Nearest Neighbors

Euclidean Distance

Compute the Euclidean distance between two points as √Σ(aᵢ−bᵢ)². A single loop accumulates the squared difference per dimension; math.sqrt at the end. Library: np.linalg.norm(np.array(a) - np.array(b)) — single call, same formula. RESULT: distance (rounded).

By hand

With NumPy

np.linalg.norm computes the L2 norm of the difference vector — identical to the loop formula for Euclidean distance.

naive.py 
import math
a = [1, 2, 3]
b = [4, 6, 3]
n = len(a)
sq_sum = 0.0
for i in range(n):
    diff = a[i] - b[i]
    sq_sum = sq_sum + diff * diff
dist = math.sqrt(sq_sum)
print('RESULT:', round(dist, 4))
library.py 
import numpy as np
from dalib.display import set_display
set_display()

a = [1, 2, 3]
b = [4, 6, 3]
dist = float(np.linalg.norm(np.array(a) - np.array(b)))
print('a:', a)
print('b:', b)
print('RESULT:', round(dist, 4))

a: [1, 2, 3]
b: [4, 6, 3]
RESULT: 5.0

Implementation notes

  • Euclidean distance generalizes the Pythagorean theorem to n dimensions. For 2D points it reduces to √((x₂−x₁)²+(y₂−y₁)²).
  • The loop accumulates diff*diff (explicit multiply) so each squared term appears as a distinct step in the replay trace, keeping the squaring operation visible rather than folded into a single expression.
  • Scale sensitivity: dimensions with large magnitudes dominate. Standardize features before computing distances — see standardize-features (ch01).
  • Cross-reference: knn-classify-majority (this chapter) applies this distance to find nearest neighbors.