Download The irrationality of pi by Anne Serban

Irrationality of π
Anne Serban
History of π
The number pi is a constant of the ratio of the circumference of a circle to the diameter.
Several ancient cultures had formulas for the area of a circle. Some scholars believe that the Jewish nation
knew that pi is 3 because of the verse in 1 Kings 7:23 which reads ”And he made the Sea of cast bronze, ten
cubits from one brim to the other; it was completely round. Its height was five cubits, and a line of thirty
cubits measured its circumference.” as well as a similar verse in 2 Chronicles 4:2.
The ancient Babylonians would take the square of its radius three times to calculate the area of a circle. This
gives pi a value of 3. One Babylonian tablet (1900 - 1680 BC) indicates a value of 3.125 for pi.
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The Rhind Papyrus (1650 BC) shows that the Egyptians would use the formula: A = ( 8d
9 ) where d is the
diameter of the circle, to calculate the area; therefore the resulting value of pi is 3.1605.
Archimedes of Syracuse (287 - 212 BC) was the first person to calculate pi ; he did this by inscribing a polygon
in a circle and circumscribing another polygon outside the circle. Archimedes used polygons with increasing
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numbers of sides up to and including a 96-gon. His calculations show that pi is between 22
7 and 71 .
A Chinese mathematician by the name of Zu Chongzhi (429 - 501 BC) found the ratio between the circumfer355
ence to the diameter to be 113
. His calculations give the value of pi up to nine decimals.
In Wales during the year 1706, William Jones, a self-taught mathematician, began to use the Greek symbol π
for pi ; in 1737, Leonhard Euler publicized the character by utilizing it in his work.
In 1761, Johann Heinrich Lambert, a Swiss mathematician, proved that π is irrational. Today, several other
proofs of π’s irrationality exist. More than one hundred years later, in 1882, Ferdinand von Lindemann, a
German mathematician, proved the transcendence of π.
Uses of π
”Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry,
especially those concerning circles, ellipses or spheres.” The following two formulas give the circumference and
area of a circle with a known radius r.
C = 2πr
A = πr2
The next two formulas give the surface area and volume of a sphere with a known radius r.
S = 4πr2
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V = πr3
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”Pi appears in areas of mathematics and the sciences having little to do with the geometry of circles, such as
number theory and statistics. It is also found in cosmology, thermodynamics, mechanics and electromagnetism.
The ubiquity of π makes it one of the most widely known mathematical constants both inside and outside the
scientific community.”
Basics
Derivatives
If f (x) is the original function, then by
1. Leibniz’s notation
d
dx f (x)
would be the first derivative of the function.
2. Lagrange’s notation
f 0 (x) is the first derivative of the function. For the second and third derivatives, another apostrophe
is added. For a higher order derivatives, the order of the derivative is put in paranthesis between the
function and the argument. e.g f (4) (x) is the fouth derivative of the function f (x).
f (n) (x) is the nth derivative
The derivative of an integer constant is 0.
The derivative of sin θ = cos θ
The derivative of cos θ = − sin θ
The chain rule says that the derivative of a composite function, a function with the variable being another
function, is the derivative of the outer function with the inner function unchanged multiplied by the derivative
of the inner function. (f (g(x)))0 = f 0 (g(x)) ∗ g 0 (x)
Antiderivatives
Rb
An antiderivative can also be called an integral, which is written as a f (x)dx where a is the lower limit of
integration, b is the upper limit of integration, f (x) is the integrand or the function to be integrated, and the
variable after d is the variable of integration.
The antiderivative of − sin θ = cos θ
The antiderivative of cos θ = sin θ
Exponents
Rules
1. am an = am+n
2. (ab)m cm = (abc)m
Types of Numbers
There exist several sets of numbers in mathematics. Although there are more sets, only the ones relevant to
the proof will be discussed.
