Why Integer Distance, Why the Mean - k-Means, One Honest Step

The exactness comes from two choices: compare squared distances and update with rational means. That keeps this one step in the exact register.

highlighted = computed this step

Why squared distance

Squared distance preserves the same nearest-centroid order as distance, but it avoids square roots. In this dataset the shown d^2 values are integers.

d^2\text{ comparison is exact}

Why the mean stays exact

Each updated coordinate is a sum divided by the assigned count. That gives 2/3 on the left and 20/3 on the right.

\text{new coordinates}=2/3,20/3

Summary

The distance comparison and centroid update are both exact arithmetic. No decimal approximation is needed for this step.

\text{integer }d^2\text{ and rational means}