# Euclidean length

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

The distance between two points in the plane is defined as the nonnegative square root of the sum of squared coordinate differences.

mko-euclidean-length · definition · v0.5.0



## Mathematical statement

In the Euclidean plane, if P=(x₁,y₁) and Q=(x₂,y₂), their distance is defined as the nonnegative square root of the sum of the squared coordinate differences.





## Representative formula (exact TeX)

d(P,Q)=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}





## Assumptions


- P and Q are points in the two-dimensional Euclidean plane.
- The coordinates x₁, y₁, x₂, and y₂ are real numbers.
- The square root is the nonnegative principal value, so distance is nonnegative.





## Symbol scope


- d — distance function
- P — first point in the plane
- Q — second point in the plane
- x₁,y₁ — coordinates of P
- x₂,y₂ — coordinates of Q





## Intuition, definition, and misconceptions

The distance formula treats horizontal and vertical displacement as perpendicular components and takes the nonnegative square root of their squared sum.

When P and Q coincide, both coordinate differences are zero, so the distance is zero. Exchanging P and Q does not change the distance.

A program can use hypot or a square root to compute a concrete distance, but floating-point representations are not identical to mathematical real numbers.

A common mistake is to ignore the nonnegative principal square root or to add coordinate differences without squaring them first.





## Adopted derivation guide

[Euclidean length: computing the distance between two points exactly from coordinates](https://ocme.evemisslab.com/en/derivations/derivation-mko-euclidean-length-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites







## Verification boundary

Concrete coordinate tests validate only the reference implementation; they do not prove that the distance function satisfies every metric axiom for all real points.

not_formalized


- [evidence-sha256-de740ccfca05c5027587999af29adeead880cfe6ebaa28063d95ad584875e31e](https://ocme.evemisslab.com/data/evidence/de740ccfca05c5027587999af29adeead880cfe6ebaa28063d95ad584875e31e.json)





## Next obligation

Connect the entry to Mathlib EuclideanSpace and dist, and record a theorem relating the two-dimensional coordinate formula to the abstract distance.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/euclidean-length.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-euclidean-length/index.md)