Pythagoras of
Samos
(d. between
500 and 490
BC) thought
that all
numbers are
rational.
NOT ALL REAL NUMBERS
ARE RATIONAL
Mika Seppälä: Square Root 2 is not Rational
Hippasus of
Metapontum, a
student of
Pythagoras,
showed, in the 5th
Century BC that the
square root of two
is not a rational
number.
Mika Seppälä: Square Root 2 is not Rational
INTEGERS, RATIONAL AND REAL NUMBERS
Notations
{
}
Natural Numbers
= 0,1,2,3,…
2
1
{
}
= 0, ±1, ±2, ±3,…
Integers
⎧p
⎫
= ⎨ p ∈ ,q ∈ ,q > 0, gcd p,q = 1⎬
⎪⎭
⎩⎪ q
p and q have no
Rational Numbers
common factors.
( )
1
Mika Seppälä: Square Root 2 is not Rational
INTEGERS, RATIONAL AND
REAL NUMBERS
Theorem
There are no rational numbers r such
that r2 = 2.
INTEGERS, RATIONAL AND
REAL NUMBERS
Extending the set of rational numbers by
irrational numbers, the equation r2 = 2 has
solutions. From a practical point of view,
irrational numbers are represented by
infinitely long decimal numbers.
Notation
Real Numbers
= { r | r rational or irrational }.
Mika Seppälä: Square Root 2 is not Rational
Mika Seppälä: Square Root 2 is not Rational
SQUARE ROOT 2 IS NOT RATIONAL
SQUARE ROOT 2 IS NOT RATIONAL
Theorem
Theorem
Proof
There are no rational numbers r
such that r2 = 2.
Assume the contrary.
Then there are positive integers p and q
such that p2/q2 = 2 and p and q do not
have common factors.
This means that p2 = 2q2. We prove the
theorem by showing that this is not possible.
Mika Seppälä: Square Root 2 is not Rational
There are no rational numbers r
such that r2 = 2.
Proof
(cont’d)
If p2 = 2q2, the area B of
the large brown square
is twice the area G of
the green square.
p
Mika Seppälä: Square Root 2 is not Rational
G
q
B
SQUARE ROOT 2 IS NOT RATIONAL
SQUARE ROOT 2 IS NOT RATIONAL
Theorem
Theorem
There are no rational numbers r
such that r2 = 2.
Proof
(cont’d)
There are no rational numbers r
such that r2 = 2.
Proof
(cont’d)
Observe that since p and q
do not have common
p
factors, they are the
smallest numbers satisfying
p2 = 2q2.
G
B
q
Hence B and G in the
picture are the smallest
p
squares with integer side
lengths such that the area of
B is twice the area of G.
G
B
q
Mika Seppälä: Square Root 2 is not Rational
Mika Seppälä: Square Root 2 is not Rational
SQUARE ROOT 2 IS NOT RATIONAL
SQUARE ROOT 2 IS NOT RATIONAL
Theorem
Theorem
There are no rational numbers r
such that r2 = 2.
Proof
(cont’d)
p
Now move a copy of the green
square to the upper right hand
corner of the larger square.
G
q
B
p
There are no rational numbers r
such that r2 = 2.
Proof
(cont’d)
The intersection I of the two
copies of the green square is the
square I with the area I.
A
I
p
A
q
Mika Seppälä: Square Root 2 is not Rational
p-q
G
q
B
p
A
I
A
q
Mika Seppälä: Square Root 2 is not Rational
p-q
SQUARE ROOT 2 IS NOT RATIONAL
SQUARE ROOT 2 IS NOT RATIONAL
Theorem
Theorem
There are no rational numbers r
such that r2 = 2.
Proof
(cont’d)
I = 2A is now impossible, since
G and B were the smallest
squares with B = 2G.
There are no rational numbers r
such that r2 = 2.
Proof
(cont’d)
The construction implies that
I = 2A.
Side length of A = p − q, that of I = 2q − p.
p
G
p
B
A
I
p
A
p-q
q
q
G
q
B
p
A
I
A
p-q
q
Mika Seppälä: Square Root 2 is not Rational
Mika Seppälä: Square Root 2 is not Rational
SQUARE ROOT 2 IS NOT RATIONAL
SQUARE ROOT 2 IS NOT RATIONAL
Theorem
Theorem
Proof of
Hippasus
There are no rational numbers r
such that r2 = 2.
Assume the contrary. Then there are
positive integers p and q , which do not
have common factors, such that
2=
p
.
q
Mika Seppälä: Square Root 2 is not Rational
Proof of
Hippasus
There are no rational numbers r
such that r2 = 2.
It follows that p2 = 2q2.
Hence p2 must be even. But this is
possible only if p is even (because clearly
2 is not an integer).
Since p is even, it is of the form p = 2n
for some integer n.
Mika Seppälä: Square Root 2 is not Rational
SQUARE ROOT 2 IS NOT RATIONAL
SQUARE ROOT 2 IS NOT RATIONAL
Theorem
Theorem
Proof of
Hippasus
There are no rational numbers r
such that r2 = 2.
The equation p2 = 2q2 then implies that
(2n)2 = 2q2.
I.e.
4n2 = 2q2.
Which yields 2n2 = q2.
Mika Seppälä: Square Root 2 is not Rational
OTHER IRRATIONAL NUMBERS
The following are famous irrational numbers:
e, π ,eπ .
1
The number n
m
is irrational or an integer.
er is irrational for rational r ≠ 0.
Mika Seppälä: Square Root 2 is not Rational
Proof of
Hippasus
There are no rational numbers r
such that r2 = 2.
The equation 2n2 = q2 means that q is even.
Hence both p and q must be even.
This is impossible, because we assumed that p
and q do not have common factors.
Mika Seppälä: Square Root 2 is not Rational