Editing Hamming distance (section)

== Properties ==
For a fixed length ''n'', the Hamming distance is a [[Metric (mathematics)|metric]] on the set of the [[word (formal language theory)|words]] of length ''n'' (also known as a [[Hamming space]]), as it fulfills the conditions of non-negativity, symmetry, the Hamming distance of two words is 0 if and only if the two words are identical, and it satisfies the [[triangle inequality]] as well:<ref name="Robinson2003" /> Indeed, if we fix three words ''a'', ''b'' and ''c'', then whenever there is a difference between the ''i''th letter of ''a'' and the ''i''th letter of ''c'', then there must be a difference between the ''i''th letter of ''a'' and ''i''th letter of ''b'', or between the ''i''th letter of ''b'' and the ''i''th letter of ''c''. Hence the Hamming distance between ''a'' and ''c'' is not larger than the sum of the Hamming distances between ''a'' and ''b'' and between ''b'' and ''c''. The Hamming distance between two words ''a'' and ''b'' can also be seen as the [[Hamming weight]] of ''a'' − ''b'' for an appropriate choice of the − operator, much as the difference between two integers can be seen as a distance from zero on the number line.{{clarify|date=June 2020}}

For binary strings ''a'' and ''b'' the Hamming distance is equal to the number of ones ([[Hamming weight|population count]]) in ''a'' [[Exclusive or|XOR]] ''b''.<ref name="Warren_2013" /> The metric space of length-''n'' binary strings, with the Hamming distance, is known as the ''Hamming cube''; it is equivalent as a metric space to the set of distances between vertices in a [[hypercube graph]]. One can also view a binary string of length ''n'' as a vector in <math>\mathbb{R}^{n}</math> by treating each symbol in the string as a real coordinate; with this embedding, the strings form the vertices of an ''n''-dimensional [[hypercube]], and the Hamming distance of the strings is equivalent to the [[Manhattan distance]] between the vertices.