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.


The white area in the top square is (a2)/2.So the white area in the lower square is 2a2.


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.


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.


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 21/2.

• 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 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.


• 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.




• 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.


• Proving a negative statement usually must be done by assuming the logical opposite and arriving at a contradictory conclusion.


• 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.


Proof : Assume, to the contrary, that √2 is rational.


Proof : Assume, to the contrary, that √2 is rational.So we can write √2= n/m with n and m positive integers.


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.


AE=AB=m


AE=AB=nEC=AC-AE


AE=AB=nEC=AC-AE=n-m


AE=AB=nEC=AC-AE=n-mBD=DE


AE=AB=nEC=AC-AE=n-mBD=DE=EC=n-m

But, ifBD=DE=EC=n-m

and BC=m,

But, ifBD=DE=EC=n-m

and BC=m, thenDC=BC-BD

But, ifBD=DE=EC=n-m

and BC=m, thenDC=BC-BD=m-(n-m)

But, ifBD=DE=EC=n-m

and BC=m, thenDC=BC-BD=m-(n-m)=2m-n.

But, ifBD=DE=EC=n-m

and BC=m, thenDC=BC-BD=m-(n-m)=2m-n.

But, ifBD=DE=EC=n-m

and BC=m, thenDC=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!

Documents

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