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Sunlight Exerts Pressure

Falling Eastward

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How do Police Radars really work?

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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
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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 square root of two

Prove that square root of twois an irrational number.

The number, square root of two, 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 square root of two is rational so that we may write

square root of two = a/b

for a and b = any two integers. To show that square root of two 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
2b2 = a2

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
b2 = 4n2 + 4n + 1

Substituting these expressions into eq. 2a, we obtain

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

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 square root of two = a/b cannot be found. We are forced to conclude that square root of two is irrational.

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