∞
Kristin DeVleming
University of Washington
What is Infinity?
A mathematical
notion of
something being
endless or
unbounded
Is infinity a number?
∞+1=?
∞+n=?
∞+∞=?
∞×∞=?
∞÷∞=?
The Hilbert Hotel
The hotel is full! But it always says vacancy.
The Hilbert Hotel
What if another person shows up? What about
n people?
∞+1=∞
∞+n=∞
The Hilbert Hotel
What if a bus with infinitely many seats pulls up?
∞+∞=∞
The Hilbert Hotel
What if infinitely many buses with infinitely
many seats show up?
The Hilbert Hotel
All odd rooms are empty
The Hilbert Hotel
The Hilbert Hotel
The Hilbert Hotel
∞×∞=∞
The Hilbert Hotel
• Alternatively, put
hotel guests in
room 2n
• Put the passenger
in seat n of bus k
into room 2n3k
• This leaves some
rooms empty!
The Hilbert Hotel
• What if an infinite number of ferries, each
carrying infinitely many buses, came to the
hotel?
• Put the passenger in seat n of bus k of ferry f
in room 2n3k5f
• Still leaves rooms empty!
More Strange Behavior
• What is ∞ - ∞ ? Is it 0?
• From before:
–∞+1=∞
–∞+2=∞
– Subtract: 0 + 1 = 0
1 = 0 ??????
More Strange Behavior
• What is ∞ - ∞ ? Is it ∞ ?
• Depends on how you subtract
More Strange Behavior
• What is ∞ - ∞ ?
– Correct answer: not a number!
More Strange Behavior
• What is ∞ ÷ ∞ ? Is it 1?
– If you have a cake with infinitely many slices, and
give one slice to each of infinitely many people,
they each get one slice, so ∞ ÷ ∞ = 1
– What if you gave each person two slices?
Then ∞ ÷ ∞ = 2!
– Correct answer: not a number!
Countable Sets
• Natural numbers: { 1, 2, 3, 4, … }
– These form an infinite set
– We can count or enumerate them
• If a set has the same size as the natural
numbers, we say it is countably infinite
Countable Sets
• Are the whole numbers countably infinite?
{ 0, 1, 2, 3, … }
1
2
3
4
…
Yes.
Countable Sets
• Are the integers countably infinite?
{ … , -3, -2, -1, 0, 1, 2, 3, … }
7
5
3
1
2
4
6
Yes.
Countable Sets
• Are the rational numbers countable?
– Rational numbers: fractions
𝑎
𝑏
(a,b)
numerators (a)
denominators (b)
The rooms we put
the guests in
correspond to the
way we pair them up
to the natural
numbers!
Countable Sets
• Are the real numbers countable?
• If so, each number would get assigned to a
room in the Hilbert hotel:
The Real Numbers
Room 1
7.98303948….
Room 2
4.21783339….
Room 3
0.22839400….
Room 4
1.61716839….
Room 5
0.99847201….
Room 6
2.03309502….
Room 7
5.55632489….
The Real Numbers
Room 1
7.98303948….
Room 2
4.21783339….
Room 3
0.22839400….
Room 4
1.61716839….
Room 5
0.99847201….
Room 6
2.03309502….
Room 7
5.55632489….
7.227494….
The Real Numbers
Room 1
7.98303948….
Room 2
4.21783339….
Room 3
0.22839400….
Room 4
1.61716839….
Room 5
0.99847201….
Room 6
2.03309502….
Room 7
5.55632489….
7.227494….
3.518523….
Not in any room!
The Real Numbers
• Conclusion: the real numbers are NOT
countable; they are “bigger” than the infinity
we have talked about so far.
Uncountable Sets
• We say the real numbers are uncountable or
uncountably infinite since they are bigger
than the natural numbers.
• Is there a set that is bigger than countable but
smaller than uncountable? Impossible to
know!
Other Fun Facts
• Even though there are more real numbers
than rational numbers, between any two real
numbers you can always find a rational one
• There are surfaces with finite volume but
infinite surface area (but not the opposite!)
• The number of different infinities is so big that
it cannot be described by a single infinity