Editing Word metric (section)

== Examples ==

=== The group of integers <math>\mathbb{Z}</math> ===

The group of [[integers]] <math>\mathbb{Z}</math> is generated by the set {-1,+1}. The integer -3 can be expressed as -1-1-1+1-1, a word of length 5 in these generators. But the word that expresses -3 most efficiently is -1-1-1, a word of length 3. The distance between 0 and -3 in the word metric is therefore equal to 3. More generally, the distance between two integers ''m'' and ''n'' in the word metric is equal to |''m''-''n''|, because the shortest word representing the difference ''m''-''n'' has length equal to |''m''-''n''|.

=== The group <math>\mathbb{Z} \oplus \mathbb{Z}</math> ===

For a more illustrative example, the elements of the group <math>\mathbb{Z}\oplus\mathbb{Z}</math> can be thought of as [[vector (geometric)|vectors]] in the [[Cartesian coordinate system|Cartesian plane]] with integer coefficients. The group <math>\mathbb{Z}\oplus\mathbb{Z}</math> is generated by the standard unit vectors <math>e_1 = \langle1,0\rangle</math>, <math>e_2 = \langle0,1\rangle</math> and their inverses <math> -e_1=\langle-1,0\rangle </math>, <math> -e_2=\langle0,-1\rangle </math>. The [[Cayley graph]] of <math>\mathbb{Z}\oplus\mathbb{Z}</math> is the so-called [[taxicab geometry]]. It can be pictured in the plane as an infinite square grid of city streets, where each horizontal and vertical line with integer coordinates is a street, and each point of <math>\mathbb{Z}\oplus\mathbb{Z}</math> lies at the intersection of a horizontal and a vertical street. Each horizontal segment between two vertices represents the generating vector <math> e_1 </math> or <math> -e_1</math>, depending on whether the segment is travelled in the forward or backward direction, and each vertical segment represents <math> e_2 </math> or <math> -e_2</math>. A car starting from <math>\langle1,2\rangle</math> and travelling along the streets to <math>\langle-2,4\rangle</math> can make the trip by many different routes. But no matter what route is taken, the car must travel at least |1 - (-2)| = 3 horizontal blocks and at least |2 - 4| = 2 vertical blocks, for a total trip distance of at least 3 + 2 = 5. If the car goes out of its way the trip may be longer, but the minimal distance travelled by the car, equal in value to the word metric between <math>\langle1,2\rangle</math> and <math>\langle-2,4\rangle</math> is therefore equal to 5.

In general, given two elements <math> v = \langle i,j\rangle </math> and <math> w = \langle k,l\rangle </math> of <math>\mathbb{Z}\oplus\mathbb{Z}</math>, the distance between <math> v </math> and <math> w </math> in the word metric is equal to <math> |i-k| + |j-l| </math>.