Hilbert's Hotel Paradox Calculator
Explore the infinite hotel paradox and discover how infinity works in mathematics
Infinite Hotel Manager
How many new guests need accommodation?
Which current guest do you want to relocate?
Room Assignment Results
Current Guest Movement
New Guest Assignment
Mathematical Formula
Current guest's new room: Original room + Number of new guests
1 + 1 = 2
Scenario Analysis
✅ Simple Solution: All current guests move up by 1 room numbers.
🏨 Vacated Rooms: Rooms 1 through 1 are now available for new guests.
♾️ Infinite Capacity: This works because the hotel has infinitely many rooms.
Example Calculation
Classic Example: Guest in Room 1397
Question: Where should the guest in room 1397 move when infinite new guests arrive?
Scenario: Infinitely many new guests
Formula: New room = 2 × Original room
Calculation: 2 × 1397 = 2794
Answer
Guest moves to room 2794
This frees up all odd-numbered rooms (1, 3, 5, 7, ...) for the infinite new guests!
Infinity Concepts
Countable Infinity
Same size as natural numbers (1, 2, 3, ...)
Hotel rooms are countably infinite
Bijection
One-to-one correspondence between sets
Each guest maps to exactly one room
Aleph-null
Cardinality of countable infinity
Size of the set of natural numbers
Paradox Insights
Infinity + Infinity = Infinity
Infinite sets can have proper subsets of the same size
Different infinities exist (countable vs uncountable)
Prime factorization ensures unique room assignments
Understanding Hilbert's Hotel Paradox
What is the Infinite Hotel Paradox?
Hilbert's Hotel Paradox is a thought experiment proposed by mathematician David Hilbert to illustrate the counterintuitive properties of infinite sets. The paradox involves a hotel with infinitely many rooms, all of which are occupied.
The Paradox Explained
Even though the hotel is completely full, it can still accommodate new guests - whether finite or infinite in number. This seems impossible with finite hotels but becomes possible when dealing with infinite sets.
Mathematical Significance
- •Demonstrates properties of countable infinity
- •Shows that ∞ + n = ∞ and ∞ + ∞ = ∞
- •Illustrates bijections between infinite sets
- •Introduces concepts of transfinite arithmetic
The Three Scenarios
1. Finite New Guests
Solution: Shift all current guests by n rooms
Simple but demonstrates infinite capacity
2. Infinite New Guests
Solution: Even-odd room redistribution
Uses the fact that even numbers are as numerous as all natural numbers
3. Infinite Buses
Solution: Prime power method
Uses unique prime factorization to handle nested infinities
Key Insight
The hotel can always accommodate more guests because infinity is not a number but a concept representing unboundedness. Adding to infinity doesn't change its size!