# Why does 'proper length' exist as a notion?

$$\text{proper time}= \tau= \sqrt{dt^2-d\mathbf{s}^2}$$

$$\text{proper length}= L= \sqrt{-dt^2+d\mathbf{s}^2}$$

What tangible benefit is brought about by calling $i \tau$ 'proper length' (applying when $\Im(L)=0$ (the spacetime intervals are spacelike))?

Could one extend the notion of the interval to simulatneusly cover both proper time and distance? Call this $\text{Interval}=dt^2-d\mathbf{s}^2$, which can be imaginary. Is it impossible to merge proper time and length because they are fundamentally different things, as proper time is a explicitly measurable quantity, but proper distance must be determined implicitly?

My problem is that I don't understand the rationale behind having proper length, rather than a cheap mathematical trick to make the interval real when $dt^2-d\mathbf{s}^2<0$.

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