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
@shortstories we want
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
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
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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
