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 CONTENTS Introduction 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 Irrationality of

Problem:
Prove that is an irrational number.

Solution:
The number, , is irrational, ie., it cannot be expressed as a ratio of integers a and b. To prove that this statement is true, let us assume that is rational so that we may write

 = a/b 1

for a and b = any two integers. To show that is irrational, we must show that no two such integers can be found. We begin by squaring both sides of eq. 1:

 2 = a2/b2 2
 or 2b2 = a2 2a.

From eq. 2a, we must conclude that a2 (and, therefore, a) is even; b2 (and, therefore, b) may be even or odd. If b is even, the ratio a2/b2 may be immediately reduced by canceling a common factor of 2. If b is odd, it is possible that the ratio a2/b2 is already reduced to smallest possible terms. We assume that b2 (and, therefore, b) is odd.

Now, we set a = 2m, and b = 2n + 1, and require that m and n be integers (to ensure integer values of a and b). Then

 a2 = 4m2 3
 and b2 = 4n2 + 4n + 1 4

Substituting these expressions into eq. 2a, we obtain

 2(4n2 + 4n + 1) = 4m2 5
 or 4n2 + 4n + 1 = 2m2 6

The L.H.S. of eq. 6 is an odd integer. The R.H.S., on the other hand, is an even integer. There are no solutions for eq. 6. Therefore, integer values of a and b which satisfy the relationship = a/b cannot be found. We are forced to conclude that is irrational.