Beta is a real number
0 < Beta < 1
F = square root of ( 1 - Beta Squared / ( 1 + Beta Squared ) )
Name at least one rational number value of Beta that gives me a rational number value for F
Equation for special relativity longitudinal doppler shift with object moving away at speed V where Beta = V / C
If my formula is wrong let me know
I want to find one or more numbers that are easy to use without a calculator for an examp!e problem or example problems
I will not give you credit
β :
i) β ∈ ℚ
ii) let r = √((1 - β²) ÷ (1 + β²)); r ∈ ℚ
iii) β ∈ (0,1) # assuming exclusive because the above holds trivially for both 0 and 1
suppose β = p ÷ q; q, p ∈ ℤ⁺, q > p
we have
r = √(((q² - p²) ÷ q²) ÷ ((q² + p²) ÷ q²)), to prove r ∈ ℚ
equivalently,
r = √((q² - p²) ÷ (q² + p²))
equivalently,
r = √(q² - p²) ÷ √(q² + p²)
i'm no mathematician but this feels like something with no rational solutions
4/5 is a solution for the numerator but it is not a solution for the denominator and both the numerator and denominator must be rational in order for the numerator divided by the denominator to be rational
false. take a = 2 ↑ (1 ÷ 3), b = 2 ↑ (2 ÷ 3)
a ∉ ℚ ∧ b ∉ ℚ ∧ a · b ∈ ℚ
@HatkeshiatorTND @shortstories Even like the square root of two times the square root of two
@KingOfWhiteAmerica @HatkeshiatorTND
Ok I was wrong about the claim that the product of two irrational numbers is always an irrational number
If Beta = 4/5 = 0.8
Then 1 - Beta^2 = 0.36
The square root of 0.36 = 0.6 which is a rational number
But
1 + Beta^2 = 16/25 + 25/25 = 41/25
The square root of 41/25 is not a rational number
The square root of a rational number times an irrational number is an irrational number
the recripical of an irrational number is an irrational number
the product of two irrational numbers is an irrational number
the product of two rational numbers is a rational number
So 4/5 does not work for Beta