Why numbering should start at zero

Elaborating on the reasons given in Dijkstra's memo on the topic, we observe the following:

1-based indexing

Ranges are of the form 1 ≤ i < N + 1.
The last element of a sequence has index N.
How do we represent the index of the last element if N is not known? We need a new symbol!
There is no obvious choice for this symbol, so we choose something like end (Matlab). Ugly.

Ranges are of the form 0 ≤ i < N.
The last element of a sequence has index N – 1.
How do we represent the index of the last element if N is not known? We need a new symbol!
But now there is an obvious choice: -1. It comes directly from the expression for the index of the last element, N-1.
And, we can now count backwards using only the usual numerical symbols, so that the index of the second-to-last element is -2, etc., instead of something ugly like end-1.