Transcript What is the ratio of the diagonal to the edge of a perfect

What is the ratio of the length of the
diagonal of a perfect square to an edge?
What is the ratio of the length of the
diagonal of a perfect square to an edge?
What is the ratio of the length of the
diagonal of a perfect square to an edge?
The white area in the top
square is (a2)/2.
What is the ratio of the length of the
diagonal of a perfect square to an edge?
The white area in the top
square is (a2)/2.
So the white area in the
lower square is 2a2.
What is the ratio of the length of the
diagonal of a perfect square to an edge?
The white area in the top
square is (a2)/2.
So the white area in the
lower square is 2a2. But
this area can also be
expressed as b2.
What is the ratio of the length of the
diagonal of a perfect square to an edge?
The white area in the top
square is (a2)/2.
So the white area in the
lower square is 2a2. But
this area can also be
expressed as b2.
Thus, b2 = 2a2.
What is the ratio of the length of the
diagonal of a perfect square to an edge?
The white area in the top
square is (a2)/2.
So the white area in the
lower square is 2a2. But
this area can also be
expressed as b2.
Thus, b2 = 2a2.
Or, (b/a)2 = 2.
We conclude that the
ratio of the diagonal to
the edge of a square is the
square root of 2, which
can be written as √2 or
1/2
2 .
• So √2 is with us whenever a perfect square is.
• So √2 is with us whenever a perfect square is.
• For a period of time, the ancient Greek
mathematicians believed any two distances
are commensurate (can be co-measured).
• So √2 is with us whenever a perfect square is.
• For a period of time, the ancient Greek
mathematicians believed any two distances
are commensurate (can be co-measured).
• For a perfect square this means a unit of
measurement can be found so that the side
and diagonal of the square are both integer
multiples of the unit.
• This means √2 would be the ratio of two
integers.
• This means √2 would be the ratio of two
integers.
• A ratio of two integers is called a rational
number.
• This means √2 would be the ratio of two
integers.
• A ratio of two integers is called a rational
number.
• To their great surprise, the Greeks discovered
√2 is not rational.
• This means √2 would be the ratio of two
integers.
• A ratio of two integers is called a rational
number.
• To their great surprise, the Greeks discovered
√2 is not rational.
• Real numbers that are not rational are now
called irrational.
• This means √2 would be the ratio of two
integers.
• A ratio of two integers is called a rational
number.
• To their great surprise, the Greeks discovered
√2 is not rational.
• Real numbers that are not rational are now
called irrational.
• We believe √2 was the very first number
known to be irrational. This discovery forced
a rethinking of what “number” means.
• We will present a proof that √2 is not
rational.
• We will present a proof that √2 is not
rational.
• Proving a negative statement usually must be
done by assuming the logical opposite and
arriving at a contradictory conclusion.
• We will present a proof that √2 is not
rational.
• Proving a negative statement usually must be
done by assuming the logical opposite and
arriving at a contradictory conclusion.
• Such an argument is called a proof by
contradiction.
Theorem: There is no rational number whose
square is 2.
Theorem: There is no rational number whose
square is 2.
Proof : Assume, to the
contrary, that √2 is rational.
Theorem: There is no rational number whose
square is 2.
Proof : Assume, to the
contrary, that √2 is rational.
So we can write √2= n/m
with n and m positive
integers.
Theorem: There is no rational number whose
square is 2.
Proof : Assume, to the
contrary, that √2 is rational.
So we can write √2= n/m
with n and m positive
integers.
Among all the fractions
representing √2, we select
the one with smallest
denominator.
So if √2 is rational (√2= n/m)
then an isosceles right
triangle with legs of length m
will have hypotenuse of
length n= √2m.
n = √2m
So if √2 is rational (√2= n/m)
then an isosceles right
triangle with legs of length m
will have hypotenuse of
length n= √2m.
Moreover, for a fixed unit,
we can take ΔABC to be the
smallest isosceles right
triangle with integer length
sides.
n = √2m
Now, for the basic trick.
Bisect the angle at A and fold
the edge AB along the edge
AC.
Now, for the basic trick.
Bisect the angle at A and fold
the edge AB along the edge
AC.
Now, for the basic trick.
Bisect the angle at A and fold
the edge AB along the edge
AC.
This creates a new triangle
ΔDEC with the angle at E
being a right angle and the
angle at C still being 45⁰.
Now, for the basic trick.
Bisect the angle at A and fold
the edge AB along the edge
AC.
This creates a new triangle
ΔDEC with the angle at E
being a right angle and the
angle at C still being 45⁰.
AE=AB=m
Now, for the basic trick.
Bisect the angle at A and fold
the edge AB along the edge
AC.
This creates a new triangle
ΔDEC with the angle at E
being a right angle and the
angle at C still being 45⁰.
AE=AB=n
EC=AC-AE
Now, for the basic trick.
Bisect the angle at A and fold
the edge AB along the edge
AC.
This creates a new triangle
ΔDEC with the angle at E
being a right angle and the
angle at C still being 45⁰.
AE=AB=n
EC=AC-AE=n-m
Now, for the basic trick.
Bisect the angle at A and fold
the edge AB along the edge
AC.
This creates a new triangle
ΔDEC with the angle at E
being a right angle and the
angle at C still being 45⁰.
AE=AB=n
EC=AC-AE=n-m
BD=DE
Now, for the basic trick.
Bisect the angle at A and fold
the edge AB along the edge
AC.
This creates a new triangle
ΔDEC with the angle at E
being a right angle and the
angle at C still being 45⁰.
AE=AB=n
EC=AC-AE=n-m
BD=DE=EC=n-m
But, if
BD=DE=EC=n-m
and BC=m,
But, if
BD=DE=EC=n-m
and BC=m, then
DC=BC-BD
But, if
BD=DE=EC=n-m
and BC=m, then
DC=BC-BD=m-(n-m)
But, if
BD=DE=EC=n-m
and BC=m, then
DC=BC-BD=m-(n-m)=2m-n.
But, if
BD=DE=EC=n-m
and BC=m, then
DC=BC-BD=m-(n-m)=2m-n.
But, if
BD=DE=EC=n-m
and BC=m, then
DC=BC-BD=m-(n-m)=2m-n.
Since n and m are integers,
n-m and 2m-n are integers
and ΔDEC is an isosceles
right triangle with integer
side lengths smaller than
ΔABC .
This contradicts our choice of
ΔABC as the smallest
isosceles right triangle with
integer side lengths for a
given fixed unit of length.
This contradicts our choice of
ΔABC as the smallest
isosceles right triangle with
integer side lengths for a
given fixed unit of length.
This means our assumption
that √2 is rational is false.
Thus there is no rational
number whose square is 2.
QED
This beautiful proof was adapted from Tom
Apostol: “Irrationality of the Square Root of
Two: A Geometric Proof”, American
Mathematical Monthly,107, 841-842 (2000).
This beautiful proof was adapted from Tom
Apostol: “Irrationality of the Square Root of
Two: A Geometric Proof”, American
Mathematical Monthly,107, 841-842 (2000).
Behold, √2 is irrational!