What is the ratio of the length of the diagonal of a perfect square to an edge

1
What is the ratio of the length of the diagonal
of a perfect square to an edge?
2
What is the ratio of the length of the diagonal
of a perfect square to an edge?
3
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.

4
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.

5
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.

6
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.

7
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.

8
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 v2 or 21/2.
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• So v2 is with us whenever a perfect square is.

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• So v2 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).

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• So v2 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.

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• This means v2 would be the ratio of two integers.

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• This means v2 would be the ratio of two
  integers.
• A ratio of two integers is called a rational
  number.

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• This means v2 would be the ratio of two
  integers.
• A ratio of two integers is called a rational
  number.
• To their great surprise, the Greeks discovered v2
  is not rational.

15

• This means v2 would be the ratio of two
  integers.
• A ratio of two integers is called a rational
  number.
• To their great surprise, the Greeks discovered v2
  is not rational.
• Real numbers that are not rational are now called
  irrational.

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• This means v2 would be the ratio of two
  integers.
• A ratio of two integers is called a rational
  number.
• To their great surprise, the Greeks discovered v2
  is not rational.
• Real numbers that are not rational are now called
  irrational.
• We believe v2 was the very first number known to
  be irrational. This discovery forced a rethinking
  of what number means.

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• We will present a proof that v2 is not rational.

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• We will present a proof that v2 is not rational.
• Proving a negative statement usually must be done
  by assuming the logical opposite and arriving at
  a contradictory conclusion.

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• We will present a proof that v2 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.

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Theorem There is no rational number whose square
is 2.
21
Theorem There is no rational number whose square
is 2.

• Proof Assume, to the contrary, that v2 is
  rational.

22
Theorem There is no rational number whose square
is 2.

• Proof Assume, to the contrary, that v2 is
  rational.
• So we can write v2 n/m with n and m positive
  integers.

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Theorem There is no rational number whose square
is 2.

• Proof Assume, to the contrary, that v2 is
  rational.
• So we can write v2 n/m with n and m positive
  integers.
• Among all the fractions representing v2, we
  select the one with smallest denominator.

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• So if v2 is rational (v2 n/m) then an isosceles
  right triangle with legs of length m will have
  hypotenuse of length n v2m.

n v2m
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• So if v2 is rational (v2 n/m) then an isosceles
  right triangle with legs of length m will have
  hypotenuse of length n v2m.
• Moreover, for a fixed unit, we can take ?ABC to
  be the smallest isosceles right triangle with
  integer length sides.

n v2m
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• Now, for the basic trick. Bisect the angle at A
  and fold the edge AB along the edge AC.

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• Now, for the basic trick. Bisect the angle at A
  and fold the edge AB along the edge AC.

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• Now, for the basic trick. Bisect the angle at A
  and fold the edge AB along the edge AC.
• This creates a new triangle ?DEC with the angle
  at E being a right angle and the angle at C still
  being 45.

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• Now, for the basic trick. Bisect the angle at A
  and fold the edge AB along the edge AC.
• This creates a new triangle ?DEC with the angle
  at E being a right angle and the angle at C still
  being 45.
• AEABm

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• Now, for the basic trick. Bisect the angle at A
  and fold the edge AB along the edge AC.
• This creates a new triangle ?DEC with the angle
  at E being a right angle and the angle at C still
  being 45.
• AEABn
• ECAC-AE

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• Now, for the basic trick. Bisect the angle at A
  and fold the edge AB along the edge AC.
• This creates a new triangle ?DEC with the angle
  at E being a right angle and the angle at C still
  being 45.
• AEABn
• ECAC-AEn-m

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• Now, for the basic trick. Bisect the angle at A
  and fold the edge AB along the edge AC.
• This creates a new triangle ?DEC with the angle
  at E being a right angle and the angle at C still
  being 45.
• AEABn
• ECAC-AEn-m
• BDDE

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• Now, for the basic trick. Bisect the angle at A
  and fold the edge AB along the edge AC.
• This creates a new triangle ?DEC with the angle
  at E being a right angle and the angle at C still
  being 45.
• AEABn
• ECAC-AEn-m
• BDDEECn-m

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• But, if
• BDDEECn-m
• and BCm,

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• But, if
• BDDEECn-m
• and BCm, then
• DCBC-BD

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• But, if
• BDDEECn-m
• and BCm, then
• DCBC-BDm-(n-m)

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• But, if
• BDDEECn-m
• and BCm, then
• DCBC-BDm-(n-m)2m-n.

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• But, if
• BDDEECn-m
• and BCm, then
• DCBC-BDm-(n-m)2m-n.

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• But, if
• BDDEECn-m
• and BCm, then
• DCBC-BDm-(n-m)2m-n.
• Since n and m are integers, n-m and 2m-n are
  integers and ?DEC is an isosceles right triangle
  with integer side lengths smaller than ?ABC .

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• This contradicts our choice of ?ABC as the
  smallest isosceles right triangle with integer
  side lengths for a given fixed unit of length.

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• This contradicts our choice of ?ABC as the
  smallest isosceles right triangle with integer
  side lengths for a given fixed unit of length.
• This means our assumption that v2 is rational is
  false.
• Thus there is no rational number whose square is
  2.
• QED

42

• 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).

43

• 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, v2 is irrational!