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Fermi's Piano Tuner Problem

How Old is Old?

If the Terrestrial Poles were to Melt...

Sunlight Exerts Pressure

Falling Eastward

What if an Asteroid Hit the Earth

Using a Jeep to Estimate the Energy in Gasoline

How do Police Radars really work?

How "Fast" is the Speed of Light?

How Long is a Light Year?

How Big is a Trillion?

"Seeing" the Earth, Moon, and Sun to Scale

Of Stars and Drops of Water

If I Were to Build a Model of the Cosmos...

A Number Trick

Designing a High Altitude Balloon

Pressure in the Vicinity of a Lunar Astronaut Space Suit due to Outgassing of Coolant Water

Calendar Calculations

Telling Time by the Stars - Sidereal Time

Fields, an Heuristic Approach

The Irrationality of

The Irrationality of

The Number (i)i

Estimating the Temperature of a Flat Plate in Low Earth Orbit

Proving that (p)1/n is Irrational when p is a Prime and n>1

The Transcendentality of

Ideal Gases under Constant Volume, Constant Pressure, Constant Temperature and Adiabatic Conditions

Maxwell's Equations: The Vector and Scalar Potentials

A Possible Scalar Term Describing Energy Density in the Gravitational Field

A Proposed Relativistic, Thermodynamic Four-Vector

Motivational Argument for the Expression-eix=cosx+isinx

Another Motivational Argument for the Expression-eix=cosx+isinx
Calculating the Energy from Sunlight over a 12 hour period
Calculating the Energy from Sunlight over actual full day
Perfect Numbers-A Case Study
Gravitation Inside a Uniform Hollow Sphere
Further note on Gravitation Inside a Uniform Hollow Sphere
Pythagorean Triples
Black Holes and Point Set Topology
Additional Notes on Black Holes and Point Set Topology
Field Equations and Equations of Motion (General Relativity)
The observer in modern physics
A Note on the Centrifugal and Coriolis Accelerations as Pseudo Accelerations - PDF File
On Expansion of the Universe - PDF File

The Transcendentality of pi

By definition, the number pi is the ratio of the circumference to the diameter of a circle. This ratio is the same for all circles.

pi is an irrational number. It cannot be represented as the ratio of two integers, regardless of the choice of integers. Equivalently, it cannot be represented as an unending, periodic decimal.

pi is also a transcendental number. It is not a root of any algebraic equation of the form

a0 + a1x + a2x2 + … + anxn = 0

where the ai are all rational numbers and n is finite. For comparison, square root of two is also an irrational number. But square root of two is not transcendental since it is a root of the equation

x2 - 2 = 0.

Both pi and square root of two are irrational, but only pi is transcendental. What makes the difference? One important argument is that a line of length square root of two can be constructed using classical techniques (i.e., using compass and straight-edge in a finite number of steps). But, because of the way pi is defined, a line of length pi cannot be so constructed. (A curve can. Mark off a unit segment. Bisect the segment. Using the midpoint as center, scribe the appropriate circle. The circle has length pi.)

To illustrate, let us first consider square root of two. To make a construction that produces a line of this length, we begin with two unit-length segments placed end to end so that one segment is at right angles with the other. We then, connect the free ends to complete a right triangle. The new line has length square root of two

Now, consider pi. A circle of unit diameter has its circumference = pi. Draw a unit circle, and locate its center. From the center produce a set of n radial lines each separated from its neighbor by an angle 2pi/n.


Circle described by above paragraph.

Approximation for n = 9


Line of length 9 (AB) graphic


 Isoseles Triangle created by points A,B, and C.  R=1/2.

 Isoseles Triangle

Now, connect the ends with straight line segments to form a set of isosceles triangles. The sum of the lengths of these straight-line segments approaches the circumference of the circle as n approaches infinity (see figure for n = 9).

To construct a line of length pi, we have but to produce the length AB n times along any line. In the figure, n = 9, AB is a typical straight line segment completing an isosceles triangle, and Circumference approximately 9(AB) (see figure).

Now, let us use some algebra to calculate the length AB in the general case. We begin by redrawing part of our previous picture.

One isosceles triangle (deltaABC with one vertex at the center C and two more vertices on the circle at A and B) is selected (see figure). The angle subtended at C is 2pi/n. We wish to calculate the length AB, and then to estimate how good an approximation the value n(AB) is to the actual circumference of the circle.

The line CM bisects the angle 2pi/n, and meets the line AB at right angles. Thus, the triangle CMA is a right triangle, and the length AM = ½(AB).

AM/(Radius) = AM/(½) = sin (pi/n)

three dots (therefore) AM = ½ sin (pi/n)

& AB = sin (pi/n).

Finally, n(AB) = n sin (pi/n).

Since the actual circumference of the circle is pi, we now write

n(AB) = {pin sin (pi/n)}/pi

= pi {(sin (pi/n))/(pi/n)}

= pi {(sin (omega))/(omega)}

where theta= pi/n.

The value n(AB) differs from pi by the multiplicative factor (sin (theta))/(theta), with theta = pi/n. Notice that, in the limit as n arrow infinity, theta arrow 0 and (sin (theta))/(theta) arrow 1. The value n(AB) does indeed approach pi in the limit. But, while the limit exists, the actual function f(theta) = (sin (theta))/(theta) does not exist at theta = 0. It becomes the indeterminate form 0/0. Thus, although n(AB) may approach the actual circumference to any arbitrary precision we might desire, the actual value n(AB) = pi can never be obtained from any construction of the type outlined above.

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