Euclidean length

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

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

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.