Euclidean Distance in KNN Explained with Simple Examples

K-Nearest Neighbors (KNN) & Euclidean Distance

📚 Table of Contents

• Introduction
• Euclidean Distance
• Mathematical Deep Dive
• Worked Example
• KNN Classification
• Higher Dimensions
• CLI Example
• Key Takeaways
• Related Articles

📖 Introduction

KNN is a simple algorithm that classifies data based on similarity.

💡 Core Idea: Nearby points tend to belong to the same class.

📏 Euclidean Distance

Formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

This represents the straight-line distance between two points.

📐 Mathematical Deep Dive

This comes from the Pythagorean theorem:

\[ a^2 + b^2 = c^2 \]

• a = horizontal difference
• b = vertical difference
• c = distance

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

🔽 Why squaring?

It removes negative values and ensures distance is always positive.

📊 Worked Example

• A (1,2)
• B (2,3)
• C (4,5)
• New (3,3)

\[ d(A) = \sqrt{(3-1)^2 + (3-2)^2} = \sqrt{5} \approx 2.23 \]

\[ d(B) = 1 \]

\[ d(C) = \sqrt{5} \approx 2.23 \]

🤖 KNN Classification

• Closest: B
• Second: A

Prediction: Class 1

📦 Higher Dimensions

\[ d = \sqrt{\sum_{i=1}^{n}(x_i - y_i)^2} \]

This allows KNN to work with multiple features.

💻 CLI Example

Code

import numpy as np

def distance(p1, p2):
    return np.sqrt(np.sum((np.array(p1) - np.array(p2))**2))

print(distance([3,3],[1,2]))
print(distance([3,3],[2,3]))
print(distance([3,3],[4,5]))

Output

$ python knn.py
2.23
1.0
2.23

🔽 Explanation

Computes squared differences → sums → square root.

🎯 Key Takeaways

• Distance = similarity measure
• KNN uses nearest neighbors
• Works in any dimension
• Based on geometry

📘 Final Thoughts

Euclidean distance converts data into measurable relationships, making machine learning possible.