Natural numbers, which are somtimes called the counting numbers, denoted by the symbol N, refer to the
numbers {1, 2, 3, 4, 5, 6, ...}
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Whole numbers, denoted by W, are all of the natural numbers as well as zero. Whole numbers can be
written as {0, 1, 2, 3, 4, 5, 6, ...}
Integer numbers are represented by Z because I is used for another set of numbers. The letter Z comes from
the German word ”zahlen” meaning numbers. Integer numbers are positive and negative numbers without a
fraction or decimal component. They are {..., −3, −2, −1, 0, 1, 2, 3, ...}
Rational numbers are symbolized by Q because R is used for a different set of numbers. The letter Q comes
from the Italian ”quoziente” which means quotient. The quotient is the result of dividing one number by
another. Because of its etymology, rational numbers are numbers which can be expressed as one integer over
another non-zero integer. Q = a/b where b is not 0
Irrational numbers are not distinguished by a special character; this set is composed of all the numbers
which are not rational i.e. numbers which cannot be written as one integer over another non-zero integer.
Binomial Theorem
The binomial theorem states that (x + y)n can be expanded into a sum with terms of the form axb y c where
b + c = n The coefficients a can be arranged for varying values of n and b to form Pascal’s triangle.
Pascal’s Triangle
n = 0:
1
n = 1:
1
n = 2:
1
n = 3:
1
n = 4:
1
2
3
1
4
1
3
6
1
4
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Fundamental Theorem of Calculus
This theorem has two parts.
Rx
Part 1: If f is a continuous function on [a, b], then the function g(x) = a f(t) dt is continuous on [a, b],
differentiable on (a, b), and g 0 (x) = f (x). In other words, the derivative of an integral of a continuous function
is the original function evaluated at x.
d
dx
Z
x
f(t) dt = f (x)
a
Part 2: If F is an antiderivative of f , then the integral from a to b of the function f is equal to the antiderivative
evaluated at b minus the antiderivative of a.
Z
b
f(x) dx = F (b) − F (a)
a
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Irrationality of π
Assume that pi is rational, then, by the definition of a rational number, π =
and b.
a
b
for relatively prime integers a
Define a polynomial function of power 2n that depends on a and b where n is a positive integer.
f (x) =
xn (a − bx)n
n!
This function has some important properties.
1. f(0) = 0
f (0) =
0n (a − b ∗ 0)n
=0
n!
Because the first term in the numerator is 0 to the power of a positive integer, the entire numerator is
0. 0 over the factorial of any positive number is 0.
2. f(π) = 0
Because π was defined as the rational number ab , then it can be replaced in the equation.
( a )n (a − b( ab ))n
a
f (π) = f ( ) = b
=
b
n!
( ab − x)n (a − a)n
=
n!
( a − x)n ∗ 0n
=0
= b
n!
=
Because the second term in the numerator is 0 to the power of a positive integer, the entire numerator
is 0. 0 over the factorial of any positive number is 0.
3. f(π - x) = f( ab - x) = f(x)
Because π was defined as the rational number ab , then it can be replaced in the equation.
( a − x)n (a − b( ab − x))n
a
=
f ( − x) = b
b
n!
( ab − x)n (a − (a − bx))n
=
n!
( a − x)n (a − a + bx)n
= b
=
n!
( a − x)n (bx)n
= b
=
n!
( a − x)n bn xn
= b
=
n!
(b ab − bx)n xn
=
=
n!
(a − bx)n xn
=
= f (x)
n!
=
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4. f (k) (0) is an integer
f (x) =
xn (a−bx)n
n!
is the original function. By multiplying both sides with n! we get
n!f (x) = xn (a − bx)n
Because of the binomial theorem, we know that (a − bx)n can be written as a polynomial with the last
term being bn xn . All of the terms are multiplied by xn ; the last term becomes bn xn+n . Now every term
is to a power between n and 2n.
If we take the derivative k times and 1 ≤ k < n, then all of the terms have x. By substitution, this
means that f (k) (0) = 0.
If we take the derivative k times and k > 2n, then all of the terms will be integer constants. Because
the derivative of an integer constant is 0, f (k) (0) = 0.
Finally, if we take the derivative k times and n ≤ k ≤ 2n, then all of the terms have x except for the
last, which will be an integer.
Therefore, f (k) (0) will be an integer for any k.
5. f 0 (x) = −f 0 (π − x)
Because f (x) = f (π − x), by using the chain rule we get that:
f 0 (x) = f (π − x)0 =
= f 0 (π − x) ∗ (π − x)0 =
= f 0 (π − x) ∗ (−1) =
= −f (π − x)
So, in general, f (k) (x) = (−1)k f (k) (π − x).
