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In the very first chapter in my class "mathematical analysis 1" I've seen something called the triangle inequality, which is $||a| - |b|| \leq |a \pm b| \leq |a| + |b|$. Now the thing is that I do understand why this is true, but i fail to see what this actually has to do with triangles. It was never explained nor proven. I'd like to know how this inequality relates to triangles to get a better understanding of it, because it's also used a lot further on. Would someone be able to explain this?

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    $\begingroup$ The name makes more sense in the more general case where you replace scalars with vectors and absolute value with euclidean norm (or any other norm). $\endgroup$
    – AnonymousCoward
    Dec 24, 2018 at 12:32
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    $\begingroup$ A plot like this might be useful: upload.wikimedia.org/wikipedia/commons/0/08/… (NB: this is for the case where $a,b$ are vectors and $|a|$ denote the length of the vector $a$) $\endgroup$
    – Winther
    Dec 24, 2018 at 12:34

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In Euclidean space with Cartesian coordinates, let $+$ denote vector addition, and let $O$ denote the origin. Choose points $a,b$, and form a triangle with vertices $a,b,O$. One side of this triangle has side length $|a|$, another has side length $|b|$, and the third has side length $|a-b|$. Now as we all know, the shortest path between two points is a straight line. So, the shortest path between the two points $a,b$ has length $|a-b|$. There is a longer path one can take which goes around the other two sides, the length of that path is $|a| + |b|$. So, $|a-b| \le |a| + |b|$.

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Suppose $a$ and $b$ are complex numbers, so that they correspond in the Argand-Cauchy plane, to two sides of a triangle, the third side corresponding to $a-b$. Then these inequalities correspond to well-known from middle school inequalities about triangles – namely that the length of a side of a triangle is between the difference and the sum of the lengths of the two other sides.

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