the triangle inequality
The triangle inequality is the rule |z + w| <= |z| + |w|: the length of the sum of two complex numbers is never more than the sum of their lengths. The name comes from the geometry. Draw z and w as arrows; adding them tip-to-tail builds a triangle whose third side is z + w. The inequality is just the everyday fact that one side of a triangle can never be longer than the other two added together — the straight route from start to finish is the shortest.
Equality, |z + w| = |z| + |w|, happens exactly when z and w point in the same direction (one is a non-negative real multiple of the other), so the triangle flattens into a straight line. There is also a useful lower bound, the reverse triangle inequality: |z - w| >= the absolute value of (|z| - |w|). This says two numbers of very different size cannot have a tiny difference — it is the tool you reach for when you need to show something stays bounded away from zero.
Modest as it looks, this is one of the most-used tools in all of analysis. It is exactly what lets you 'split and bound' — estimate a complicated sum or integral by estimating its pieces — and it underwrites the very definition of limits and continuity in the complex plane. The ML-inequality for contour integrals, the comparison tests for series, and countless convergence proofs all rest on this one humble line.
With z = 3 + 4i (so |z| = 5) and w = -1 (so |w| = 1): z + w = 2 + 4i, with |z + w| = the square root of 20, about 4.47. Indeed 4.47 <= 5 + 1 = 6, and also 4.47 >= |5 - 1| = 4. Both bounds hold.
Sum of arrows: the third side never exceeds the other two added — and never falls below their difference.
It bounds the modulus of a SUM, not of a product: for products you instead get the exact equality |z w| = |z| |w|. Confusing the additive bound with the multiplicative identity is a common slip.