6. f (k) (π) is an integer
f (k) (x) = (−1)k f (k) (π − x), by replacing x with 0, we get that f (k) (0) = (−1)k f (k) (π − 0). Because
f (k) (0) is an interger, we know that f (k) (π) is an integer as well.
Now we are going to define another function as the alternating sum of f (x) and the first even n derivatives.
g(x) = f (x) − f (2) (x) + f (4) (x)... + (−1)j f (2j) (x) + ... + (−1)n f (2n) (x)
This function also has some useful properties.
1. g(0) is an integer
This is known because f (0) is an integer, and f (k) (0) is an integer, for any k. Thus, the sum of integers
is an integer.
2. g(π) is an integer
Because f (π) is an integer, and f (k) (π) is an integer, for any k, the sum of integers is an integer.
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3. g(x) + g (2) (x) = f (x)
g(x) = f (x) − f (2) (x) + f (4) (x)... + (−1)j f (2j) (x) + ... + (−1)n f (2n) (x)
g (2) (x) = f (2) (x) − f (4) (x) + ... + (−1)j f (2j) (x) + ... + (−1)n f (2n+2) (x)
In the equation g (2) (x) = f (2) (x)−f (4) (x)+...+(−1)n f (2n+2) (x) the last term is 0 because any derivative
of f (x) of order greater than 2n is 0. Notice that all of the terms of the first equation appear in the
second equation but with the opposite sign, except for f (x). Therefore g(x) + g (2) (x) = f (x).
4.
d
0
dx (g (x) sin x
− g(x) cos x) = f (x) sin x
By the chain rule, we get that:
d 0
(g (x) sin x − g(x) cos x) = g (2) (x) sin x + g 0 (x) cos x − (g 0 (x) cos x + g(x) − sin x) =
dx
g (2) (x) sin x + g 0 (x) cos x − (g 0 (x) cos x − g(x) sin x) =
g (2) (x) sin x + g 0 (x) cos x − g 0 (x) cos x + g(x) sin x) =
g (2) (x) sin x + g(x) sin x) =
(g (2) (x) + g(x)) sin x = f (x) sin x
Because the derivative of (g (2) (x) + g(x)) sin x = f (x) sin x, it follows anti-derivative of f (x) sin x = (g (2) (x) +
g(x)) sin x
By the Fundamental Theorem of Calculus
Z
π
f(x)(sin x) dx = g 0 (π) sin π − g(π) cos π − (g 0 (0) sin 0 − g(0) cos 0) =
0
= g 0 (π) sin π − g(π) cos π − g 0 (0) sin 0 + g(0) cos 0 =
= −g(π)(−1) + g(0)(1) =
= g(π) + g(0) =
Due to the fact that g(π) and g(0) are integers, their sum is an integer.
For any value of x where 0 < x < π,
1. sin(0) < sin x < sin(π)
0 < sin x < 1
Multiplying by f (x), we get 0 < f (x) sin x < f (x)
2. 0 < xn < π n
3. In the function f (x) =
xn (a−bx)n
,
n!
(a − bx) < a. Consequently (a − bx)n < an
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Combining all of this information together we get that 0 < f (x) sin x <
an π n
n!
an π n
n→∞ n!
0 < f (x) sin x < lim
limn→∞
an π n
n!
will be positive and arbitrarily small. So f (x) sin x is a integer between 0 and 1.
This is not possible; the assumption that π is rational is wrong.
Therefore π is irrational.
Q.E.D.
References
1. Bible
2. http://mindyourdecisions.com/blog/2013/11/08/proving-pi-is-irrational-a-step-by-step-guide-to-a
3. http://www.rapidtables.com/math/number/exponent.htm
4. https://www.exploratorium.edu/pi/history_of_pi/
5. http://www.ualr.edu/lasmoller/pi.html?
6. http://www.ms.uky.edu/~lee/ma502/pi/MA502piproject.html
7. http://projecteuclid.org/download/pdf_1/euclid.bams/1183510788
8. http://www.pi314.net/eng/lambert.php
9. http://sixthform.info/maths/files/pitrans.pdf
10. https://en.wikipedia.org/wiki/Pi
11. https://en.wikipedia.org/wiki/List_of_types_of_numbers
12. https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational
13. https://en.wikipedia.org/wiki/Binomial_theorem
14. https://en.wikipedia.org/wiki/Ferdinand_von_Lindemann
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