Distance
Elementary
Think of two objects, like two houses on a street. The distance between the two houses is how far apart the houses are. No distance is negative.
High school
Consider two points, either on the number line or in the plane. On the number line, the distance between $a$ and $b$ is $\abs{a - b}$. In the plane, the distance between $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$.
Undergraduate (metric spaces)
Consider a metric space with metric $d$, and points $x$ and $y$ in it. The distance between $x$ and $y$ is the value $d(x, y)$.
Examples
Examples
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$\abs{3 - 7} = 4$
Elementary
$4$ is the distance from $7$ to $3$ (and from $3$ to $7$) along the real number line.
Non-examples
Things that may be mistaken for examples.
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$3 - 7 = -4$
Elementary
$3 - 7 = -4$ is not a distance, because a distance cannot be negative.
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Walking $3$ m east and then $3$ m west
Elementary
The distance traveled is $6$ m, not $0$ (the distance between the start and end points is $0$).
Distinctions involving this term
- Types of difference: Many quantities compare two things by “how far apart” they are. A difference is the result of subtraction and can be negative. No distance is negative. An error compares an approximation with the truth, a change compares the end with the start, and a set difference removes elements.