important inequalities: the triangle inequality

Can the distance between two complex numbers be shorter than the difference of their moduli, or longer than the sum of those moduli? A modulus is only a length. Until the two tips are on the board, both comparisons are still just symbols.

Every real number is also a complex number, so a picture drawn in the plane still covers a pair of reals.

Two tips, three lengths

The extra assumption on this board is that $z$ and $w$ are complex: each one is a point of the plane, and the origin is the common start of the two arrows. Drag $Z$ or $W$.

The segment between the tips is $|z-w|$. The three marks underneath start on the same vertical line, so the horizontal position is the length. The upper mark is $||z|-|w||$, the mark on the axis is $|z-w|$, and the lower mark is $|z|+|w|$. Slide either tip. The middle mark stays between the other two.

That is the triangle inequality for two complex numbers,

$||z|-|w||\le |z-w|\le |z|+|w|.$

The picture had to come first. The left-hand length is how far the two moduli already disagree, and the right-hand length is the longest path that still visits both arrows. The straight segment is what remains when you refuse the detour.

Drag until $O$, $Z$, and $W$ lie on one line. If $Z$ and $W$ sit on the same side of $O$, the middle mark meets the upper one. If they sit on opposite sides, the middle mark meets the lower one. That is what this board does when the three points are collinear.

The broken path is longer

The same comparison adds more than two lengths. For complex numbers $x_1,\dots ,x_n$,

$\left|{\sum }_{k=1}^n{x}_k\right|\le {\sum }_{k=1}^n|x_k|.$

A strict inequality is too strong. Take every $x_k$ real and positive: both sides are the same number, so equality holds.

The board freezes the sum at three steps. That is an extra assumption of the figure, not of the inequality. The solid path is the sum of the moduli. The dashed segment is the modulus of the sum. Drag a corner and the dashed segment stays at most as long as the path. Lay $A$, $B$, and $C$ in that order on the positive $x$-axis, every step a positive real, and the path meets the dashed segment. That is the equality a strict inequality leaves out.

The same comparison, with the sum replaced by an integral, reads

$\left|{\int }_{G}f(x)dx\right|\le {\int }_G|f(x)|dx,$

whenever the integral of the modulus exists. The region is whatever the integral runs over. The board above is still three complex steps: the straight total stays at most the total of the lengths.

A modulus turns a complex number back into a length, and a sum of lengths is at least the straight length. Nothing here says which of two positive numbers deserves to be called a mean. That order is means of positive numbers. The comparison of $(1+x{)}^n$ with the line $1+nx$ is a different hypothesis, on Bernoulli's inequality.