What does length in 1-dimension mean?

Question

I'm trying to understand what is norm in one dimension. A norm in two dimensions I do understand - it's a length of a vector, which makes sense. But, according to Wikipedia a norm is positive length and $1$-norm is just the absolute value. I don't get the intuition of what the length in one dimension means.

Answer 1

If you understand what a norm is in dimension $2$, then you should know that, if $V$ is a vector space of finite dimension, a norm on $V$ is a function $\lVert {}\cdot{} \rVert \colon V \to \mathbb{R}_{\geq 0}$ which essentially associates to each vector $v \in V$ its length. When $V \cong \mathbb{R}$ you just denote $\lVert {}\cdot{}\rVert$ by $\lvert {}\cdot{}\rvert$. Geometrically, $\lvert x-y \rvert $ is the distance between $x$ and $y$ in $\mathbb{R}$.

Answer 2

You may think of length of a real number as the distance from the real number zero. For example length of $x=-5$ is the distance from $-5$ to $0$ which is $5$

Note that this is just the absolute value of the point.

Thus you may say norm in one dimension is the same as absolute value.