Are transcendental numbers a closed system?
Let’s take each in turn. Transcendental numbers: These are not closed under any of the 4 operations. Addition: Let n be any integer. Then π + n – π = n. Why is n – π transcendental? If it were algebraic, n -(n – π), being the difference of 2 algebraic numbers would be algebraic. So π would also be algebraic, contradiction. Subtraction: e-e = 0. Multiplication: e*1/e = 1. Division: π/π = 1. Imaginary numbers are not closed under any of the 4 operations either. Addition: a+bi + a -bi = 2a is real. Subtraction: bi-bi = 0 which is real Multiplication: (a+bi)(a-bi) = a²+b² which is real. Division: i/i = 1 which is real. Look at Wikipedia for lots of fascinating information about these kinds of numbers. There are still many open questions about them. For instance, we know that e^π is transcendental, but we know nothing about π^e.
