Irrationality of √(j(j+1)) with a Consequence for Loop Quantum Gravity

Key Points

• The aim is to prove the irrationality of √(n(n+2)) for positive integers, extending that result to half-integers in loop quantum gravity.
• Presented a concise proof that √(n(n+2)) is irrational for every positive integer n.
• Derived the corollary that √(j(j+1)) is irrational for half-integer j ≥ 1/2.
• Establishes that √(n(n+2)) is irrational for all positive integers n.
• Confirms that the quantity 8π√(j(j+1)) is transcendental for every spin in loop quantum gravity.